Symplectic Geometric Algorithms for Hamiltonian Systems 7534135958, 9787534135958

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Kang Feng Mengzhao Qin Symplectic Geometric Algorithms for Hamiltonian Systems

Kang Feng Mengzhao Qin

Symplectic Geometric Algorithms for Hamiltonian Systems With 62 Figures

ZHEJIANG PUBLISHING UNITED GROUP ZHEJIANG SCIENCE AND TECHNOLOGY PUBLISHING HOUSE

Authors Kang Feng (1920-1993) Institute of Computational Mathematics and Scientific/ Engineering Computing Beijing 100190, China

Mengzhao Qin Institute of Computational Mathematics and Scientific/ Engineering Computing Beijing 100190, China Email: [email protected]

ISBN 978-7-5341-3595-8 Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou ISBN 978-3-642-01776-6 ISBN 978-3-642-01777-3 (eBook) Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009930026 ¤ Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Frido Steinen-Broo, EStudio Calamar, Spain Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

“. . . In the late 1980s Feng Kang proposed and developed so-called symplectic algorithms for solving equations in Hamiltonian form. Combining theoretical analysis and computer experimentation, he showed that such methods, over long times, are much superior to standard methods. At the time of his death, he was at work on extensions of this idea to other structures . . . ”

Peter Lax

Cited from SIAM News November 1993

Kang Feng giving a talk at an international conference

“ A basic idea behind the design of numerical schemes is that they can preserve the properties of the original problems as much as possible . . . Different representations for the same physical law can lead to different computational techniques in solving the same problem, which can produce different numerical results . . .” Kang Feng (1920 – 1993) Cited from a paper entitled “How to compute property Newton’s equation of motion”

Prize certificate

Author’s photograph taken in Xi’an in 1989

Foreword

Kang Feng (1920–1993), Member of the Chinese Academy of Sciences, Professor and Honorary Director of the Computing Center of the Chinese Academy of Sciences, famous applied mathematician, founder and pioneer of computational mathematics and scientific computing in China. It has been 16 years since my brother Kang Feng passed away. His scientific achievements have been recognized more and more clearly over time, and his contributions to various fields have become increasingly outstanding. In the spring of 1997, Professor Shing-Tung Yau, a winner of the Fields Medal and a foreign member of the Chinese Academy of Sciences, mentioned in a presentation at Tsinghua University, entitled “The development of mathematics in China in my view”, that “there are three main reasons for Chinese modern mathematics to go beyond or hand in hand with the West. Of course, I am not saying that there are no other works, but I mainly talk about the mathematics that is well known historically: Professor Shiingshen Chern’s work on characteristic class, Luogeng Hua’s work on the theory of functions of several complex variables, and Kang Feng’s work on finite elements.” This high evaluation of Kang Feng as a mathematician (not just a computational mathematician) sounds so refreshing that many people talked about it and strongly agreed with it. At the end of 1997, the Chinese National Natural Science Foundation presented Kang Feng et al. with the first class prize for his other work on a symplectic algorithm for Hamiltonian systems, which is a further recognition of his scientific achievements (see the certificate on the previous page). As his brother, I am very pleased. Achieving a major scientific breakthrough is a rare event. It requires vision, ability and opportunity, all of which are indispensable. Kang Feng has achieved two major scientific breakthroughs in his life, both of which are very valuable and worthy of mention. Firstly, from 1964 to 1965, he proposed independently the finite element method and laid the foundation for the mathematical theory. Secondly, in 1984, he proposed a symplectic algorithm for Hamiltonian systems. At present, scientific innovation has become the focus of discussion. Kang Feng’s two scientific breakthroughs may be treated as case studies in scientific innovation. It is worth emphasizing that these breakthroughs were achieved in China by Chinese scientists. Careful study of these has yet to be carried out by experts. Here I just describe some of my personal feelings. It should be noted that these breakthroughs resulted not only from the profound mathematical knowledge of Kang Feng, but also from his expertise in classical physics and engineering technology that were closely related to the projects. Scientific breakthroughs are often cross-disciplinary. In addition, there is often a long period of time before a breakthrough is made-not unlike a long time it takes for a baby to be born, which requires the accumulation of results in small steps.

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The opportunity for inventing the finite element method came from a national research project, a computational problem in the design of the Liu Jia Xia dam. For such a concrete problem, Kang Feng found a basis for solving of the problem using his sharp insight. In his view, a discrete computing method for a mathematical and physical problem is usually carried out in four steps. Firstly, one needs to know and define the physical mechanism. Secondly, one writes the appropriate differential equations accordingly. In the third step, design a discrete model. Finally, one develops the numerical algorithm. However, due to the complexity of the geometry and physical conditions, conventional methods cannot always be effective. Nonetheless, starting from the physical law of conservation or variational principle of the matter, we can directly relate to the appropriate discrete model. Combining the variational principle with the spline approximation leads to the finite element method, which has a wide range of adaptability and is particularly suited to deal with the complex geometry of the physical conditions of computational engineering problems. In 1965, Kang Feng published his paper entitled “Difference schemes based on the variational principle”, which solved the basic theoretical issues of the finite element method, such as convergence, error estimation, and stability. It laid the mathematical foundation for the finite element method. This paper is the main evidence for recognition by the international academic community of our independent development of the finite element method. After the Chinese Cultural Revolution, he continued his research in finite element and related areas. During this period, he made several great achievements. I remember that he talked with me about other issues, such as Thom’s catastrophe theory, Prigogine’s theory of dissipative structures, solitons in water waves, the Radon transform, and so on. These problems are related to physics and engineering technology. Clearly he was exploring for new areas and seeking a breakthrough. In the 1970s, Arnold’s “Mathematical Method of Classical Mechanics” came out. It described the symplectic structure for Hamiltonian equations, which proved to be a great inspiration to him and led to a breakthrough. Through his long-term experience in mathematical computation, he fully realized that different mathematical expressions for the same physical law, which are physically equivalent, can perform different functions in scientific computing (his students later called this the “Feng’s major theorem”). In this way, for classical mechanics, Newton’s equations, Lagrangian equations and Hamiltonian equations will show a different pattern of calculations after discretization. Because the Hamiltonian formulation has a symplectic structure, he was keenly aware that, if the algorithm can maintain the geometric symmetry of symplecticity, it will be possible to avoid the flaw of artificial dissipation of this type of algorithm and design a high-fidelity algorithm. Thus, he opened up a broad way for the computational method of the Hamiltonian system. He called this way the “Hamiltonian way”. This computational method has been used in the calculation of the orbit in celestial mechanics, in calculations for the particle path in accelerator, as well as in molecular dynamics. Later, the scope of its application was expanded. For example, it has also been widely used in studies of the atmosphere and earth sciences and elsewhere. It

Foreword

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has been effectively applied in solving the GPS observation operator, indicating that Global Positioning System data can be dealt with in a timely manner. This algorithm is 400 times more efficient than the traditional method. In addition, a symplectic algorithm has been successfully used in the oil and gas exploration fields. Under the influence of Kang Feng, international research on symplectic algorithm has become popular and flourishing, nearly 300 papers have been published in this field to date. Kang Feng’s research work on the symplectic algorithm has been well-known and recognized internationally for its unique, innovative, systematic and widespread properties, for its theoretical integrity and fruitful results. J. Lions, the former President of the International Mathematics Union, spoke at a workshop when celebrating his 60th birthday: “This is another major innovation made by Kang Feng, independent of the West, after the finite element method.” In 1993 one of the world’s leading mathematicians, P.D. Lax, a member of the American Academy of Sciences, wrote a memorial article dedicated to Kang Feng in SIAM News, stating that “In the late 1980s, Kang Feng proposed and developed so-called symplectic algorithms for solving evolution equations . . .. Such methods, over a long period, are much superior to standard methods.” E. J. Marsden, an internationlly wellknown applied mathematician, visited the computing institute in the late 1980s and had a long conversation with Kang Feng. Soon after the death of Kang Feng, he proposed the multi-symplectic algorithm and extended the characteristics of stability of the symplectic algorithm for long time calculation of Hamiltonian systems with infinite dimensions. On the occasion of the commemoration of the 16th anniversary of Kang Feng’s death and the 89th anniversary of his birth, I think it is especially worthwhile to praise and promote what was embodied in the lifetime’s work of Kang Feng — “ independence in spirit, freedom in thinking”. 1 Now everyone is talking about scientific innovation, which needs a talented person to accomplish. What type of person is needed most? A person who is just a parrot or who has an “independent spirit, freely thinking”? The conclusion is self-evident. Scientific innovation requires strong academic atmosphere. Is it determined by only one person or by all of the team members? This is also self-evident. From Kang Feng’s scientific career, we can easily find that the key to the problem of scientific innovation is “independence in spirit, freedom in thinking”, and that needs to be allowed to develop and expand. Kang Feng had planned to write a monograph about a symplectic algorithm for Hamiltonian systems. He had accumulated some manuscripts, but failed to complete it because he died too early due to sickness. Fortunately, his students and Professor Mengzhao Qin (see the photo on the previous page), one of the early collaborators, spent 15 years and finally completed this book based on Kang Feng’s plan, realizing his wish. It is not only an authoritative exposition of this research field, but also an 1

Yinke Chen engraved on a stele in 1929 in memory of Guowei Wang in campus of Tsinghua University.

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exposure of the academic thought of a master of science, which gives an example of how an original and innovative scientific discovery is initiated and developed from beginning to end in China. We would also like to thank Zhejiang Science and Technology Publishing House, which made a great contribution to the Chinese scientific cause through the publication of this manuscript. Although Kang Feng died 16 years ago, his scientific legacy has been inherited and developed by the younger generation of scientists. His scientific spirit and thought still elicit care, thinking and resonance in us. He is still living in the hearts of us.

Duan, Feng Member of Chinese Academy of Sciences Nanjing University Nanjing September 20, 2009

Preface It has been 16 years since Kang Feng passed away. It is our honor to publish the English version of Symplectic Algorithm for Hamiltonian Systems, so that more readers can see the history of the development of symplectic algorithms. In particular, after the death of Kang Feng, the development of symplectic algorithms became more sophisticated and there have been a series of monographs published in this area, e.g., Sanz-Serna & M.P. Calvo’s Numerical Hamiltonian Problems published in 1994 by Chapman and Hall Publishing House; E. Hairer, C. Lubich and G. Wanner’s Geometrical Numerical Integration published in 2001 by Springer Verlag; B. Leimkuhler and S. Reich’s Simulating Hamiltonian Dynamics published in 2004 by Cambridge University Press. The symplectic algorithm has been developed from ordinary differential equations to partial differential equations, from a symplectic structure to a multi-symplectic structure. This is largely due to the promotion of this work by J. Marsden of the USA and T. Bridge and others in Britain. Starting with a symplectic structure, J. Marsden first developed the Lagrange symplectic structure, and then to the multi-symplectic structure. He finally proposed a symplectic structure that meets the requirement of the Lagrangian form from the variational principle by giving up the boundary conditions. On the other hand, T. Bridge and others used the multisymplectic structure to derive directly the multi-symplectic Hamilton equations, and then constructed the difference schemes that preserve the symplectic structure in both time and space. Both methods can be regarded as equivalent in the algorithmic sense. Now, in this monograph, most of the content refers only to ordinary differential equations. Kang Feng and his algorithms research group working on the symplectic algorithm did some foundation work. In particular, I would like to point out three negative theorems: “ non-existence of energy preserving scheme”, “ non-existence of multistep linear symplectic scheme”, and “ non-existence of volume-preserving scheme form rational fraction expression”. In addition, generating function theory is not only rich in analytical mechanics and Hamilton–Jacobi equations. At the same time, the construction of symplectic schemes provides a tool for any order accuracy difference scheme. The formal power series proposed by Kang Feng had a profound impact on the later developed “ backward error series” work ,“ modified equation” and “ modified integrator”. The symplectic algorithm developed very quickly, soon to be extended to the geometric method. The structure preserving algorithm (not only preserving the geometrical structure, but also the physical structure, etc.) preserves the algebraic structure to present the Lie group algorithm, and preserves the differential complex algorithm. Many other prominent people have contributed to the symplectic method in addition to those mentioned above. There are various methods related to structure preserving algorithms and for important contributions the readers are referred to R. McLachlan & GRW Quispel “ Geometric integration for ODEs” and T. Bridges & S. Reich “ Numerical methods for Hamiltonian PDEs”. The book describes the symplectic geometric algorithms and theoretical basis for a number of related algorithms. Most of the contents are a collection of lectures given

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by Kang Feng at Beijing University. Most of other sections are a collection of papers which were written by group members. Compared to the previous Chinese version, the present English one has been improved in the following respects. First of all, to correct a number of errors and mistakes contained in the Chinese version. Besides, parts of Chapter 1 and Chapter 2 were removed, while some new content was added to Chapter 4, Chapter 7, Chapter 8, Chapter 9 and Chapter 10. More importantly, four new chapters — Chapter 13 to Chapter 16 were added. Chapter 13 is devoted to the KAM theorem for the symplectic algorithm. We invited Professor Zaijiu Shang , a former PhD student of Kang Feng to compose this chapter. Chapter 14 is called Variational Integrator. This chapter reflects the work of the Nobel Prize winner Professor Zhengdao Li who proposed in the 1980s to preserve the energy variational integrator, but had not explained at that time that it had a Lagrange symplectic type, which satisfied the Lagrange symplectic structure. Together with J. Marsden he proposed the variational integrator trail connection, which leads from the variational integrator. Just like J. Marsden, he hoped this can link up with the finite element method. Chapter 15 is about Birkhoffian Systems, describing a class of dissipative structures for Birkohoffian systems to preserve the dissipation of the Birkhoff structure. Chapter 16 is devoted to Multisymplectic and Variational Integrators, providing a summary of the widespread applications of multisymplectic integrators in the infinitely dimensional Hamiltonian systems. We would also like to thank every member of the Kang Feng’s research group for symplectic algorithms: Huamo Wu, Daoliu Wang, Zaijiu Shang, Yifa Tang, Jialin Hong, Wangyao Li, Min Chen, Shuanghu Wang, Pingfu Zhao, Jingbo Chen, Yushun Wang, Yajuan Sun, Hongwei Li, Jianqiang Sun, Tingting Liu, Hongling Su, Yimin Tian; and those who have been to the USA: Zhong Ge, Chunwang Li, Yuhua Wu, Meiqing Zhang, Wenjie Zhu, Shengtai Li, Lixin Jiang, and Haibin Shu. They made contributions to the symplectic algorithm over different periods of time. The authors would also like to thank the National Natural Science Foundation, the National Climbing Program projects, and the State’s Key Basic Research Projects for their financial support. Finally, the authors would also like to thank the Mathematics and Systems Science Research Institute of the Chinese Academy of Sciences, the Computational Mathematics and Computational Science and Engineering Institute, and the State Key Laboratory of Computational Science and Engineering for their support. The editors of this book have received help from E. Hairer, who provided a template from Springer publishing house. I would also like to thank F. Holzwarth at Springer publishing house and Linbo Zhang of our institute, and others who helped me successfully publish this book. For the English translation, I thank Dr. Shengtai Li for comprehensive proofreading and polishing, and the editing of Miss Yi Jin. For the English version of the publication I would also like to thank the help of the Chinese Academy of Sciences Institute of Mathematics. Because Kang Feng has passed away, it may not be possible to provide a comprehensive representation of his academic thought, and the book will inevitably contain some errors. I accept the responsibility for any errors and welcome criticism and corrections.

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We would also like to thank Springer Beijing Representation Office and Zhejiang Science and Technology Publishing House, which made a great contribution to the Chinese scientific cause through the publication of this manuscript. We are especially grateful to thank Lisa Fan, W. Y. Zhou, L. L. Liu and X. M. Lu for carefully reading and finding some misprints, wrong signs and other mistakes. This book is supported by National Natural Science Foundation of China under grant No.G10871099 ; supported by the Project of National 863 Plan of China (grant No.2006AA09A102-08); and supported by the National Basic Research Program of China (973 Program) (Grant No. 2007CB209603).

Mengzhao Qin Institute of Computational Mathematics and Scientific Engineering Computing Beijing September 20, 2009

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.

Preliminaries of Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.1 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1.1.1 Differentiable Manifolds and Differentiable Mapping . . . . 40 1.1.2 Tangent Space and Differentials . . . . . . . . . . . . . . . . . . . . . . 43 1.1.3 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.1.4 Submersion and Transversal . . . . . . . . . . . . . . . . . . . . . . . . . 51 1.2 Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 1.2.1 Tangent Bundle and Orientation . . . . . . . . . . . . . . . . . . . . . . 56 1.2.2 Vector Field and Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 1.3 Exterior Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 1.3.1 Exterior Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 1.3.2 Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 1.4 Foundation of Differential Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 1.4.1 Differential Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 1.4.2 The Behavior of Differential Forms under Maps . . . . . . . . 80 1.4.3 Exterior Differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 1.4.4 Poincar´e Lemma and Its Inverse Lemma . . . . . . . . . . . . . . . 84 1.4.5 Differential Form in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 1.4.6 Hodge Duality and Star Operators . . . . . . . . . . . . . . . . . . . . 88 1.4.7 Codifferential Operator δ . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 1.4.8 Laplace–Beltrami Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 90 1.5 Integration on a Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 1.5.1 Geometrical Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 1.5.2 Integration and Stokes Theorem . . . . . . . . . . . . . . . . . . . . . . 93 1.5.3 Some Classical Theories on Vector Analysis . . . . . . . . . . . 96 1.6 Cohomology and Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 1.7 Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 1.7.1 Vector Fields as Differential Operator . . . . . . . . . . . . . . . . . 99 1.7.2 Flows of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 1.7.3 Lie Derivative and Contraction . . . . . . . . . . . . . . . . . . . . . . . 103 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

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Symplectic Algebra and Geometry Preliminaries . . . . . . . . . . . . . . . . . . 113 2.1 Symplectic Algebra and Orthogonal Algebra . . . . . . . . . . . . . . . . . . . 113 2.1.1 Bilinear Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 2.1.2 Sesquilinear Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 2.1.3 Scalar Product, Hermitian Product . . . . . . . . . . . . . . . . . . . . 117 2.1.4 Invariant Groups for Scalar Products . . . . . . . . . . . . . . . . . . 119 2.1.5 Real Representation of Complex Vector Space . . . . . . . . . 121 2.1.6 Complexification of Real Vector Space and Real Linear Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 2.1.7 Lie Algebra for GL(n, F) . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 2.2 Canonical Reductions of Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . 128 2.2.1 Congruent Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 2.2.2 Congruence Canonical Forms of Conformally Symmetric and Hermitian Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 130 2.2.3 Similar Reduction to Canonical Forms under Orthogonal Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 2.3 Symplectic Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 2.3.1 Symplectic Space and Its Subspace . . . . . . . . . . . . . . . . . . . 137 2.3.2 Symplectic Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 2.3.3 Lagrangian Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 2.3.4 Special Types of Sp(2n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 2.3.5 Generators of Sp(2n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 2.3.6 Eigenvalues of Symplectic and Infinitesimal Matrices . . . 158 2.3.7 Generating Functions for Lagrangian Subspaces . . . . . . . . 160 2.3.8 Generalized Lagrangian Subspaces . . . . . . . . . . . . . . . . . . . 162 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

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Hamiltonian Mechanics and Symplectic Geometry . . . . . . . . . . . . . . . . . 165 3.1 Symplectic Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 3.1.1 Symplectic Structure on Manifolds . . . . . . . . . . . . . . . . . . . 165 3.1.2 Standard Symplectic Structure on Cotangent Bundles . . . . 166 3.1.3 Hamiltonian Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 167 3.1.4 Darboux Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 3.2 Hamiltonian Mechanics on R2n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 3.2.1 Phase Space on R2n and Canonical Systems . . . . . . . . . . . 169 3.2.2 Canonical Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 172 3.2.3 Poisson Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 3.2.4 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 3.2.5 Hamilton–Jacobi Equations . . . . . . . . . . . . . . . . . . . . . . . . . 182 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

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Symplectic Difference Schemes for Hamiltonian Systems . . . . . . . . . . . . 187 4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 4.1.1 Element and Notation for Hamiltonian Mechanics . . . . . . 187

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4.1.2

Geometrical Meaning of Preserving Symplectic Structure ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 4.1.3 Some Properties of a Symplectic Matrix . . . . . . . . . . . . . . . 190 4.2 Symplectic Schemes for Linear Hamiltonian Systems . . . . . . . . . . . 192 4.2.1 Some Symplectic Schemes for Linear Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 4.2.2 Symplectic Schemes Based on Pad´e Approximation . . . . . 193 4.2.3 Generalized Cayley Transformation and Its Application . . 197 4.3 Symplectic Difference Schemes for a Nonlinear Hamiltonian System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 4.4 Explicit Symplectic Scheme for Hamiltonian System . . . . . . . . . . . . 203 4.4.1 Systems with Nilpotent of Degree 2 . . . . . . . . . . . . . . . . . . 204 4.4.2 Symplectically Separable Hamiltonian Systems . . . . . . . . . 205 4.4.3 Separability of All Polynomials in R2n . . . . . . . . . . . . . . . 207 4.5 Energy-conservative Schemes by Hamiltonian Difference . . . . . . . . 209 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 5.

The Generating Function Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5.1 Linear Fractional Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5.2 Symplectic, Gradient Mapping and Generating Function . . . . . . . . 215 5.3 Generating Functions for the Phase Flow . . . . . . . . . . . . . . . . . . . . . 221 5.4 Construction of Canonical Difference Schemes . . . . . . . . . . . . . . . . . 226 5.5 Further Remarks on Generating Function . . . . . . . . . . . . . . . . . . . . . . 231 5.6 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 5.7 Convergence of Symplectic Difference Schemes . . . . . . . . . . . . . . . . 239 5.8 Symplectic Schemes for Nonautonomous System . . . . . . . . . . . . . . . 242 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

6.

The Calculus of Generating Functions and Formal Energy . . . . . . . . . . 249 6.1 Darboux Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 6.2 Normalization of Darboux Transformation . . . . . . . . . . . . . . . . . . . . . 251 6.3 Transform Properties of Generator Maps and Generating Functions 255 6.4 Invariance of Generating Functions and Commutativity of Generator Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 6.5 Formal Energy for Hamiltonian Algorithm . . . . . . . . . . . . . . . . . . . . 264 6.6 Ge–Marsden Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

7.

Symplectic Runge–Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 7.1 Multistage Symplectic Runge–Kutta Method . . . . . . . . . . . . . . . . . . . 277 7.1.1 Definition and Properties of Symplectic R–K Method . . . . 277 7.1.2 Symplectic Conditions for R–K Method . . . . . . . . . . . . . . . 281 7.1.3 Diagonally Implicit Symplectic R–K Method . . . . . . . . . . 284 7.1.4 Rooted Tree Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 7.1.5 Simplified Conditions for Symplectic R–K Method . . . . . 297

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Symplectic P–R–K Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 7.2.1 P–R–K Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 7.2.2 Symplified Order Conditions of Explicit Symplectic R–K Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 7.3 Symplectic R–K–N Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 7.3.1 Order Conditions for Symplectic R–K–N Method . . . . . . . 319 7.3.2 The 3-Stage and 4-th order Symplectic R–K–N Method . 323 7.3.3 Symplified Order Conditions for Symplectic R–K–N Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 7.4 Formal Energy for Symplectic R–K Method . . . . . . . . . . . . . . . . . . . 333 7.4.1 Modified Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 7.4.2 Formal Energy for Symplectic R–K Method . . . . . . . . . . . 339 7.5 Definition of a(t) and b(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 7.5.1 Centered Euler Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 7.5.2 Gauss–Legendre Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 7.5.3 Diagonal Implicit R–K Method . . . . . . . . . . . . . . . . . . . . . . 347 7.6 Multistep Symplectic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 7.6.1 Linear Multistep Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 7.6.2 Symplectic LMM for Linear Hamiltonian Systems . . . . . . 348 7.6.3 Rational Approximations to Exp and Log Function . . . . . . 352 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 8.

Composition Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 8.1 Construction of Fourth Order with 3-Stage Scheme . . . . . . . . . . . . . 365 8.1.1 For Single Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 8.1.2 For System of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 8.2 Adjoint Method and Self-Adjoint Method . . . . . . . . . . . . . . . . . . . . . 372 8.3 Construction of Higher Order Schemes . . . . . . . . . . . . . . . . . . . . . . . 377 8.4 Stability Analysis for Composition Scheme . . . . . . . . . . . . . . . . . . . . 388 8.5 Application of Composition Schemes to PDE . . . . . . . . . . . . . . . . . . 396 8.6 H-Stability of Hamiltonian System . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

9.

Formal Power Series and B-Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 9.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 9.2 Near-0 and Near-1 Formal Power Series . . . . . . . . . . . . . . . . . . . . . . . 409 9.3 Algorithmic Approximations to Phase Flows . . . . . . . . . . . . . . . . . . . 414 9.3.1 Approximations of Phase Flows and Numerical Method . 414 9.3.2 Typical Algorithm and Step Transition Map . . . . . . . . . . . . 415 9.4 Related B-Series Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 9.4.1 The Composition Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 9.4.2 Substitution Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 9.4.3 The Logarithmic Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

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10.

Volume-Preserving Methods for Source-Free Systems . . . . . . . . . . . . . . 443 10.1 Liouville’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 10.2 Volume-Preserving Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 10.2.1 Conditions for Centered Euler Method to be Volume Preserving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 10.2.2 Separable Systems and Volume-Preserving Explicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 10.3 Source-Free System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 10.4 Obstruction to Analytic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 10.5 Decompositions of Source-Free Vector Fields . . . . . . . . . . . . . . . . . . 452 10.6 Construction of Volume-Preserving Schemes . . . . . . . . . . . . . . . . . . . 454 10.7 Some Special Discussions for Separable Source-Free Systems . . . . 458 10.8 Construction of Volume-Preserving Scheme via Generating Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 10.8.1 Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 10.8.2 Construction of Volume-Preserving Schemes . . . . . . . . . . . 464 10.9 Some Volume-Preserving Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 467 10.9.1 Volume-Preserving R–K Methods . . . . . . . . . . . . . . . . . . . . 467 10.9.2 Volume-Preserving 2-Stage P–R–K Methods . . . . . . . . . . . 471 10.9.3 Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 10.9.4 Some Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

11.

Contact Algorithms for Contact Dynamical Systems . . . . . . . . . . . . . . . 477 11.1 Contact Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 11.1.1 Basic Concepts of Contact Geometry . . . . . . . . . . . . . . . . . 477 11.1.2 Contact Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 11.2 Contactization and Symplectization . . . . . . . . . . . . . . . . . . . . . . . . . . 484 11.3 Contact Generating Functions for Contact Maps . . . . . . . . . . . . . . . . 488 11.4 Contact Algorithms for Contact Systems . . . . . . . . . . . . . . . . . . . . . . 492 11.4.1 Q Contact Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 11.4.2 P Contact Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 11.4.3 C Contact Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 11.5 Hamilton–Jacobi Equations for Contact Systems . . . . . . . . . . . . . . . 494 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

12.

Poisson Bracket and Lie–Poisson Schemes . . . . . . . . . . . . . . . . . . . . . . . . 499 12.1 Poisson Bracket and Lie–Poisson Systems . . . . . . . . . . . . . . . . . . . . . 499 12.1.1 Poisson Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 12.1.2 Lie–Poisson Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 12.1.3 Introduction of the Generalized Rigid Body Motion . . . . . 505 12.2 Constructing Difference Schemes for Linear Poisson Systems . . . . 507 12.2.1 Constructing Difference Schemes for Linear Poisson Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

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Construction of Difference Schemes for General Poisson Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 12.2.3 Answers of Some Questions . . . . . . . . . . . . . . . . . . . . . . . . . 511 12.3 Generating Function and Lie–Poisson Scheme . . . . . . . . . . . . . . . . . 514 12.3.1 Lie–Poisson–Hamilton–Jacobi (LPHJ) Equation and Generating Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 12.3.2 Construction of Lie–Poisson Schemes via Generating Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 12.4 Construction of Structure Preserving Schemes for Rigid Body . . . . 523 12.4.1 Rigid Body in Euclidean Space . . . . . . . . . . . . . . . . . . . . . . 523 12.4.2 Energy-Preserving and Angular Momentum-Preserving Schemes for Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 12.4.3 Orbit-Preserving and Angular-Momentum-Preserving Explicit Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 12.4.4 Lie–Poisson Schemes for Free Rigid Body . . . . . . . . . . . . 530 12.4.5 Lie–Poisson Scheme on Heavy Top . . . . . . . . . . . . . . . . . . . 535 12.4.6 Other Lie–Poisson Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 538 12.5 Relation Among Some Special Group and Its Lie Algebra . . . . . . . . 543 12.5.1 Relation Among SO(3), so(3) and SH1 , SU (2) . . . . . . . 543 12.5.2 Representations of Some Functions in SO(3) . . . . . . . . . . 545 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 13.

KAM Theorem of Symplectic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 549 13.1 Brief Introduction to Stability of Geometric Numerical Algorithms 549 13.2 Mapping Version of the KAM Theorem . . . . . . . . . . . . . . . . . . . . . . 551 13.2.1 Formulation of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 551 13.2.2 Outline of the Proof of the Theorems . . . . . . . . . . . . . . . . . 554 13.2.3 Application to Small Twist Mappings . . . . . . . . . . . . . . . . . 558 13.3 KAM Theorem of Symplectic Algorithms for Hamiltonian Systems559 13.3.1 Symplectic Algorithms as Small Twist Mappings . . . . . . . 560 13.3.2 Numerical Version of KAM Theorem . . . . . . . . . . . . . . . . . 564 13.4 Resonant and Diophantine Step Sizes . . . . . . . . . . . . . . . . . . . . . . . . . 568 13.4.1 Step Size Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 13.4.2 Diophantine Step Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 13.4.3 Invariant Tori and Further Remarks . . . . . . . . . . . . . . . . . . . 574 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578

14.

Lee-Variational Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 14.1 Total Variation in Lagrangian Formalism . . . . . . . . . . . . . . . . . . . . . . 581 14.1.1 Variational Principle in Lagrangian Mechanics . . . . . . . . . 581 14.1.2 Total Variation for Lagrangian Mechanics . . . . . . . . . . . . . 583 14.1.3 Discrete Mechanics and Variational Integrators . . . . . . . . . 586 14.1.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 14.2 Total Variation in Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . 591 14.2.1 Variational Principle in Hamiltonian Mechanics . . . . . . . . 591

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14.2.2 Total Variation in Hamiltonian Mechanics . . . . . . . . . . . . . 593 14.2.3 Symplectic-Energy Integrators . . . . . . . . . . . . . . . . . . . . . . . 596 14.2.4 High Order Symplectic-Energy Integrator . . . . . . . . . . . . . 600 14.2.5 An Example and an Optimization Method . . . . . . . . . . . . . 603 14.2.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 14.3 Discrete Mechanics Based on Finite Element Methods . . . . . . . . . . . 606 14.3.1 Discrete Mechanics Based on Linear Finite Element . . . . . 606 14.3.2 Discrete Mechanics with Lagrangian of High Order . . . . . 608 14.3.3 Time Steps as Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 14.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 15.

Structure Preserving Schemes for Birkhoff Systems . . . . . . . . . . . . . . . . 617 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 15.2 Birkhoffian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 15.3 Generating Functions for K(z, t)-Symplectic Mappings . . . . . . . . . 621 15.4 Symplectic Difference Schemes for Birkhoffian Systems . . . . . . . . . 625 15.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 15.6 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639

16.

Multisymplectic and Variational Integrators . . . . . . . . . . . . . . . . . . . . . . 641 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 16.2 Multisymplectic Geometry and Multisymplectic Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 16.3 Multisymplectic Integrators and Composition Methods . . . . . . . . . . 646 16.4 Variational Integrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 16.5 Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658

Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669

Introduction The main theme of modern scientific computing is the numerical solution of various differential equations of mathematical physics bearing the names, such as Newton, Euler, Lagrange, Laplace, Navier–Stokes, Maxwell, Boltzmann, Einstein, Schr¨odinger, Yang-Mills, etc. At the top of the list is the most celebrated Newton’s equation of motion. The historical, theoretical and practical importance of Newton’s equation hardly needs any comment, so is the importance of the numerical solution of such equations. On the other hand, starting from Euler, right down to the present computer age, a great wealth of scientific literature on numerical methods for differential equations has been accumulated, and a great variety of algorithms, software packages and even expert systems has been developed. With the development of the modern mechanics, physics, chemistry, and biology, it is undisputed that almost all physical processes, whether they are classical, quantum, or relativistic, can be represented by an Hamiltonian system. Thus, it is important to solve the Hamiltonian system correctly.

1. Numerical Method for the Newton Equation of Motion In the spring of 1991, the first author [Fen92b] presented a plenary talk on how to compute the numerical solution of Newton classical equation accurately at the Annual Physics Conference of China in Beijing. It is well known that numerically solving so-called mathematics-physics equations has become a main topic in modern scientific computation. The Newton equation of motion is one of the most popular equations among various mathematics-physics equations. It can be formulated as a group of second-order ordinary differential equations, f = ma = m¨ x. The computational methods of the differential equations advanced slowly in the past due to the restriction of the historical conditions. However, a great progress was made since Euler, due to contributions from Adams, Runge, Kutta, and St¨omer, etc.. This is especially true since the introduction of the modern computer for which many algorithms and software packages have been developed. It is said that the three-body problem is no longer a challenging problem and can be easily computed. Nevertheless, we propose the following two questions: 1◦ Are the current numerical algorithms suitable for solving the Newton equation of motion? 2◦ How can one calculate the Newton equation of motion more accurately? It seems that nobody has ever thought about the first issue seriously, which may be the reason why the second issue has never been studied systematically. In this book, we will study mainly the fundamental but more difficult Newton equation of motion that is in conservative form. First, the conservative Newton equation has two equivalent mathematical representations: a Lagrange variation form and a Hamiltonian form. The

2

Introduction

latter transforms the second-order differential equations in physical space into a group of the first-order canonical equations in phase space. Different representations for the same physical law can lead to different computational techniques in solving the same problem, which can produce different numerical results. Thus making a wise and reasonable choice among various equivalent mathematical representations is extremely important in solving the problem correctly. We choose the Hamiltonian formulation as our basic form in practice based on the fact that the Hamiltonian equations have symmetric and clean form, where the physical laws of the motion can be easily represented. Secondly, the Hamiltonian formulation is more general and universal than the Newton formulation. It can cover the classical, relativistic, quantum, finite or infinite dimensional real physical processes where dissipation effect can be neglected. Therefore, the success of the numerical methods for Hamiltonian equations has broader development and application perspectives. Thus, it is very surprising that the numerical algorithms for Hamiltonian equations are almost nonexistent after we have searched various publications. This motivates us to study the problem carefully to seek the answers to the previous two questions. Our approach is to use the symplectic geometry, which is the geometry in phase space. It is based on the anti-symmetric area metric, which is in contrast to the symmetric length metrics of Euclid and Riemann geometry. The basic theorem of the classic mechanics can be described as “the dynamic evolution of all Hamiltonian systems preserves the symplectic metrics, which means it is a symplectic (canonical) transformation”. Hence the correct discretization algorithms to all the Hamiltonian systems should be symplectic transformation. Such algorithms are called symplectic (canonical) algorithms or Hamiltonian algorithms. We have intentionally analyzed and evaluated the derivation of the Hamiltonian algorithm within the symplectic structures. The fact proved that this approach is correct and fruitful. We have derived a series of symplectic algorithms, found out their properties, laid out their theoretical foundation, and tested them with extremely difficult numerical experiments. In order to compare the symplectic and non-symplectic algorithm, we proposed eight numerical experiments: harmonic oscillator, nonlinear Duffing oscillator, Huygens oscillator, Cassini oscillator, two dimensional multi-crystal and semi-crystal lattice steady flow, Lissajous image, geodesic flow on ellipsoidal surface, and Kepler motion. The numerical experiments demonstrate the superiority of the symplectic algorithm. All traditional non-symplectic algorithms fail without exception, especially in preserving global property and structural property, and long-term tracking capability, regardless of their accuracy. However, all the symplectic algorithms passed the tests with long-term stable tracking capability. These tests clearly demonstrate the superiority of the symplectic algorithms. Almost all of the traditional algorithms are non-symplectic with few exceptions. They are designed for the asymptotic stable system which has dissipation mechanism to maintain stability, whereas the Hamiltonian system does not have the asymptotic stability. Hence all these algorithms inevitably contain artificial numerical dissipation, fake attractors, and other parasitics effects of non-Hamiltonian system. All these effects lead to seriously twist and serious distortion in numerical results. They can be used in short-term transient simulation, but are not suitable and can lead to wrong

Introduction

3

conclusions for long-term tracking and global structural property research. Since the Newton equation is equivalent to Hamiltonian equation, the answer to the first question is “No”, which is quite beyond expectation. The symplectic algorithm does not have any artificial dissipation so that it can congenitally avoid all non-symplectic pollution and become a “clean” algorithm. Hamiltonian system has two types of conservation laws: one is the area invariance in phase space, i.e., Liouville–Poincar´e conservation law; the other is the motion invariance which includes energy conservation, momentum and angular momentum conservation, etc. We have proved that all symplectic algorithms have their own invariance, which has the same convergence to the original theoretical invariance as the convergence order of the numerical algorithm. We have also proved that the majority of invariant tori of the near integrable system can be preserved, which is a new formulation of the famous KAM (Kolmogorov–Arnorld–Moser) theorem[Kol54b,Kol54a,Arn63,Mos62] . All of these results demonstrate that the structure of the discrete Hamiltonian algorithm is completely parallel to the conservation law, and is very close to the original form of the Hamiltonian system. Moreover, theoretically speaking, it has infinite longterm tracking capability. Hence, a correct numerical method to solve the Newton equation is to Hamiltonize the equation first and then use the Hamiltonian algorithm. This is the answer to the second question. We will describe in more detail the KAM theory of symplectic algorithms for Hamiltonian systems in Chapter 13. In the following we present some examples to compare the symplectic algorithm and other non-symplectic algorithms in solving Newton equation of motion.

(1)

Calculation of the Harmonic oscillator’s elliptic orbit

Calculation of the Harmonic oscillator’s elliptic orbit (Fig. 0.1(a)) uses Runge–Kutta method (R–K) with a step size 0.4. The output is at 3,000 steps. It shows artificial dissipation, shrinking of the orbit. Fig. 0.1(b) shows the results using Adams method with a step size 0.2. It is anti-dissipative and the orbit is scattered out. Fig. 0.1(c) shows the results of two-step central difference (leap-frog scheme). This scheme is symplectic to linear equations. The results are obtained with a step size 0.1. It shows that the results of three stages for 10,000,000 steps: the initial 1,000 steps, the middle 1,000 steps, and the final 1,000 steps. They are completely in agreement.

(2)

The elliptic orbit for the nonlinear oscillator

Fig. 0.2(a) shows the results of two-step central-difference. This scheme is nonsymplectic for nonlinear equations. The output is for step size 0.2 and 10,000 steps. Fig. 0.2(a) shows the initial 1,000 steps and Fig. 0.2(b) shows the results between 9,000 to 10,000 steps. Both of them show the distortion of the orbit. Fig. 0.2(c) is for the second-order symplectic algorithm with 0.1 step size, 1,000 steps.

4

Introduction

Fig. 0.1.

Calculation of the Harmonic oscillator’s elliptic orbit

Fig. 0.2.

Calculation of the nonlinear oscillator’s elliptic orbit

Introduction

Fig. 0.3.

(3)

5

Calculation of the nonlinear Huygens oscillator

The oval orbit of the Huygens oscillator

Using the R–K method, the two fixed points on the horizontal axes become two fake attractors. The probability of the phase point close to the two attractors is the same. The same initial point outside the separatrix is attracted randomly either to the left or to the right. Fig. 0.3(a) shows the results with a step size 0.10000005 and 900,000 steps, which approach the left attractor. Fig. 0.3(b) shows the results with a step size 0.10000004 and 900,000 steps, which approach the right attractor. Fig. 0.3(c) shows the results of the second-order symplectic algorithm with a step size 0.1. Four typical orbits are plotted and each contains 100,000,000 steps: for every orbit first 500 steps, the middle 500 steps, and the final 500 steps. They are in complete agreement.

(4)

The dense orbit of the geodesic for the ellipsoidal surface

The dense orbit of the geodesic for the ellipsoidal surface with irrational frequency ratio. The square of frequency ratio is 5/16, step size is 0.05658, 10,000 steps. Fig.0.4(a) is for the R–K method which does not tend to dense. Fig. 0.4(b) is for the symplectic algorithm which tends to dense.

6

Introduction

Fig. 0.4.

(5)

Geodesics on ellipsoid, frequency ratio

√

5 : 4, non dense (a), dense orbit (b)

The close orbit of the geodesic for the ellipsoidal surface

The close orbit of the geodesic for the ellipsoidal surface with rational frequency ratio. The frequency ratio is 11/16, step size is 0.033427, 100,000 steps and 25 cycles. Fig.0.5(a) is for the R–K method which does not lead to the close orbit. Fig. 0.5(b) is for the symplectic algorithm which leads to the close orbit.

Fig. 0.5.

(6)

Geodesics on ellipsoid, frequency ratio 11:16, non closed (a), closed orbit (b)

The close orbit of the Keplerian motion

The close orbit of the Keplerian motion with rational frequency ratio. The frequency ratio is 11/20, step size is 0.01605, 240,000 steps and 60 cycles. Fig. 0.6(a) is for the R–K method which does not lead to the close orbit. Fig. 0.6(b)is for the symplectic method which leads to the close orbit.

Introduction

Fig. 0.6.

2.

7

Geodesics on ellipsoid, frequency ratio 11:20, non closed (a), closed orbit (b)

History of the Hamiltonian Mechanics

We first consider the three formulations of the classical mechanics. Assume a motion has n degrees of freedom. The position is denoted as q = (q1 , · · · , qn ). The potential function is V = V (q). Then we have m

∂ d2 q = − V, d t2 ∂q

which is the standard formulation of the motion. It is a group of second-order differential equations in space Rn . It is usually called the standard formulation of the classical mechanics, or Newton formulation. Euler and Lagrange introduced an action on the difference between the kinetic energy and potential energy L(q, q) ˙ = T (q) ˙ − V (q) =

1 (q, ˙ M q) ˙ − V (q). 2

Using the variational principle the above equation can be written as d ∂L ∂L − = 0, d t ∂ q˙ ∂q which is called the variational form of the classical mechanics, i.e., the Lagrange form. In the 19th century, Hamilton proposed another formulation. He used the momentum p = M q˙ and the total energy H = T + V to formulate the equation of motion as ∂H ∂H p˙ = − , q˙ = , ∂q ∂p which is called Hamiltonian canonical equations. This is a group of the first-order differential equations in 2n phase space (p1 , · · · , pn , q1 , · · · , qn ). It has simple and symmetric form.

8

Introduction

The three basic formulations of the classical mechanics have been described in almost all text-books on theoretical physics or theoretical mechanics. These different mathematical formulations describe the same physics law but provide different approaches in problem solving. Thus equivalent mathematical formulation can have different effectiveness in computational methods. We have verified this in our own simulations. The first author did extensive research on Finite Element Method (FEM) in the 1960s [Fen65] which represents a systematic algorithm for solving equilibrium problem. Physical problems of this type have two equivalent formulations: Newtonian, i.e., solving the second-order elliptic equations, and variational formulation, i.e., minimization principle in energy functional. The key to the success of FEM in both theoretical and computational methods lies in using a reasonable variational formulation as the basic principle. After that, he had attempted to apply the FEM idea to the dynamic problem of continuum media mechanics, but not yet achieved the corresponding success, which appears to be difficult to accomplish even today. Therefore, the reasonable choice for computational method of dynamic problem might be the Hamiltonian formulation. Initially it is a conjecture and requires verification from the computational experiments. We have investigated how others evaluated the Hamiltonian system in history. First we should point out that Hamilton himself proposed his theory based on the geometric optics and then extended it to mechanics that appears to be a very different field. In 1834 Hamilton said, “This set of idea and method has been applied to optics and mechanics. It seems it can be applied to other areas and developed into an independent knowledge by the mathematicians”[Ham34] . This is just his expectation, and other peers in the same generation seemed indifferent to this set of theory, which was “beautiful but useless”[Syn44] to them. Klein, a famous mathematician, while giving a high appreciation to the mathematical elegance of the theory, suspected its applicability, and said: “. . . a physicist, for his problems, can extract from these theories only very little, and an engineer nothing”[Kle26] . This claim has been proved wrong at least in physics aspect in the later history. The quantum mechanics developed in the 1920s under the framework of the Hamiltonian formulation. One of the founders of the quantum mechanics, Schr¨odinger said, “Hamiltonian principle has been the foundation for modern physics . . . If you want to solve any physics problem using the modern theory, you must represent it using the Hamiltonian formulation”[Sch44] .

3.

The Importance of the Hamiltonian System

The Hamiltonian system is one of the most important systems among all the dynamics systems. All real physical processes where the dissipation can be neglected can be formulated as Hamiltonian system. Hamiltonian system has broad applications, which include but are not limited to the structural biology, pharmacology, semiconductivity, superconductivity, plasma physics, celestial mechanics, material mechanics, and partial differential equations. The first five topics have been listed as “Grand Challenges” in Research Project of American government.

Introduction

9

The development of the physics verifies the importance of the Hamiltonian systems. Up to date, it is undisputed that all real physical processes where the dissipation can be neglected can be written as Hamiltonian formulation, whether they have finite or infinite degrees of freedom. The problem with finite degrees of freedom includes celestial and man-made satellite mechanics, rigid body, and multi-body (including the robots), geometric optics, and geometric asymptotic method (including ray-tracing approximation method in wave-equation, and WKB equation of quantum mechanics), confinement of the plasma, the design of the high speed accelerator, automatic control, etc. The problem with infinite degrees of freedom includes ideal fluid dynamics, elastic mechanics, electrical mechanics, quantum mechanics and field theory, general relativistic theory, solitons and nonlinear waves, etc. All the above examples show the ubiquitous and nature of the Hamiltonian systems. It has the advantage that different physics laws can be represented by the same mathematical formulation. Thus we have confidence to say that successful development of the numerical methods for Hamiltonian system will have extremely broad applications. We now discuss the status of the numerical method for Hamiltonian systems. Hamiltonian systems, including finite and infinite dimensions, are Ordinary Differential Equations (ODE) or Partial Differential Equations (PDE) with special form. The research on the numerical method of the differential equations started in the 18th century and produced abundant publications. However, we find that few of them discuss the numerical method specifically for Hamiltonian systems. This status is in sharp contrast with the importance of the Hamiltonian system. Therefore, it is appealing and worthy to investigate and develop numerical methods for this virgin field.

4. Technical Approach — Symplectic Geometry Method The foundation for the Hamiltonian system is symplectic geometry, which is increasingly flourishing in both theory and practice. The history of symplectic geometry can be traced back to Astronomer Hamilton in the 19th century. In order to study the Newton mechanics, he introduced generalized coordinates and generalized momentums to represent the energy of the system, which is now called Hamiltonian function now. For a system with n degrees of freedom, the n generalized coordinates and momentums are spanned into a 2n phase space. Thus the Newton mechanics becomes the geometry in phase space. In terms of the modern concept, this is a kind of symplectic geometry. Later, Jacobi, Darboux, Poincar´e, Cartan, and Weyl did a lot of research on this topic from different points of view (algebra and geometry). However, the major development of the modern symplectic geometry started with the discovery of KAM theorem (1950s to the beginning of 1960s). In the 1970s, in order to research Fourier integral operator, quantum representation of the geometry, group representation theory, classification of the critical points, Lie Algebra, etc., people did a lot of work on symplectic geometry (e.g., Arnold[Arn89] , Guillemin[GS84] , Weinstein[Wei77] , Marsden[AM78] , etc.), which promoted the development in these areas. In the 1980s, the research on total

10

Introduction

symplectic geometry emerged subsequently, such as the research on “coarse” symplectic (e.g., Gromov et al.), fix point for symplectic map (e.g., Conley, Zehnder’s Arnold conjecture), the convexity of the matrix mapping (e.g., Atiyah, Guillemin, Sternberg et al.). The research on symplectic geometry is not only extremely enriched and vital, but its application is also widely applied to different areas, such as celestial mechanics, geometric optics, plasma, the design of high speed accelerators, fluid dynamics, elastic mechanics, optimal control, etc. Weyl[Wey39] said the following in his monograph on the history of the symplectic group: “I called it complex group initially. Because this name can be confused with the complex number, I suggest using symplectic, a Greek word with the same meaning.” An undocumented law for the modern numerical method is that the discretized problem should preserve the properties of the original problem as much as possible. To achieve this goal, the discretization should be performed in the same framework as the original problem. For example, the finite element method treats the discretized and original problem in the same framework of the Sobolev space so that the basic properties of the original problem, such as symmetry, positivity, and conservativity, etc., are all preserved. This not only ensures the effectiveness and reliability in practice, but also provides a theoretical foundation. Based on the above principle, the constructed numerical methods for the Hamiltonian system should preserve the Hamiltonian structure, which we call “Hamiltonian algorithm”. The Hamiltonian algorithm must be constructed in the same framework as the Hamiltonian system. In the following, we will describe the basic mathematical framework of the Hamiltonian system and derive the Hamiltonian algorithm from the same framework. This is our approach. We will use the Euclid geometry as an analogy to describe the symplectic geometry. The structure of an Euclid space Rn lies in the bilinear, symmetric, non-degenerate inner product, (x, y) = x, Iy, I = In . Since it is non-degenerate, (x, x) is alwayspositive when x = 0. Therefore we can define the length of the vector x as ||x|| = (x, x) > 0. All the linear operators that preserve the inner product, i.e., satisfy AT IA = I, form a group O(n), called the orthogonal group, which is a typical Lie group. The corresponding Lie algebra o(n) consists of all the transformation that satisfies AT + A = AI + IA = 0, which is infinitesimal orthogonal transformation. The symplectic geometry is the geometry on the phase space. The symplectic space, i.e, the symplectic structure in phase space, lies in a bilinear, anti-symmetric, and non-degenerate inner product, [x, y] = x, Jy,

J = J2n =

 O −In

which is called the symplectic inner product. When n = 1,   x [x, y] =  1 x2

 y1  , y2

In O

 ,

Introduction

11

which is the area of the parallel quadrilateral with vectors x and y as edges. Generally speaking, the symplectic inner product is an area metric. Due to the anti-symmetry of the inner product, [x, x] = 0 always holds for any vector x. Thus it is impossible to derive the concept of length of a vector from the symplectic inner product. This is the fundamental difference between the symplectic geometry and Euclid geometry. All transformations that preserve the symplectic inner product form a group, called a symplectic group, Sp(2n), which is also a typical Lie group. Its corresponding Lie algebra consists of all infinitesimal symplectic transformations B, which satisfy B T J + JB = 0. We denote it as sp(2n). Since the non-degenerate anti-symmetric matrix exists only for even dimensions, the symplectic space must be of even dimensions. The phase space exactly satisfies this condition. Overall the Euclid geometry is a geometry for studying the length, while the symplectic geometry is for studying the area. The one-to-one nonlinear transformation in the symplectic geometry is called symplectic transformation, or canonical transformation. The transformation whose Jacobian is always a symplectic matrix plays a major role in the symplectic geometry. For the Hamiltonian system, if we represent a pair of n-dim vectors with a 2n-dim vector z = (p, q), the Hamiltonian equation becomes ∂H dz = J −1 . dt ∂z Under the symplectic transformation, the canonical form of the Hamiltonian equation is invariant. The basic principle of the Hamiltonian mechanics is for any Hamiltonian system. There exists a group of symplectic transformation (i.e., the phase flow) GtH1 ,t0 that depends on H and time t0 , t1 , so that z(t1 ) = GtH1 ,t0 z(t0 ), which means that GtH1 ,t0 transforms the state at t = t0 to the state at t = t1 . Therefore, all evolutions of the Hamiltonian system are also evolutions of the symplectic transformation. This is a general mathematical principle for classical mechanics. When H is independent of t, GtH1 ,−t2 = GtH1 ,−t0 , i.e., the phase flow depends only on the difference in parameters t1 − t0 . We can let GtH = Gt,0 H . One of the most important issues for the Hamiltonian system is stability. The feature of this type of problems in geometry perspective is that its solution preserves the metrics. Thus the eigenvalue is always a purely imaginary number. Therefore, we cannot use the asymptotic stability theory of Poincar´e and Liapunov. The KAM theorem must be used. This is a theory about the total stability and is the most important breakthrough for Newton mechanics. The application of the symplectic geometry to the numerical analysis was first proposed by K. Feng [Fen85] in 1984 at the international conference on differential geometry and equations held in Beijing. It is based on a basic principle of the analytical mechanics: the solution of the system is a volume-preserved transformation (i.e., symplectic transformation) with one-parameter2 on symplectic 2

Before K.Feng’s work, there existed works of de Vorgelaere[Vog56] , Ruth[Rut83] and Menyuk[Men84] .

12

Introduction

integration. Since then, new computational methods for the Hamiltonian system have been developed and we have studied the numerical method of the Hamiltonian system from this perspective. The new methods make the discretized equations preserve the symplectic structure of the original system, i.e., to restore the original principle of the discretized Hamiltonian mechanics. Its discretized phase flow can be regarded as a series of discrete symplectic transformations, which preserve a series of phase area and phase volume. In 1988, K. Feng described his research work on the symplectic algorithm during his visit to Western Europe and gained the recognition from many prominent mathematicians. His presentation on “Symplectic Geometry and Computational Hamiltonian Mechanics” has obtained consistent high praise at the workshop to celebrate the 60th birthday of famous French mathematician Lions. Lions thought that K. Feng founded the symplectic algorithm for Hamiltonian system after he developed the finite element methods independent of the efforts in the West. The prominent German numerical mathematician Stoer said, “This is a new method that has been overlooked for a long time but should not be overlooked.” We know that we can not study the Hamiltonian mechanics without the symplectic geometry. In the meantime, the computational method of the Hamiltonian mechanics doesn’t work without the symplectic difference scheme. The classical R–K method is not suitable to solve this type of problems, because it cannot preserve the long-term stability. For example, the fourth-order R–K method obtains a completely distorted result after 200,000 steps with a step size 0.1, because it is not a symplectic algorithm, but a dissipative algorithm. We will describe in more detail the theory of symplectic geometry and symplectic algebra in Chapters 1, 2 and 3.

5.

The Symplectic Schemes

Every scheme, whether it is explicit or implicit, can be treated as a mapping from this time to the next time. If this mapping is symplectic, we call it a symplectic geometric scheme, or in short, symplectic scheme. We first search the classical difference schemes. The well-known Euler midpoint scheme is a symplectic scheme z n+1 = z n + J −1 Hz

 z n+1 + z n  2

.

The symplectic scheme is usually implicit. Only for a split Hamiltonian system, we can obtain an explicit scheme in practice by alternating the explicit and implicit stages. Its accuracy is only of first order. Symmetrizing this first-order scheme yields a second-order scheme (or so-called reversible scheme). There exist multi-stage R–K symplectic schemes among the series of R–K schemes. It is proved that the 2s-order Gauss multi-stage R–K scheme is symplectic. We will give more details on these topics in Chapters 4 , 7 and 8. The theoretical analysis and a priori error analysis will be described in Chapter 6 and 9.

Introduction

13

In addition, the first author and his group constructed various symplectic schemes with arbitrary order of accuracy using the generating function theory from the analytical mechanics perspective. In the meantime, he extended the generating function theory and Hamilton–Jacobi equations by constructing all types of generating function and the corresponding Hamilton–Jacobi equations. The generating function theory and the construction of the symplectic schemes will be introduced in Chapter 5.

6. The Volume-Preserving Scheme for Source-free System Among the various dynamical systems, one of them is called source-free dynamical system, where the divergence of the vector field is zero: dx = f (x), dt

div f (x) = 0.

  ∂xn+1 The phase flow to this system is volume-preserved, i.e., det = 1. Therefore, ∂xn the numerical solution should also be volume-preserved. We know that Hamiltonian system is of even dimensions. However, the source-free system can be of either even or odd dimensions. For the system of odd dimensions, the Euler midpoint scheme may not be volume-preserved. ABC (Arnold–Beltrami– Childress) flow is one of the examples. Its vector field has the following form: x˙ = A sin x + C cos y, y˙ = B sin x + A cos z, z˙ = C sin y + B cos x, which is a source-free system and the phase flow is volume-preserved. This is a split system and constructing the volume-preserving scheme is easy. Numerical experiments show that the volume-preserving scheme can calculate the topological structure accurately, whereas the traditional schemes can not[FS95,QZ93] . We will give more details in Chapter 10.

7.

The Contact Schemes for Contact System

There exists a special type of dynamical systems with odd dimensions. They have similar symplectic structure as the systems of even dimensions. We call them contact systems. The reader can find more details in Chapter 11. Consider the contact system in R2n+1 space

14

Introduction

⎡ ⎤ ⎡ ⎤ ⎤ x1 y1 x ⎢ ⎥ ⎢ ⎥ (2n + 1) − dim vector : ⎣ y ⎦ , where x = ⎣ ... ⎦ , y = ⎣ ... ⎦ , z = (z); z xn y ⎡ ⎤ ⎡ ⎤ ⎡n ⎤ a(x, y, z) a1 b1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (2n + 1) − dim v.f. : ⎣ b(x, y, z) ⎦ , where a = ⎣ ... ⎦ , b = ⎣ ... ⎦ , c = (c). ⎡

an

c(x, y, z)

bn

A contact system can be generated from a contact Hamiltonian function K(x, y, z): dx = −Ky + Kz x = a, dt dy = Kx = b, dt dz = Ke = c, dt

Ke (x, y, z) = K(x, y, z) − (x, Ky (x, y, z)). The contact structure in R2n+1 space is defined as ⎡

⎤ dx α = xd y + d z = [0, x, 1] ⎣ d y ⎦ . dz A transformation f is called the contact transformation if it could preserve the contact structure with a pre-factor μf . A scheme which can preserve the contact structure is called contact scheme[FW94,Shu93] . The contact schemes have potential applications in the propagation of the wave front[MF81,QC00] , the applications in thermal dynamics[MNSS91,EMvdS07] , and the characteristic method for the first-order differential equations[Arn88] . The symplectic algorithm, the volume-preserving algorithm, the contact algorithm, and the Lie–Poisson algorithm are all schemes that preserve the geometry structure of the phase space. We call these methods “geometric integration for dynamic system”[FW94,LQ95a] . The geometric integration was first introduced by the first author[FW94] and has been widely accepted and used by the international scientists. The 1996 workshop on the advance of the numerical method, held in England, mentioned the importance of the structure-preserving schemes for the dynamics system. In that workshop, a series of high-order structure preserving schemes has been proposed via the multiplicative extrapolation method[QZ92,QZ94] . We have extended the explicit schemes of Yoshida[Yos90] to all self-adjoint schemes. By using the product of the schemes and their adjoint, we have constructed very high order self-adjoint schemes. The details are described in Chapter 8. Concerning the Lie–Poisson algorithm we will describe more details in Chapter 12.

Introduction

15

8. Applications of the Symplectic Algorithms for Dynamics System (1) Applications of symplectic algorithms to large time scale system Nearly all systems of celestial mechanics and dynamic astrophysics are Hamiltonian or approximately Hamiltonian with few dissipations. Such systems can be described by canonical forms of Hamiltonian systems, which has now become one of the most important research areas of dynamical system. However, due to the complicated nonlinearity of those canonical Hamiltonian systems, few analytic solutions are available. Although sometimes approximate analytic solutions in form of power series can be obtained by the perturbation method, the long time dynamics, the quantity property, and the intrinsic nonlinearity are overlooked by such solutions. Thus, the numerical methods are required to study those problems to get more accurate and quantitative numerical solutions, which not only provide the information and images on the whole phase space of the given mechanical system for further qualitative analysis, but also lead to some important results for the system. There are two ways to analyze Hamiltonian system qualitatively. One way is to get the numerical solution of the canonical Hamiltonian system directly by the numerical methods, and the other is simpler discretization process to the equation of motion, which becomes a simple mapping question which making computing easier. The later method reduces the computational effort so that it can be performed by normal computers to study the large time scale evolution of dynamical systems. Traditional numerical methods for dynamics system can be categorized into singlestep methods, e.g. the R–K method, and multi-step methods, e.g. the widely used Adams method for the first order differential equations, and Cowell methods for the second order differential equations. However, all the methods have the artificial numerical dissipations so that the corresponding total energy of the Hamiltonian system will change linearly. This will distort basic property of Hamiltonian system and lead to wrong results for a long time computation. By quantitative analysis, we know that the dissipation of the total energy will accumulate errors of the numerical trajectories of the celestial bodies. The errors will increase at least squarely with respect to the integration time step. In the 1980s, the first author and his group established the theory of the symplectic algorithms for Hamiltonian system. The significance of this theory is not only to present a new kind of algorithms, but also to elucidate the reason for the false dissipation of the traditional methods, i.e.,that the main truncation error terms of those non-symplectic methods are dissipative terms, whereas the main truncation error terms of symplectic algorithms are not dissipative terms. Thus the numerical energy of the system will not decrease linearly, but change periodically. Due to the conservation of the symplectic structure of the system, which is the basic property, the symplectic algorithms have the long time capacity to simulate the evolution of the celestial bodies. As the energy is a very important parameter of such a system, the numerical results of symplectic algorithms, which can preserve the energy approximatively, are more

16

Introduction

reasonable. Furthermore, because the errors of the energy are controlled, the errors of numerical trajectories of celestial bodies are no longer along the track by (t − t0 )2 laws of the fast-growing, and with only a t − t0 linear growth, this to the long arc computation is extremely advantageous. For the advantages of the symplectic algorithms, nowadays they have been widely used in the study of dynamical astronomy, especially in the qualitative analysis of the evolution of solar system, e.g. to analyze the stable motion area, space distributions and trajectory resonance of little planets, long time evolution of large planets and extra-planets, and other hot topics in the dynamical astronomy.

(2)

Applications of symplectic algorithms to qualitative analysis

We first use two simple examples to illustrate the special affects of symplectic schemes on the qualitative analysis in dynamics astronomy[LZL93,JLL02,LL95,LL94,Lia97,LLZW94] . Example I. The Keplerian motions. It is the elliptical motions of two-body problem. The corresponding Hamiltonian function is: H(p, q) = T (p) + V (q), where p and q are the generalized coordinates and generalize momentum, T and V are the kinetic and potential energies. The analytic solution is a fixed ellipse. When we simulate this problem by the R–K methods and symplectic algorithms, the former ones shrink the ellipse gradually, whereas the later ones preserve the shape and size of the ellipse (see the numerical trajectories after 150 and 1000 steps respectively in Fig. 0.7(a), e=0.7 and Fig. 0.7(b), e=0.9 where e is the eccentricity of the ellipse). This means the non-symplectic R–K methods have the false energy dissipation and the symplectic algorithms preserve the main character of the Kepler problem because of the conservation of the symplectic structure. Example II. The axial symmetry galaxy’s stellar motion question. Its simplified dynamic model corresponding to the Hamiltonian function is: H(p, q) =

1 2 1 2 (p1 + p22 ) + (q12 + q22 ) + (2q12 q2 − p32 ). 2 2 3

To obtain the basic character of the dynamics of this system, we compute it with order 7 and order 8 Runge–Kutta–Fehlberg methods (denoted as RKF(7) and RKF(8) resp.), as well as the order 6 explicit symplectic algorithm (SY6). The numerical results are listed in Fig. 0.8 to Fig. 0.10. In these figures (Fig. 0.8 to Fig. 0.9), we see that the symplectic algorithm preserves the energy H very well in both of the two cases (ordered LCN= 0 and disorder region LCN> 0), while the RKF methods increase the energy with the evolution of time ΔH. In Fig. 0.10 (a) and Fig. 0.10 (b), the symplectic algorithms present numerically the basic characters of the system: the fixed curve in case of LCN= 0 and the chaos property in case of LCN> 0.

Introduction

17

Fig. 0.7.

Comparison of calculation of Keplerian motion by R–K and symplectic methods.

Fig. 0.8. LCN=0.

Curves of ΔH obtained by RKF(8)[left] and SY6 [right] both with H0 = 0.553,

Fig. 0.9. Curves of ΔH obtained by RKF(8)[left] and SY6 [right] both with H0 = 0.0148, LCN > 0

The symplectic algorithms can preserve the symplectic structure of Hamiltonian systems and the basic evolutionary property of such dynamical systems. Therefore,

18

Introduction

Fig. 0.10. Poincar´e section obtained by RKF(8)[left] with H0 = 0.553,LCN=0 and SY6 [right] with H0 = 0.0148,LCN>0

the symplectic algorithms were widely used to study the dynamical astronomy. Currently, it is a hot topic to study the dynamical evolution of the solar system, such as the long-term trajectory evolution of large planet and extra-planet, the space distribution of little planets in main zone (Kirkwood interstice phenomenon), trajectory resonance, the evolution of satellite system of a large planet, the birth and evolution of planet loops and the trajectory evolution of a little planet near Earth. All these problems require numerical simulation for a very long time, e.g. 109 years or more for the solar system. Thus, the time steps for the numerical methods shall be large enough due to limitations of our computers, while the basic property of the system should be preserved. This excludes all the non-symplectic methods, whilst just lower order symplectic algorithms are valid for the task. In recent years, many astronomers in Japan and America, e.g. Kinoshita[KYN91] , Bretit[GDC91] and Wisdom[WHT96,WH91] , have done a large amount of research on the evolution of the solar system. The following contribution of Wisdom has been widely cited. He derived the Hamiltonian function in Jacobin coordinates of the solar system as

H(p, q) =

n−1 

Hi (p, q) + εΔH(p, q),

i=1

where Hi (p, q) is corresponding Hamiltonian function for a two-body system, ε  1 is a small parameter. By splitting the Hamiltonian function, explicit symplectic algorithms with different orders can be constructed. The advantage of those symplectic algorithms is that the truncation errors are as small as order of ε than those of algorithms constructed by the ordinary splitting for the Hamiltonian function (i.e., H(p, q) = T (p) + V (q)). Even the lower order symplectic algorithms obtained by this splitting method are very effective in a study of the evolution of the solar system. Since the 1980s , Chinese astronomers have also made some progress in the applications of symplectic algorithms to the research of dynamical astronomy, such as[WH91]

Introduction

19

1◦ For the restrictive three-body system constituted by solar, the major planet and the planetoid, some new results have been obtained after studying its corresponding resonance of 1:1 orbit and the triangle libration point. These results can successfully explain the distribution of stability region [ZL94,ZLL92] of Trojan planetoid, as well the actual size of the stable region of distributed triangle libration points corresponding to several relate major planet. 2◦ Adopting the splitting method of Wisdom for the Hamiltonian function to study the long-term trajectories evolution of some little planets. H(p, q) = H0 (p, q) + εH1 (q), where H0 (p, q) is the Hamiltonian function for an integrable system, ε  1 is a little parameter. The numerical results obtained by using this splitting method are very reasonable because the energy is preserved in a controlled range and no false dissipation occurs. 3◦ Application of symplectic algorithms to galaxy system. The bar phenomenon and the evolution of stars in NGC4736 Galaxy were simulated successfully by the symplectic algorithms. 4◦ Some useful results on how to describe the evolutionary features of celestial dynamical system were obtained by further study on the symplectic integrators and the existence of their formal integrations, as well as the changes of all kinds of conservation laws. Besides the research on Hamilton systems in dynamics astronomy mentioned above, the small diffusion situation were also discussed and applied. In view of the fact that the diffusion factor is relatively weak, a mixed symplectic algorithm constituted by the explicit scheme and the centered Euler scheme is applied for the conservative part (the main part corresponding to mechanic system) and the dissipative part, which is remarkably effective, because it could maintain the features of Hamilton system in the main part of this system.

(3) Applications of symplectic algorithms to quantitative computations Because the structure could be preserved by the symplectic algorithms, the errors of their numerical energy don’t accumulate linearly. When the celestial systems are integrated by symplectic algorithms, errors of trajectories will increase linearly as t − t0 , whereas errors of the non-symplectic methods increase rapidly as (t − t0 )2 . We show some examples next. Taking the trajectory of Lageos satellite as background, we consider two mechanics system of the Earth perturbation problems. The first one just takes into account the nonspherical perturbation of the Earth and the second one takes into account the nonspherical perturbation of the Earth and the perturbation of atmospheric resistance. The former problem corresponds to a Hamiltonian system, while the later one corresponds to a quasi-Hamiltonian system because of very small dissipation. We use the RKF7(8) methods and revised order 6 symplectic algorithm (denoted as SY6) to compute the

20

Introduction

two problems, respectively. The numerical results of the errors Δ(M + ω) of main 1000 cycles trajectory are listed in Table 0.1 and Table 0.2. From the two tables, we can clearly see that the errors of the non-symplectic methods, though very small at the beginning, increase rapidly as (t − t0 )2 ; whereas the errors of symplectic algorithm increase linearly as t − t0 . The results of symplectic algorithms are much better. This indicates that though the accuracy order of symplectic algorithms is the same as for other methods, they have more application value in the quantitative computations. We also improve the RKF7(8) for energy conserving methods by compensating the energy at every time step. We denote such method as the RKH method whose numerical results are also listed in the two tables. From the results, we can see that we have made much improvement of the schemes. The results of the energy error by the RKH are almost same with those by the symplectic algorithm. Thus the RKH methods not only have high order accuracy, but also can preserve the energy approximately as the symplectic algorithms. Table 0.1.

Errors of trajectories with nonspherical perturbation of the EarthΔ(M + ω)

method

N of steps / circle

100 circles

1000 circles

10000 circles

FKF7(8)

100

1.5 E − 10

1.4 E − 08

1.3 E − 06

SY6

50

0.5 E − 09

0.6 E − 08

1.0 E − 07

RKH

100

0.9 E − 11

0.9 E − 10

0.9 E − 09

Table 0.2.

Errors of trajectories with perturbation of atmospheric resistanceΔ(M + ω)

method

N of steps / circle

100 circles

1000 circles

10000 circles

FKF7(8)

100

1.4 E − 410

1.3 E − 08

1.3 E − 06

SY6

50

0.6 E − 09

0.7 E − 08

1.0 E − 07

RKH

100

2.1 E − 11

3.5 E − 10

6.2 E − 09

(4)

Applications of symplectic algorithms to quantum systems

The governing equation of the time evolution of quantum system is the Schr¨odinger equation ∂ψ ˆ ˆ =H ˆ 0 (r) + Vˆ (t, r), i = Hψ, H (0.1) ∂t ˆ is Hermitian. where the operator H According to the basic theory of quantum mechanics, the initial state of a quantum system uniquely determines all the states after initial moment of time. That is to say, if the state function ψ(t1 , r) is given at time t1 , then the solution (so-called wave function) of Equation (0.1) is determined as

Introduction

21

ψ(t, r) = a(t, r) + ib(t, r), where functions a and b are real. Such a solution can be generated by a group of time evolutionary operators t1 ,t2 }, i.e., {UH ˆ t1 ,t2 ψ(t1 , r). ψ(t2 , r) = UH ˆ ˆ They are independent of the Every operator is unitary and depends on t1 , t2 and H. state ψ(t1 , r) at time t1 . Therefore, the time evolutions of the quantum system are evolutions of unit transformation in this sense. Every operator can induce an operator, which acts on the real function vector. The two components of the real functions vector are the real part and the image part of the wave function, i.e.,  b(t , r)   b(t , r)  2 1 t1 ,t2 = SH . ˆ a(t2 , r) a(t1 , r) t1 ,t2 The operator SH preserves the inner product and symplectic wedge product for ˆ any two real function vectors. It is simply called norm-preserving symplectic evolution. The quantum system is a Hamiltonian system (with infinite dimensions) and the time evolution of the Schr¨odinger equation can be rewritten as a canonical Hamiltonian system for the two real functions of the wave function as the generalized momentum and generalized coordinates. The norm of wave function is the conservation law of the canonical system. Thus it is reasonable to integrate such a system by the norm-preserving symplectic numerical methods. To apply such a method to the infinite dimensional system, we should first space discretize the system into a finite dimensional canonical Hamiltonian system, which also preserves the norm of wave ˆ 0 (r) for the evolutionfunction. Suppose the characteristic functions of the operator H ary Schr¨odinger equation with some given boundary conditions contain the discrete states and continuous states. ˆ is independent on time explicitly, the energy of the quanWhen the Hamiltonian H ˆ tum system ψ|, H|ψ = Z T HZ is a conservation law both for the canonical system and norm-preserving symplectic algorithm. Such a norm-preserving symplectic algorithm with the fourth order accuracy can be constructed by the order 4 diagonal Pad´e approximations to the exponential function eλ . In the following, we take an example to introduce the method to discretize the time involved Schr¨odinger equation to a canonical system[LQHD07,QZ90a] . Consider the time evolution of an atom moving in one dimensional space by the action of some strong field V (t, x) is

i

∂ψ ˆ = Hψ, ∂t

ˆ =H ˆ 0 (r) + Vˆ (t, r), H 2

ˆ 0 = − 1 ∂ + V0 (x), H 2 ∂ x2  0, 0 < x < 1, V0 (x) = ∞, x ≤ 0 or x ≥ 1.

22

Introduction

In contrast to the characteristic function expanding method, we don’t make any truncation for the wave function when discretizing the Schr¨odinger equation. Therefore, ˆ 0. the resulting canonical system contains all the characteristic states of H The numerical conservation laws of explicit symplectic algorithms will converge to the corresponding conservation laws of the system as the time step tends to zero. Thus, although numerical energy and norm of the wave function presented by explicit symplectic algorithms will not be preserved exactly, they will converge to the true energy and norm of the wave function of the system as the time step reduces. The time dependent Schr¨odinger equation (TDSE) in one dimensional space by the action of some strong field V (t, x) is i

∂ψ ˆ = Hψ, ∂t

ˆ =H ˆ 0 (x) + εVˆ (t, x), H 2

ˆ 0 = − 1 ∂ + V0 (x), H 2 ∂x2  0, 0 < x < 1, V0 (x) = ∞, x ≤ 0 or x ≥ 1. ⎧ ⎪ 0 < x < 0.5, ⎨ 2x, V (x) = 2x − 2x, 0.5 ≤ x ≤ 1, ⎪ ⎩ 0, x ≤ 0 or x ≥ 1. Using the similar method √ as before, we expand the wave function as the characterˆ 0 to discretize the TDSE. istic functions {Xn (x) = 2 sin nπx, n = 1, 2, · · ·} of H Because the Hamiltonian operator is real, the discrete TDSE is a separable linear canonical Hamiltonian system with the parameters as follows. S = (Smn ),

⎧ 1 − (−1)n 1 ⎪ ⎪ + , ⎪ ⎪ 2 n2 π 2 ⎪ ⎪ ⎪ ⎪ ⎨ 0,

vmn

Smn =

n2 π 2 δmn + εvmn , 2

m = n, |m − n| = 1, 3, 5, · · · , m−n 2

−16mn(1 − (−1) = ⎪ ⎪ (m2 − n2 )2 π 2 ⎪ ⎪

)

, |m − n| = 2, 4, 6, · · · , n = 2, 4, 6, · · · , m−n ⎪ ⎪ ⎪ −8|2mn − (−1) 2 (m2 − n2 )| ⎪ ⎩ , |m − n| = 2, 4, 6, · · · , n = 1, 3, 5, · · · . 2 2 2 2 (m − n ) π

The initial state is taken as ψ(0, x) =

1+i |X1 (x) + X2 (x)|, 2

ε = 5π 2 .

The energy of the system is conserved because the Hamiltonian does not depend on the time explicitly. E(b, a) = e0 = 42.0110165. The norm of wave function keeps unitary, i.e., N (b, a) = n0 = 1. We take the Euler midpoint rule, order 2 explicit symplectic algorithm and the order 2 R–K method to compute the problem with the same time step h = 10−3 . The numerical results are as follows:

Introduction

23

1◦ The R–K method can not preserve the energy and the norm of wave function, as evident by ER–K in Fig. 0.11(left) and NR–K in Fig. 0.11(right). 2◦ The Euler midpoint rule can preserve the energy and norm, as evident by EE in Fig. 0.11(left) and NE in Fig. 0.11(right). Note that for EE in Fig. 0.11(left), there is a very small increase at some time because of the implicity of the Euler scheme.

Fig. 0.11.

Energy [left] and norm [right] comparison among the 3 difference schemes

˜ k , ak ; h) 3◦ The explicit symplectic algorithms can preserve exactly the energy E(b k k   ˜ and norm N (b , a ; h), as evident by ES in Fig. 0.11(left) and NS in Fig. 0.11(right). If we want to get further insight into these conservation laws within smaller scales, we find that as the time steps get smaller, the numerical energy of symplectic algorithm converges to the true energy of the system e0 = 42.0110165 and the numerical norm converges to unit n0 = 1. See Table 0.3 showing the numerical energy and norm as well as their errors. The errors are defined as CE (h) = max |ESk − e0 |, k

CN (h) = max |NSk − n0 |. k

Actually, the numerical energy and norm obtained by symplectic algorithm oscillate slightly, as shown by ES and NS in Fig. 0.12. However, the amplitude of their oscillations will converge to zero, if the time step tends to zero. As the time step tends to zero, we have e(h) −→ e0 ,

CE (h) = maxk |ESk − e0 | −→ 0,

n(h) −→ n0 ,

CN (h) = maxk |NSk − n0 | −→ 0.

24

Introduction

Table 0.3.

The change of energy and norm of the wave function with the step size h

e(h)

CS (h)

n(h)

CN (h)

42.0169964

0.0445060

0.9996509

0.0003106

42.0110763

0.0004195

0.9999965

0.0000030

42.0110171

0.0000018

0.9999990

0.0000000

10

42.0110165

0.0000000

1.0000000

0.0000000

10−7

42.0110165

0.0000000

1.0000000

0.0000000

exact value

42.0110165

0.0000000

1.0000000

0.0000000

−3

10

−4

10

−5

10

−6

Fig. 0.12.

Energy E and norm N obtained from explicit symplectic scheme

In all, for a quantum system with real Hamiltonian function independent of time explicitly, the explicit symplectic algorithms can preserve the energy and norm of the wave function to any given accuracy. They overcome the main disadvantages of the traditional numerical methods. Next, we look at the quantum system with real Hamiltonian function, which is dependent on time explicitly. In this case, the resulting system after semi-discretization is an m-dimensional, separable, linear, Hamiltonian canonical system. The energy of the system is not conserved any more, but the norm of the wave function is still a quadratic conservation law. The TDSE for an atom in one dimensional space with the action of some strong field V (t, x) = εx sin(ωt) is i

∂ψ ˆ = Hψ, ∂t

ˆ =H ˆ 0 (x) + εVˆ (t, x), H

2 ˆ 0 = − 1 ∂ + V0 (x). H 2

2∂x

By the similar method as before, we expand the wave function as the characteristic √ ˆ 0 to discretize the TDSE. Because functions Xn x = 2 sin nπx(n = 1, 2, · · ·) of H

Introduction

Fig. 0.13.

25

ω = 3π 2 /2, ε = π 2 /2: Graph of norm[left]; graph of probability[right]

the Hamiltonian operator is real, the discrete TDSE is a separable linear canonical Hamiltonian system with the parameters as follows. n2 π 2

δm,n + εv(t)mn ; S(t) = (s(t)mn ), s(t)mn = 2 ⎧ sin(ωt), m = n, ⎪ ⎪ ⎪ ⎨ 0, |m − n| = 2, 4, 6, · · · , v(t)mn = ⎪ ⎪ ⎪ ⎩ 8mn sin (ω t) , |m − n| = 1, 3, 5, · · · . (m2 − n2 )2 π 2

√ The initial state is taken as ψ(0, x) = X1 (x) = 2 sin(πx). The energy of the system is not conserved in this case because the Hamiltonian depends on the time explicitly. The norm of wave function remains unitary, i.e., N (b, a) = n0 = 1. We take the Euler midpoint rule scheme, order 2 explicit symplectic algorithm and the order 2 R–K method to compute the problem with the same time step h = 4 × 10−3 . The numerical results are as follows: 1◦ The R–K method increases the norm of wave function rapidly, see NR–K in Fig. 0.13(left). It leads to unreasonable results, see in Fig. 0.13(right). 2◦ The Euler midpoint rule scheme can preserve the norm, see NE in Fig.0.13(left). These results are in good agreement with the theoretical results. See Fig. 0.13(right) π for the results for weak fields ε = . When ω = ΔE1n , i.e., resonance occurs, the 2 basic state and the first inspired state will intermix and the variation period of the energy is identical to the period of intermixing. See the corresponding results in Fig. 0.14(left) and Fig. 0.14(right). When ω = ΔE1n there will not be intermixing. See the corresponding numerical results in Fig. 0.15(left) and Fig. 0.15(right), where O is the basic state. When the field is strong, the selection rule is untenable, and no resonance occurs, but the basic state will intermix with the first, second, . . . inspired states. See the results for ω =

5π 2 3π 2 in Fig. 0.16(left) and Fig. 0.16(right) and ω = = ΔE12 4 2

in Fig. 0.17(left) and Fig. 0.17(right).

26

Introduction

Fig. 0.14.

ω = 3π 2 /2, ε = π 2 /2: Graph of probability[left]; graph of norm[right]

Fig. 0.15.

ω = 5π 2 /4, ε = 3π 2 /2: Graph of probability[left]; graph of norm[right]

3◦ The order 2 explicit symplectic algorithms can not preserve the norm exactly. The numerical norms oscillate near the unit. See NS in Fig. 0.13, where changes of numerical energy and states of intermixing obtained by symplectic algorithms are similar to the results of Euler midpoint rule scheme. We can conclude that for this system the R–K method can not preserve the norm of wave function and its results are unreasonable; the Euler scheme can preserve the norm and its results are in agreement with the theoretical results; the second order scheme obtains the numerical norm which oscillates near the unit and its energy and states of intermixing are the same as for the results of Euler scheme. Thus, the Euler scheme (an implicit symplectic scheme) and the second order explicit symplectic algorithm are good choices for studying the quantum system with the Hamiltonian dependent on time explicitly. They overcome the drawbacks of the traditional R–K methods.

(5)

Applications to computation of classical trajectories

Applications of symplectic algorithms to computation of classical trajectories of A2 B molecular reacting system [LDJW00] . To study the classical or semi-classical trajectories of the dynamical system, microscopic chemistry is an effective theory method.

Introduction

Fig. 0.16.

ω = 5π 2 /4, ε = 50π 2 : Graph of probability[left]; graph of norm[right]

Fig. 0.17.

ω = 3π 2 /2, ε = 50π 2 : Graph of probability[left]; graph of norm[right]

27

The classical trajectory method regards the atom approximatively as a point and the system as a system of some points, and advances the process of action as the classical motions of point system in potential energy plane of the electrons. It was Bunker who first applied the R–K method to computations of classical trajectory of molecular reacting system. Karplus et al. did a large number of computations by all kinds of numerical methods and screened out the R–K–G (Runge–Kutta–Gear) method to prolong the computation time from 10−15 s to 10−12 s. The R–K–G method made rapid progress in the theoretical study of reacting dynamics of microscopic chemistry and was widely used for computation of classical trajectory. However, its valid computation time is much less than 10−8 s which is necessary time for study of chemical reactions. Moreover, there were many differences between the numerical quantities and theoretical quantities of some parameters. The classical trajectory method describes the microscopic reaction system approximately as a Hamiltonian system which naturally has symplectic structure. Thus, it is expected that the symplectic algorithms will overcome the shortages of the R–K–G method and improve the numerical results. Here we take the mass of the proton as the unit mass and 4.45 × 10−14 s as unit time. Consider the classical motions of the A2 B type molecules like H2 O and SO2 moving in the electron potential energy plane of the reaction system and preserving the

28

Introduction

symmetry of C2v . Set the masses of A and B to be mA = 1 and mB = 2 resp., the center of mass of the molecule be the origin of some coordinate, the C2 axes be z axes, and the coordinates of two atoms A and the atom B be (y1 , z1 ), (y2 , z2 ) and (y3 , z3 ) reps. in the fixed coordinate system. By Banerjee’s coordinates separating method, we can get the generalized coordinates of the A2 B molecule as q1 = z1 + z2 − 2z3 ,

q2 = y 2 − y 1 ,

and the generalized mass as M1 = 0.25, M2 = 0.5, further the generalized momentum as d q1 d q2 p1 = 0.25 , p2 = 0.5 , dt dt and the kinetic energy of system as K(p) = 2p21 + p22 . The potential energy suggested by Banerjee, who introduced the symmetry C2v  and notation D = q12 + q22 , was V (q) = 5π 2 (D2 − 5D + 6.5) + 4D−1 +0.5π 2 (|q2 | − 1.5)2 + |q2 |−1 . The Hamiltonian function for the A2 B molecular system is H(p, q) = K(p) + V (q), and the canonical equations for the classical trajectories are d p1 ∂V =− = −f1 (q), dt ∂t d p2 ∂V =− = −f2 (q), dt ∂ q2

d q1 ∂K = = g1 (p), dt ∂ p1 d q2 ∂K = = g2 (p). dt ∂ p2

It is a separated Hamiltonian system, which can be integrated by explicit symplectic algorithms. We can obtain its numerical solutions of some initial values as tk = kh,

pk1 = p1 (tk ),

q1k = q1 (tk ),

pk2 = p2 (tk ),

q2k = q2 (tk ),

and further its classical trajectories of A2 B system and the changes of kinetic energy, potential energy and total energy with time by following relations: y3 = 0,

z3 = −

q1 ; 4

y2 = −y1 =

q2 , 2

z2 = z1 =

The initial values are taken as q1 (0) = 3,

3 2

q2 (0) = ;

p1 = 0,

p2 = 0.

q1 . 4

Introduction

Fig. 0.18.

29

The potential energy curve of the electronic potential function in phase space

We compute this system with order 4 explicit symplectic algorithm and R–K method. The time step is taken as h = 0.01 for both. The numerical classical trajectories, kinetic energy, potential energy and total energy are recorded. Fig. 0.18 shows the potential energy curve of the electronic potential function in phase space. If |q1 | → +∞, then V (q) → +∞; if |q2 | → 0 or |q2 | → +∞, then V (q) → +∞. By the theoretical analysis, we know that the total energy of the system will be conserved all the time, the three atoms will oscillate nearly periodically, and the whole geometry structure of the system may be reversed but kept periodic. The changes of the total energy with time are shown in Fig.0.19, where we can see that the total energy obtained by symplectic algorithms are preserved up to 6.23 × 10−9 s, whereas the R–K method reduces them rapidly with time. The motion trajectories of the system in the plane by the symplectic algorithms and R–K method are shown in Fig. 0.20 (a), (c), (e) and (b), (d), (f) resp., where we can see that the numerical results of symplectic algorithms are coincident with the theoretical results but the results of R–K method are not. We also applied the order 1 and 2 symplectic algorithms, the Euler method and the revised Euler method to compute the same problem. The conclusions are almost the same. Because all the traditional methods such as R–K methods, Adams methods and Euler methods can not preserve the symplectic structure of this microscopic system, they will bring false dissipations inevitably, which will make their numerical results meaningless after long-term computations. On the contrary, symplectic algorithms can preserve the structure and do not bring any false dissipations. Therefore, they are suitable for long-term computations and greatly improve the classical trajectory methods for studying the microscopic dynamical reactions of chemical systems.

30

Introduction

Fig. 0.19.

The changes of the total energy with the time

(6) Applications to computation of classical trajectories of diatomic system [Dea94,DLea96] Consider the classical motion of AB diatomic molecule system in electron potential energy plane. Set the masses of A and B to be m1 and m2 resp., the center of mass to be the origin of some coordinate with fixed axes Ox, the coordinates of two atoms A and B to be −x1 and x2 resp. Then the generalized coordinate is q = x2 + x1 and the m1 m2 dq . Further, the generalized momentum is p = M m1 + m2 dt p2 and the generalized kinetic energy is U (p) = . Take the potential function as the 2M

generalize mass is M =

Morse potential

V (q) = D{e−2a(q−qe ) − 2e−a(q−qe ) }, where the parameters D, a, qe were derived by E. Ley and Koo recently. Thus, the total energy for such system is H(p, q) = U (p) + V (q), and the canonical Hamiltonian system for the classical trajectory is d V (q) dp =− = −f (q), dt dt d U (p) dq = = g(p). dt dt

It is a separable system. By explicit symplectic algorithms, we can get its numerical solutions as tk = kh, pk = p(tk ), q k = q(tk ), and advance its classical trajectories of AB two-atom system as x1 =

m2 q , m1 + m2

x2 =

m1 q , m1 + m2

Introduction

31

Fig. 0.20. The motion trajectories of the system in the plane,(a) and (b) period range from 4.45 × 10−10 s to (4.45 × 10−10 + 4.45 × 10−13 )s. (c) and (d) period range from 6.23 × 10−9 s to (6.23 × 10−9 + 4.45 × 10−13 )s. (e) and (f) period range from 6.23 × 10−9 s to (6.23 × 10−9 + 4.45 × 10−13 )s. (a), (c), (e) is the symplectic algorithm path, (b), (d), (f) is the R–K method path

as well as the changes of kinetic energy, potential energy and total energy with the variation of time. We compute some states of two homonuclear molecules Li2 and N2 and two heteronuclear molecules CO and CN by using the order 1, 2 and 4 explicit symplectic algorithms and compare the numerical results of total energy and classical trajectories with the Euler method and order 2 and 4 order R–K methods. In Fig. 0.21, Fig. 0.22 and Fig. 0.23, we show the numerical results of the classical trajectories, total energy and the trajectories in p − q phase space obtained by order 4 explicit symplectic algorithm and order 4 R–K method respectively. The parame-

32

Introduction

Fig. 0.21.

Classical orbit of two homonuclear molecules Li2

Fig. 0.22.

Comparison of energy of two homonuclear molecules Li2

ters in those computations are taken as the time step h = 0.005, the initial values √ ˚ a= q(0) = qe , p(0) = 2M D − 0.0001, and D = 8541cm−1 , qe = 2.67328A, −1 ˚ = 0.1nm. The results show that the symplectic algorithms can preserve ˚ ,A 0.867A the energy after 106 time steps and the facts that the two Li atoms oscillate periodically and their trajectories in phase remain invariant are simulated by the symplectic algorithm. The results are opposite for the R–K method. The numerical total energy, and the oscillation period and amplitude of the two atoms were reduced, after 3000 time steps. Furthermore, the trajectories in the phase space became flat to q axis after 50000 time steps and lost entirely their shape as manifested in the theory analysis and experiments (Fig. 0.21, Fig. 0.22, Fig. 0.23). The results of the other molecules N2 , CO and CN are similar. Thus, we can draw the conclusion that the symplectic algorithms can preserve the symplectic structure and the basic properties of the microscopic system. Therefore they are capable of long time computations for such systems.

Introduction

Fig. 0.23.

(7)

33

The trajectories in p − q phase space

Applications to atmospheric and geophysical science

Recently, the symplectic algorithms have been applied to study the observation operator of the global positioning system (GPS) by Institute of Atmospheric Physics of the Chinese Academy of Science[WZJ95,WJX01] . Numerical weather forecasting needs very large amount of atmospheric information from GPS. One of the key problems in this field is how to reduce largely the computational costs and to compute it accurately for a long time. The symplectic algorithms provide rapid and accurate numerical algorithms for them to deal with the information of GPS efficiently. The computational costs of the symplectic algorithms are one four hundredth of the costs of traditional algorithms. For the complicated nonlinear system of atmosphere and ocean, symplectic algorithms can preserve its total energy, total mass, total potential so well that the relative errors of potential height is below 0.0006 (see Fig. 0.24). Another application of symplectic algorithms to geophysics is carried out by Institute of Geophysics to prospect for the oil and natural gas[GLCY00,LLL01a,LLL01b,LLL99] , which has obtained several great achievements. For example, the spread waves of earthquake under the framework of Hamiltonian system and the corresponding symplectic algorithms have been investigated. Moreover, “the information of oil reserves and geophysics and its process system ” has been produced, and the task of prospecting for 1010 m3 of natural gas, which has obtained. Fig. 0.25 shows the numerical results of prestack depth migration in the area of Daqing Xujiaweizi by applying symplectic algorithms to Marmousi model. Recently, Liuhong et.al. proposed a new method[LYC06] to calculate the depth extrapolation operator via exponential of pseudo-differential operator in lateral varied medium. The method offers the phase of depth extrapolation operator by introducing lateral differential to velocity, which in fact is an application of Lie group method.

34

Introduction

Fig. 0.24.

The relative errors of potential height is below 0.0006 after 66.5 days

Fig. 0.25. Numerical results of prestack depth migration in the area of Daqing Xujiaweizi obtained by applying symplectic algorithms to Marmousi model

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Chapter 1. Preliminaries of Differentiable Manifolds

Before introducing the concept of differentiable manifold, we first explain what mapping is. Given two sets X, Y, and a corresponding principle, if for any x ∈ X, there exists y = f (x) ∈ Y to be its correspondence, then f is a mapping of the set X into the set Y , which is denoted as f : X → Y. X is said to be the domain of definition of f , and f (x) = {f (x) | x ∈ X} ⊂ Y is said to be the image of f . If f (X) = Y , then f is said to be surjective or onto; if f (x) = f (x ) ⇒ x = x , then f is said to be injective (one-to-one); if f is both surjective and injective (i.e., X and Y have a one-to-one correspondence under f ), f is said to be bijective. For a bijective mapping f , if we define x = f −1 (y), then f −1 : Y → X is said to be the inverse mapping of f . In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). For example, for two groups G and G and a mapping f : G → G , a → f (a), if f (a, b) = f (a) · f (b), ∀a, b ∈ G, then f is said to be a homomorphism from G to G . A homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structures, i.e., properties such as identity element, inverse element, and binary operations. An isomorphism is a bijective homomorphism. If f is a G → G homomorphic mapping, and also a one-to-one mapping from G to G , then f is said to be a G → G isomorphic mapping. An epimorphism is a surjective homomorphism. Given two topological spaces (x, τ ) and (y, τ ), if the mapping f : X → Y is one-to-one, and both f and its inverse mapping f −1 : Y → X are continuous, then f is said to be a homeomorphism. If f and f −1 are also differentiable, then the mapping is said to be diffeomorphism. A monomorphism (sometimes called an extension) is an injective homomorphism. A homomorphism from an object to itself is said to be an endomorphism. An endomorphism that is also an isomorphism is said to be an automorphism. Given two manifolds M and N , a bijective mapping f from M to N is called a diffeomorphism if both f : M → N and its inverse f −1 : N → M are differentiable (if these functions are r times continuously differentiable, f is said to be a C r -diffeomorphism). Many differential mathematical methods and concepts are used in classical mechanics and modern physics: differential equations, phase flow, smooth mapping, manifold, Lie group and Lie algebra, and symplectic geometry. If one would like to construct a new numerical method, one needs to understand these basic theories and concepts. In this book, we briefly explain manifold, symplectic algebra, and symplectic geometry. In a series of books[AM78,Che53,Arn89,LM87,Ber00,Wes81] can be found these materials.

40

1. Preliminaries of Differentiable Manifolds

1.1 Differentiable Manifolds The concept of manifold is an extension of Euclidean space. Roughly speaking, a manifold is an abstract mathematical space where every point has a neighborhood that resembles Euclidean space (homeomorphism). Differentiable manifold is one of the manifolds that can have differentiable structures.

1.1.1 Differentiable Manifolds and Differentiable Mapping Definition 1.1. A Hausdorff space M with countable bases is called an n-dimensional topological manifold, if for any point in M there exists an open neighborhood homeomorphic to an open subset of Rn . Remark 1.2. Let (U, ϕ), (V, ψ) be two local coordinate systems (usually called chart) on the topological manifold M . (U, ϕ), (V, ψ) are said to be compatible, if U ∩ V = Ø, or the change of coordinates ϕ ◦ ψ −1 and ψ ◦ ϕ−1 are smooth when U ∩ V = Ø. Definition 1.3. A chart is a domain U ⊂ Rn together with a 1 to 1 mapping ϕ : W → U of a subset W of the manifold M onto U . ϕ(x) is said to be the image of the point x ∈ W ⊂ M on the chart U . Definition 1.4. A collection of charts ϕi : Wi → Ui is an atlas on M if 1◦ Any two charts are compatible. 2◦ Any point x ∈ M has an image on at least one chart. Remark 1.5. If a smooth atlas on a topological manifold M possesses with its all compatible local coordinate systems (chart), then this smooth atlas is called the maximum atlas. Definition 1.6. If an n-dimensional topological manifold M is equipped with the maximal smooth atlas A, then (M, A) is called the n-dimensional differentiable manifold, and A is called the differentiable structure on M . Definition 1.7. Two atlases on M are equivalent if their union is also an atlas (i.e., if any chart of the first atlas is compatible with any chart of the second). Remark 1.8. Suppose M is the n-dimensional topological manifold, A = {(Uλ , ϕλ )} is a smooth atlas on M . Then there exists a unique differentiable structure A∗ , which contains A. Hence, a smooth atlas determines a unique differentiable structure on M . The local coordinate system will be called (coordinate) chart subsequently. Definition 1.9. A differentiable manifold structure on M is a class of equivalent atlases. Definition 1.10. A differentiable manifold M is a set M together with a differentiable manifold structure on it. A differentiable manifold structure is induced on set M if an atlas consisting of compatible charts is prescribed.

1.1 Differentiable Manifolds

41

Below are examples of differentiable manifold. Example 1.11. Rn is an n-dimensional differentiable manifold. Let A ={(Rn , I)}, where I is the identity mapping. Example 1.12. S n is an n-dimensional differentiable manifold. We only discuss the n = 1 case. Let U1 = {(u1 , u2 ) ∈ S 1 |u1 > 0},

U2 = {(u1 , u2 ) ∈ S 1 |u1 < 0},

U3 = {(u1 , u2 ) ∈ S 1 |u2 > 0},

U4 = {(u1 , u2 ) ∈ S 1 |u2 < 0}.

Define ϕi : Ui → (−1, 1), such that (s.t.) ϕi (u1 , u2 ) = u2 ,

i = 1, 2;

ϕi (u1 , u2 ) = u1 ,

i = 3, 4.

Note that on ϕ1 (U1 ∩ U3 )   2 2 2 2 1 − (u2 )2 ϕ3 ◦ ϕ−1 1 : u −→ ( 1 − (u ) , u ) −→ is smooth, then A ={(Uk , ϕk )} is a smooth atlas on S 1 . Example 1.13. RP n is an n-dimensional differentiable manifold. Let Uk = {[(u1 , · · · , un+1 )] | (u1 , · · · , un+1 ) ∈ S n , uk = 0},

k = 1, · · · , n + 1

defines ϕk : Uk → Int B n (1), s.t. ϕk ([(u1 , · · · , un+1 )]) = uk |uk |−1 (u1 , · · · , uk−1 , uk+1 , · · · , un+1 ), n    1 n n where B (1) = (u , · · · , u ) ∈ R  (ui )2 ≤ 1 . It is easy to prove that A n

={(Uk , ϕk )} is a smooth atlas on RP n .

i=1

Example 1.14. Let M, N be m- and n-dimensional differentiable manifolds, respectively, then M ×N is a m+n dimensional differentiable manifold (product manifold). Suppose A = {(Uα , ϕα )}, B = {(Vα , ψα )} are smooth atlases on M, N respectively. Denote A × B={(Uα × Vλ, ϕα × ψλ )}, where ϕα × ψλ : Uα × Vλ → ϕα (Uα ) × ψλ (Vλ ),(ϕα × ψλ )(p, q) = ϕα (p), ψλ (q) , (p, q) ∈ Uα × Vλ , then A × B is a smooth atlas on M × N . Definition 1.15. Let M, N be m- and n-dimensional differentiable manifolds, respectively. A continuous mapping f : M → N is called C k differentiable at p ∈ M , if the local representation f = ψ ◦ f ◦ ϕ−1 : ϕ(U ) → ψ(V ) is C k differentiable for the charts (U, ϕ), (V, ψ) corresponding to points p and f (p), and f (U ) ⊂ V . If f is C k differentiable in each p ∈ M , then f is called C k differentiable, or called C k mapping. See Fig. 1.1.

42

1. Preliminaries of Differentiable Manifolds

Fig. 1.1.

A differentiable mapping

Example 1.16. Let M1 , M2 be m- and n-dimensional differentiable manifolds, respectively. Define θ1 : M1 × M2 → M1 , θ2 : M1 × M2 → M2 , such that θ1 (p, q) = p,

θ2 (p, q) = q,

∀ (p, q) ∈ M1 × M2 ,

then θ1 , θ2 are all smooth mappings. If the charts on M1 , M2 are denoted by (U, ϕ), (V, ψ), then it is easy to show that (U × V, ϕ × ψ) is the chart on M1 × M2 . Thus, the local coordinate expression of θ1 , θ1 = ϕ ◦ θ1 ◦ (ϕ × ψ)−1 : (ϕ × ψ)(U × V ) −→ ϕ(U ),

θ1 (u, v) = u

is a smooth mapping. Therefore, θ1 is a smooth mapping. Likewise, θ2 is also a smooth mapping. Example 1.17. Let M, N1 , N2 be differentiable manifolds. f1 : M −→ N1 ,

f2 : M −→ N2

k

are C -mapping. Define f : M −→ N1 × N2 ,

f (p) = (f1 (p), f2 (p)),

∀ p ∈ M,

then f is a C k -mapping. ∀ p0 ∈ M , let N1 contain the chart (V, ψ) of f1 (p0 ), and let N2 contain the chart (W, χ) of f2 (p0 ), and M contain the chart (U, ϕ) of p0 . Assume f1 (U ) ⊂ V, f2 (U ) ⊂ W , and f1 : ψ ◦ f1 ◦ ϕ−1 : ϕ(U ) −→ ψ(V ), f2 : χ ◦ f2 ◦ ϕ−1 : ϕ(U ) −→ χ(W ) are all C k -mapping, and (V × W, ϕ × χ) is a chart that contains (f1 (p0 ), f2 (p0 )) = f (p0 ) on the product manifold N1 × N2 , which satisfies f (U ) ⊂ V × W . Then we have f = (ψ × χ) ◦ f ◦ ϕ−1 : ϕ(U ) −→ (ψ × χ)(U × W ), i.e., f is C k -mapping.

f = (f1 , f2 ),

1.1 Differentiable Manifolds

43

Remark 1.18. According to the definition, if f : M → N, g : N → L are C k mappings, then g ◦ f : M → L is also a C k -mapping. Definition 1.19. Let M, N be differentiable manifolds, f : M → N is a homeomorphism. If f, f −1 are smooth, then f is called diffeomorphism from M to N . If there exists a diffeomorphism between differentiable manifolds M and N , then M and N are called differentiable manifolds under diffeomorphism, denoted by M  N . If we define two smooth atlases (R, I), (R, ϕ) on R, and ϕ : R → R, ϕ(u) = u3 , √ −1 because the change of coordinates I ◦ ϕ (u) = 3 u in u = 0 is not differentiable, then (R, I) and (R, ϕ) determine two different differentiable structures A, A on R. However, if we define f : (R, A) → (R, A ), {f (u)} = u3 , then (R, A)  (R, A ). In fact, there exist examples that are not homeomorphism in a differentiable manifold, like the famous Milnor exotic sphere.

1.1.2 Tangent Space and Differentials In order to establish the differential concept for differentiable mapping on a differentiable manifold, we first need to extend the concept of tangent of curve and tangent plane of surface in Euclidean space. If we take the tangent vector in Euclidean space not simply as a vector with size and direction, but as a linear mapping, which satisfies the Leibniz rule, from the differentiable functional space to R, then the definition of tangent vector can be given similarly for a manifold. Let M be the m-dimensional differentiable manifold, p ∈ M be a fixed point. Let C ∞ (p) be the set of all smooth functions that are defined in some neighborhood of p. Define operations on M that have the following properties: (f + g)(p) = f (p) + g(p), (αf )(p) = αf (p), (f g)(p) = f (p)g(p). Definition 1.20. A tangent vector Xp at the point p ∈ M is a mapping Xp : C ∞ −→ R, that has the following properties: 1◦ Xp (f ) = Xp (g), if f, g ∈ C ∞ (p) are consistent in some neighborhood of the point p. 2◦ Xp (αf + βg) = αXp (f ) + βXp (g), ∀ f, g ∈ C ∞ (p), ∀ α, β ∈ R. 3◦ Xp (f g) = f (p)Xp (g) + g(p)Xp (f ), ∀ f, g ∈ C ∞ (p) (which is equivalent to the derivative operation in Leibniz rule). Denote Tp M ={All tangent vectors at the point p ∈ M } and define operation: (Xp + Yp )(f ) = Xp (f ) + Yp (f ), (kXp )(f ) = kXp (f ), ∀ f ∈ C ∞ (p).

44

1. Preliminaries of Differentiable Manifolds

It is easy to verify that Tp M becomes the vector space that contains the above operation, which is called the tangent space at the point p of the differential manifold M. Remark 1.21. By definition of the tangent vector, it is easy to know that if f is the constant function, Xp (f ) = 0 for Xp ∈ Tp M . Lemma 1.22. Let (U, ϕ) be the chart that contains p ∈ M , and let x1 , · · · , xm , ϕ(p) = (a1 , · · · , am ) be the coordinate functions. If f ∈ C ∞ (p), then there exists a function gi in some neighborhood W of p ∈ M , such that f (q) = f (p) +

m 

(xi (q) − ai )gi (q),

∀ q ∈ W,

i=1

 

and gi (p) =









∂f  ∂f  ∂  ∂f ◦ ϕ−1  .  where i  = i  (f ) =  i ∂x p ∂x p ∂x p ∂ui ϕ(p)

Proof. Assume ϕ(p) = O ∈ Rm , and f is well defined in some neighborhood of p. Let W = ϕ−1 (B m ). Then ∀ q ∈ W and we have f (q) − f (p) = f ◦ ϕ−1 (u) − f ◦ ϕ−1 (O). After calculation, we obtain f (q) − f (p) =

m 

ui g i (u),

i=1

where g i (u) =

 1 ∂f ◦ ϕ−1 0

∂ui

(su1 , · · · , sum ) d s (i = 1, · · · , m). Let g i (ϕ(q)) =

gi (q), then gi is smooth on W , and satisfies m 

xi (q)gi (q), i=1   ∂f ◦ ϕ−1  ∂f  gi (p) = g i (O) =  = i . i

f (q) = f (p) +

∂u

O

∂x

p



Hence lemma is proved.

   ∂  ∂  ∂f ◦ ϕ−1  ∞ Theorem 1.23. Define : C (p) → R, (f ) = , ∀f ∈    ∂xi p ∂xi p ∂ui ϕ(p)  ∂  C ∞ (p), then i  (i = 1, · · · , m) is a group of bases for Tp M . Therefore, dim Tp M ∂x

p

= m, and for Xp ∈ Tp M, we have Xp =

m  i=1

Xp (xi )



∂   . ∂xi p

1.1 Differentiable Manifolds

45

Proof. ∀ Xp ∈ Tp M, as f ∈ C ∞ (p). By Lemma 1.22, we know f = f (p) +

m 

(xi − ai )gi ,

i=1

then m  ∂f  ∂  Xp (f ) = Xp [(x − a )gi ] = Xp (x ) i  = Xp (xi ) i  (f ). ∂x p i=1 ∂x p i=1 i=1 m 

i

m 

i

i

i The  decomposed coefficients, {Xp (x )}, of Xp with respect to (w.r.t.) the bases

∂   (i = 1, · · · , m) are called coordinates of the tangent vector Xp w.r.t. the ∂xi p



chart(U, ϕ).

Remark 1.24. By Theorem 1.23 we know: if the coordinates of Xp w.r.t. chart (U, ϕ) are defined as (Xp (x1 ), · · ·, Xp (xm )), then Tp M and Rm are isomorphisms,  and the  ∂ m basis for Tp M corresponds exactly to the standard basis for R , i.e.,  → ei = ∂xi p (0, · · · , 1, 0, · · · , 0). 1. Definition and properties of differentials of mappings The definition of differentials of a mapping is as follows: Definition 1.25. Let f : M → N be a smooth mapping. ∀ p ∈ M, Xp ∈ Tp M, we define f∗p : Tp M → Tf (p) N that satisfies: f∗p (Xp )(g) = Xp (g ◦ f ),

∀ g ∈ C ∞ (f (p)).

This linear mapping f∗p is called the differential of f at the p ∈ M . Definition 1.26. The differential of the identity mapping I is an identity mapping, i.e., I∗p : Tp M → Tp M . Remark 1.27. Let M, N, L be differentiable manifolds, p ∈ M , and f : M → N, g : N → L are smooth mappings, then (g ◦ f )∗p = g∗f (p) ◦ f∗p . Remark 1.28. If f : M → N is a diffeomorphism, then f∗p : Tp M → Tf (p) N is a isomorphism. Proposition 1.29. Let x1 , · · · , xm , y 1 , · · · , y n be the coordinate functions of (U, ϕ), (V, ψ) respectively, then  f∗p where fj = y j ◦ f .

 

∂   ∂xi p

=

n   ∂fj   i j=1



∂  ,  ∂x p ∂y j f (p)

46

1. Preliminaries of Differentiable Manifolds

Proof. Since

 f∗p

 





∂  ∂  ∂fk   (y k ) = i  (y k ◦ f ) = i  , ∂xi p ∂x p ∂x p

therefore, by Theorem 1.23 we have  f∗p

 

∂   ∂xi p

n    ∂fj  ∂y k    i j

=

i,j=1

∂x

p ∂y

n   ∂fj   i

=

i,j=1

f (p)



∂  (y k ).  ∂x p ∂y j f (p)



Therefore the proposition is completed. Let Xp =

n 

αi

i=1

have

n    ∂  j ∂  , f (X ) = β , by Proposition 1.29, we   ∗p p ∂xi p ∂y j f (p) j=1

⎛

∂f1 ⎞ β1 ⎜ ∂x1 ⎜ .. ⎟ ⎜ .. ⎝ . ⎠=⎜ . ⎝ ∂fn βn

⎛

 This matrix

∂fi ∂xj

∂x1

 n×m

···

⎞

∂f1 ⎛ 1 ⎞ α ∂xm ⎟

.. .

···

∂fn ∂xm

⎟⎜ ⎟⎝ ⎠

.. ⎟ . . ⎠ αm

is the Jacobian matrix of f at p w.r.t. charts (U, ϕ), (V, ψ).

Its rank rkp f is called the rank of f : M → N at the p. From the above equations, we can easily observe that f∗p is equivalent to Df(ϕ(p)) under the assumption of isomorphism, where Df(ϕ(p)) is the differential at ϕ(p) of the local representation of f , f = ψ ◦ f ◦ ϕ−1 . 2. Geometrical meaning of differential of mappings A smooth on M is a smooth mapping c : (a, b) → M . The tangent vector,   curve  d  c∗t0 on Tc(t0 ) M is called the velocity vector of c at t0 . Let f : M → N be a  dt

t0

smooth mapping. Then, f ◦ c is a smooth curve on N that passes f (p). By composite differentiation, we have d     d   = f∗p0 ◦ c∗t0 , (f ◦ c)∗t0   d t t=t0 d t t=t0 i.e., f∗p0 transforms the velocity vector of c at t0 to the velocity vector of f ◦ c at t0 .

1.1.3 Submanifolds The extension of the curve and surface on Euclidean space to the differentiable manifold is the submanifold. In the following section, we focus on the definitions of three submanifolds and their relationship. First, we describe a theorem.

1.1 Differentiable Manifolds

1.

47

Inverse function theorem

Theorem 1.30. Let M, N be m-dimensional differentiable manifolds, f : M → N is a smooth mapping, p ∈ M . If f∗p : Tp M → Tf (p) N is an isomorphism, then there exists a neighborhood, W of p ∈ M , such that 1◦ f (W ) is a neighborhood of f (p) in N. 2◦ f |W : W → f (W ) is a diffeomorphism (this theorem is an extension of the inverse function theorem for a manifold). Proof. Consider charts (U, ϕ) on M about p ∈ M and (V, ψ) on N about f (p) ∈ N , so that f (U ) ⊂ V . Then, the local representation f = ψ ◦ f ◦ ϕ−1 : ϕ(U ) → ψ(V ) is a smooth mapping. Since f∗p : Tp M → Tf (p) N is an isomorphism, Dfˆ(ϕ(p)) : Rm → Rm is also an isomorphism. By the inverse function theorem, there exists a neighborhood O of ϕ(p) ∈ Rm such that f(O) is a neighborhood of ψ(f (p)) in Rm , and f : O → f(O) is a diffeomorphism. O has to be chosen appropriately. Let O ⊂ ϕ(U ), and f(O) ⊂ ψ(V ). Let W = ϕ−1 (O). Then, W is the neighborhood of p, which meets our requirement.  Remark 1.31. Given a chart (V, ψ) on N , f (p) ⊂ V , choose ϕ = ψ ◦ f and some neighborhood U of p, such that ϕ(U ) ⊂ V . By 2◦ of Theorem 1.30, f |W : W → f (W ) is a diffeomorphism and (U, ϕ) is a chart on M . Hence f = ψ ◦ f ◦ ϕ−1 = I is an identity mapping from ϕ(U ) to ψ(V ). Example 1.32. Suppose f : R → S 1 , defined by f (t) = (cos t, sin t). Using the chart of Example 1.11, we obtain    π π , kπ + cos t, t ∈ kπ − ,  f (t) = 2 2 − sin t, t ∈ (kπ, (k + 1)π). Obviously, f (t) = 0, ∀ t ∈ R. However, f : R → S 1 is not injective. Thus, f is not a diffeomorphism. This example shows that f∗p : Tp M → Tf (p) N isomorphism and f : M → N homeomorphism at some neighborhood of p are only local properties. We have discussed the case where f∗p : Tp M → Tf (p) N is an isomorphism. In the following section we turn to the case when f∗p is injective. 2. Immersion Definition 1.33. Let M, N be differentiable manifolds, and f : M → N a smooth mapping, and p ∈ M . If f∗p : Tp M → Tf (p) N is injective (i.e., rkp f = m), then f is said to immerse at p. If f immerses at every p ∈ M , then f is called an immersion. Below are some examples of immersion. Example 1.34. Let U ∈ Rm be an open subset, α : U → Rn , α(u1 , · · · , um ) = (u1 , · · · , um , 0, · · · , 0). By definition, α is obviously an immersion, and is often called a model immersion.

48

1. Preliminaries of Differentiable Manifolds

Proposition 1.35. Let M, N be m- and n-dimensional differentiable manifolds respectively; f : M → N is a smooth mapping, p ∈ M . If f immerses at p, then there exist charts (U, ϕ) on M about p ∈ M and (V, ψ) on N about f (p) ∈ N in which the coordinate description f = ψ ◦ f ◦ ϕ−1 : ϕ(U ) → ψ(V ) has the form f(u1 , · · · , um ) = (u1 , · · · , um , 0, · · · , 0). Proof. Choose charts (U1 , ϕ1 ) and (V1 , ψ1 ) appropriately so that p ∈ U1 , f (p) ∈ V1 , ϕ1 (p) = 0 ∈ Rm , ψ1 (f (p)) = 0 ∈ Rn and f (U1 ) ⊂ V1 . Since f immerses    ∂ fi  at p, the rank of Jacobian Jf(0) =  is m, where f = (f1 , · · · , fn ). We can ∂uj 0 assume that the first m rows in the Jacobian matrix Jf(0) are linearly independent. Then, define a mapping for ϕ1 (U1 ) × Rn−m → Rn = Rm × Rn−m by g(u, v) = f(u) + (0, v). It is easy to prove that g(u,!0) = f(u) maps origin 0 to itself in Rn and the rank of 0 Jg (O ) = Jf(O) is n, where 0 denotes a m × (n − m) zero matrix, and by In−m the inverse function theorem, g is a diffeomorphism from a neighborhood of origin of Rm to a neighborhood of origin of Rn . Shrink U1 , V1 so that they become U, V , and −1  let ϕ = ϕ1 |U, ψ = g −1 ◦ (ψ|V ). Since ψ ◦ f ◦ ϕ−1 = g −1 ◦ ψ1 ◦ f ◦ ϕ−1 ◦f = 1 =g g(u, 0), the proposition is proved.  Remark 1.36. By definition of immersion, if f : M → N immerses at p ∈ M , then f immerses in some neighborhood of p. Remark 1.37. By Proposition 1.35, f limited in some neighborhood of p has a local injective expression f = (u1 , · · · , um , 0, · · · , 0). Then, f limited in some neighborhood of p is injective. Note that this is only a local injection, not total injective. Definition 1.38. Let N, N  be differentiable manifolds, N  ⊂ N . If the inclusion map i : N  → N is an immersion, then N  is said to be an immersed submanifold of N. Remark 1.39. Suppose f is an immersion and injective (such f would henceforth be called injective immersion), M is a smooth atlas A = {(Uα , ϕα )}. Denote f A = {(f (Uα ), ϕα ◦ f −1 )}. Then, it is easy to prove that {f (M ), f A} is a differentiable manifold. Since f has a local expression f = ϕα ◦ f −1 ◦ f ◦ ϕ−1 α = I : ϕα (Uα ) → ϕα (Uα ), f : M → f (M ) is a diffeomorphism, i.e., f∗p : Tp M → Tf (p) f (M ) is an isomorphism. Since f is an immersion, the inclusion map i : f (M ) → N is also an immersion. Hence, f (M ) is an immersed submanifold of N . From the following example, we can see that the manifold topology of an immersed submanifold may be inconsistent with its subspace topology and can be very complex.

1.1 Differentiable Manifolds

49

Example 1.40. T 2 = S 1 × S 1 = {(z1 , z2 ) ∈ C × C | |z1 | = |z2 | = 1}. Define f : R → T 2 , s.t. f (t) = (e2πit , e2πiα ), where α is an irrational number. We can prove that f (R), which is a differentiable manifold derived from f , is an immersed submanifold of T 2 , and f (R) is dense in T 2 . We may regard T 2 as a unit square on the plane of R2 , which is also a 2D manifold that has equal length on opposite sides. It can be represented by a pair of ordered real numbers (x, y), where x, y are mod Z real numbers. Define ϕ : R2 −→ S 1 × S 2 ,

1

2

ϕ(u1 , u2 ) = (e2πiu , e2πiu )

and define“ ∼ ” : (u1 , u2 )∼ (v1 , v 2 ) ⇔ u1 = v1 (mod Z), u2 = v 2 (mod Z), and 1 1 1 1 let W = u10 − , u10 + × u20 − , u20 + . Then, (ϕ(W ), ϕ−1 ) is a chart 2

2

2

2

of T 2 = S 1 × S 1 that contains f (t0 ). Choose a neighborhood U of t0 ∈ R so that f (U ) ⊂ ϕ(W ). Then, the local expression of f , f = (t, αt) is an immersion at t0 . It is easy to prove that if ϕ−1 f (R) is dense in R2 , then f (R) is dense in T 2 = S 1 × S 1 . By definition, f is injective. It is concluded that the topology of T 2 is different from the topology of f (U ), which derives from f (R). 3. Regular submanifolds The type of submanifold given below has a special relationship to its parent differential manifold, which is similar to that of Euclidean space and its subspace. Definition 1.41. Let M  ⊂ M have the subspace topology, and k be some nonnegative integer, 0 ≤ k ≤ m. If there exists a chart (U, ϕ) of M that contains p in every p ∈ M  , so that 1◦ ϕ(p) = O ∈ Rm . 2◦ ϕ(U ∩ M  ) = {(u1 , · · · , um ) ∈ ϕ(U ) | uk+1 = · · · = um = 0}. Then M  is said to be a k-dimensional regular submanifold of M , and the chart is called submanifold chart. Let A = {(Uα , ϕα )} be a set that contains all submanifold charts on M . Denote "α = Uα ∩ M  , ϕ "α ), π : Rk × Rm−k → Rk . " "α , ϕ "α )}, where U "α = π ◦ (ϕα |U A={(U  "α is an open set of M  , and ϕ "α → ϕ "α ) "α : U "α (U Since M has the subspace topology, U "α = M  , and hence "α ) ⊂ Rk . Moreover, Uα # U "α to ϕ "α (U is a homeomorphism for U "α , ϕ "β , ϕ "α ∩ U "β = Ø, we have "α ), (U "β ) ∈ A" and U A" is an atlas of M  . ∀ (U 1 k −1 1 k ϕ $β ◦ ϕ "−1 α (u , · · · , u ) = π ◦ ϕβ ◦ ϕα (u , · · · , u , 0, · · · , 0).

Obviously, A" is a smooth atlas of M  , which determines a differentiable structure of M  . Thus, M  is a k-dimensional differentiable manifold. Below is an example of regular submanifold. Example 1.42. Let M, N be m- and n-dimensional differentiable manifolds respectively, and f : M → N be a smooth mapping. Then, the graph of f gr(f ) = {(p, f (p)) ∈ M × N | p ∈ M } is an m-dimensional closed submanifold of M × N ( closed regular submanifold ).

50

1. Preliminaries of Differentiable Manifolds

Proof. Consider charts (U, ϕ), (V, ψ), p0 ∈ U, f (p0 ) ∈ V, ϕ(p0 ) = O ∈ Rm , ψ(f (p0 )) = O ∈ Rn , f (U ) ⊂ V , and define G : ϕ(U ) × ψ(V ) → Rn+m = Rm × Rn , so that G(u, v) = (u, v − f(u)). It is easy to prove that G(gr(f)) = {(u, O  ) | u ∈ ϕ(U )}, and the rank of 

%

JG (O, O ) =

Im O −Df(O) In

&

is n + m. Since G(O, O ) = (O, O ), G homeomorphically maps some neighborhood " of (O, O ) on ϕ(U ) × ψ(V ) to some neighborhood V" of (O, O ) on Rn+m . Denote U " ), W = (ϕ × ψ)−1 (U

χ = G ◦ (ϕ × ψ)|W.

Then, (W, χ) is a chart of M × N that contains (p0 , f (p0 )), and χ(p0 , f (p0 )) = (O, O ) ∈ Rn+m , χ(W ∩ gr(f )) = {(u, v) ∈ χ(W )|v = 0}. The proof can be obtained.



Remark 1.43. If N  is a regular submanifold N , f : M → N is a smooth mapping, f (M ) ⊂ N  , then f : M → N  is also a smooth mapping. Let (U, ϕ), (V, ψ) be " is a induced chart of N  from N . Then, by the fact that N  is charts of N , then (V" , ψ) a regular submanifold of N , we know ψ ◦ f ◦ ϕ−1 (u) = (ψ" ◦ f ◦ ϕ−1 (u), 0). Then, the smoothness of f : M → N leads to the smoothness of f : M → N  . Remark 1.44. Let M  be a k-dimensional regular submanifold of M , then i : M  → M is the inclusion mapping. Take a submanifold chart (U, ϕ) of M that induces the " ) → ϕ(U ) has the form " , ϕ) "−1 : ϕ( "U chart (U " of M  . Then, i = ϕ ◦ i ◦ ϕ i (u1 , · · · , uk ) = (u1 , · · · , uk , 0, · · · , 0). Thus, i∗p : Tp M  → Tp M is injective, which means that the regular submanifold is definitely an immersed submanifold. 4. Embedded submanifolds Definition 1.45. Let f : M → N be an injective immersion. If f : M → f (M ) is a homeomorphism, where f (M ) has the subspace topology of N , then f (M ) is an embedded submanifold of N . Proposition 1.46. Suppose f : M → N is an embedding, then f (M ) is a regular submanifold of N , and f : M → f (M ) is a diffeomorphism.

1.1 Differentiable Manifolds

51

Proof. Since f is an embedding, f is an immersion ,∀ q ∈ f (M ), ∃ p ∈ M so that f (p) = q. Let charts (U, ϕ), (V, ψ), p ∈ U, f (p) ∈ V so that ϕ(p) = O ∈ Rm , ψ(q) = O ∈ Rn , f (U ) ⊂ V , and f(u1 , · · · , um ) = (u1 , · · · , um , · · · , 0). Since f : M → f (M ) is a homeomorphism, if U is an open subset of M , then f (U ) is an open subset of f (M ), and there exists an open subset W1 ⊂ N so that f (U ) = W1 ∩ f (M ). Denote W = V ∩ W1 , χ = ψ|W . Then, χ(q) = O ∈ Rn , and χ(W ∩ f (M )) = {(u1 , · · · , un ) ∈ χ(W ) | um+1 = · · · = un = 0}, i.e., (W, χ) is a submanifold chart of N that contains q, which also means that f (M ) $, χ is a regular submanifold of N . Let (W ") be a chart of f (M ) induced from (W, χ). −1 Then from χ ◦ f ◦ ϕ (u) = (" χ ◦ f ◦ ϕ−1 (u), 0), we conclude that f : M → f (M ) is a diffeomorphism.  Remark 1.47. If f is an immersion, then we can appropriately choose the charts of M, N , such that f has the local expression f(u1 , · · ·, um ) = (u1 , · · · , um , · · · , 0). Therefore, it is easy to see that f can be an injective immersion in the neighborhood U of p, and f : U → f (U ) is a homeomorphism. Obviously, f (U ) has the induced subspace topology from N . Therefore, f | U : U → N is an embedding. Definition 1.48. Let X, Y be two topological spaces, and f : X → Y be continuous. If for every compact subset K in Y , we have f −1 (K) to be a compact subset in X, then f is said to be a proper mapping. Proposition 1.49. Let f : M → N be an injective immersion. If f is a proper mapping, then f is an embedding. Proof. It would be sufficient to prove f −1 : f (M ) → M is continuous. Assume there / f (W ),but exist an open set W of M and a sequence of points {qi } of f (M ) s.t. qi ∈ {qi } converges to some point qi of f (W ). Denote pi = f −1 (qi ), p0 = f −1 (q0 ), p0 ∈ W . Since {q0 , qi }(i = 1, 2, · · ·) is compact, and f is a proper mapping, {p0 , pi }(i = 1, 2, · · ·) is a compact set of M . Let p1 ∈ M be the convergence of {pi }. Since f is continuous, {f (pi )} converges to f (p1 ), i.e., f (p1 ) = f (p0 ). Thus, p0 = p1 . Therefore, when i is large enough, there exists pi ∈ W , and qi = f (pi ) ∈ f (W ). This / f (W ).  is in contradiction with qi ∈ Remark 1.50. Let f be an injective immersion. If M is compact, then f is a proper mapping. By Proposition 1.49, f is an embedding.

1.1.4 Submersion and Transversal Below we will discuss the local property of f when f∗p : Tp M → Tf (p) N is surjective. Definition 1.51. f is smooth, p ∈ M . If f∗p : Tp M → Tf (p) N is surjective, then f is a submersion at p; if f is a submersion at every p ∈ M , then f is said be a submersion. Similar to the proposition for f that immerses at p, we have the following proposition.

52

1. Preliminaries of Differentiable Manifolds

Proposition 1.52. Given a smooth f and p ∈ M , if f submerses at p, then there exists chart (U, ϕ) on M about p and (V, ψ) on N about f (p) ∈ N in which f = ψ ◦ f ◦ ϕ−1 : ϕ(U ) → ψ(V ) has the form f(u1 , · · · , um ) = (u1 , · · · , un ). Proof. Take charts (U1 , ϕ1 ), (V, ψ1 ), p ∈ U1 , f (p) ∈ V, ϕ1 (p) = O ∈ Rm , ψ1 (f (p))    ∂ fi  = O ∈ Rn and f (U1 ) ⊂ V . Since f is a submersion, Jf(O) =  has rank ∂uj O n, where f = (f1 , · · · , fn ). We assume that the first n rows of Jf(O) are linearly independent. Let g : ϕ1 (U1 ) → ψ(V ) × Rm−n satisfy g(u1 , · · · , um ) = (f(u1 , · · · , um ), un+1 , · · · , um ). ! Jf(O) Then, g(O) = O, and Jg (O) = has rank n. By the inverse function O Im−n theorem, g(O) maps a neighborhood W at O diffeomorphically to a neighborhood of −1 = g(W ) ⊂ ψ(V ) × Rm−n . Let U = W ∩ U1 , ϕ = g ◦ (ϕ1 |U ), then ψ ◦ f ◦ ϕ−1 1 ◦g −1 m n 1 m 1 n β ◦ g ◦ g = β, and β : R → R , β(u , · · · , u ) = (u , · · · , u ) is a projection  from Rm → Rn . Remark 1.53. By Definition 1.51, if f : M → N is a submersion at p ∈ M , then f is a submersion in some neighborhood of p. Remark 1.54. If f : M → N is a submersion, then f is an open mapping (i.e., open set mapping to an open set). Furthermore, f (M ) is an open subset of N . Let G be an open subset of M , ∀ q ∈ f (G). There exists a p ∈ G, s.t. f (p) = q. Since f is a submersion, there exist charts (U, ϕ), (V, ψ), p ∈ U, q ∈ V , s.t. U ⊂ G, and f : ϕ(U ) → ψ(V ), f(u1 , · · · , um ) = (u1 , · · · , un ). Let H = β(ϕ(U )), where β(u1 , · · · , um ) = (u1 , · · · , un ), s.t. H ⊂ ψ(V ). Thus, ψ −1 (H) is a neighborhood of q ∈ N , ψ −1 (H) ⊂ f (G), i.e., f (G) is an open subset of N . Next, we consider under what condition would f −1 (q0 ) be a regular submanifold of M , and ∀ q0 ∈ N be fixed. Definition 1.55. Given f : M → N is smooth, p ∈ M , if f∗p : Tp M → Tf (p) N is surjective, then p is said to be a regular point of f (i.e., f submerses at p), otherwise p is said to be a critical point of f , and q ∈ N is called a regular value of f , if q ∈ / f (M ) or q ∈ f (M ), but each p ∈ f −1 (q) is a regular point of f ; otherwise, q is called a critical value of f . Remark 1.56. When dim M < dim N , as a result of dim Tp M = dim M < dim N = dim Tf (p) N , for q ∈ f (M ), p ∈ f −1 (q), p cannot be a regular point of f . Hence, q ∈ N is a regular value of f ⇔ q ∈ / f (M ). Theorem 1.57. Let f : M → N be smooth, q ∈ N ; if q is a regular value of f , and f −1 (q) = Ø, then f −1 (q) is an (m − n)-dimensional regular submanifold of M . Moreover, ∀p ∈ f −1 (q), Tp {f −1 (q)} = ker f∗p .

1.1 Differentiable Manifolds

53

Proof. Since q is a regular value, ∀ p ∈ f −1 (q), f submerses at p by definition. By the Proposition 1.52, there exist charts (U, ϕ), (V, ψ), p ∈ U, f (p) = q ∈ V, ϕ(p0 ) = O ∈ Rm , ψ(q) = O ∈ Rn , and ψ ◦ f ◦ ϕ−1 (u1 , · · · , um ) = f(u1 , · · · , um ) = (u1 , · · · , un ), ϕ{U ∩ f −1 (q)} = {(u1 , · · · , um ) ∈ ϕ(U ) | u1 = · · · = un = 0}, i.e., (U, ϕ) is a submanifold chart of M that contains p. Therefore, f −1 (q) is a regular submanifold of M , and dim f −1 (q) = m − n. Note that f |f −1 (q) : f −1 (q) → M, f |f −1 (q) = f ◦ i, i : f −1 (q) → M is an inclusion mapping. Since f |f −1 (q) = q is a constant mapping, f∗p ◦i∗p = (f |f −1 (q) )∗p = 0, i.e., i∗p (Tp {f −1 (q)}) ⊂ ker f∗p . Furthermore, because q is a regular value of f , f∗p (Tp M ) = Tq N , and dim ker f∗p = dim Tp M − dim f∗p (Tp M ) = m − n = dim f −1 (q). Therefore, we have Tp {f −1 (q)} = ker f∗p .  Remark 1.58. Given f : M → N is smooth, dim M = dim N, M is compact. If q ∈ N is a regular value of f , then f −1 (q) = Ø or f −1 (q) consists of finite points. By Theorem 1.57, if f −1 (q) = Ø, then f −1 (q) is a 0-dimensional regular submanifold of M . By definition, we have ϕ(U ∩ f −1 (q)) = O ∈ Rm , i.e., every point in f −1 (q) is an isolated point. Moreover due to the compactness of f −1 (q), f −1 (q) must consist of finite points. Below, we give some applications of Theorem 1.57. Example 1.59. Let f : Rn+1 → R, and f (u1 , · · · , un+1 ) =

n+1 

(ui )2 .

i=1

From the Jacobian matrix of f at (u1 , · · · , un+1 ), we know f is not a submersion at (u1 , · · · , un+1 ) ⇔ u1 = · · · = un+1 = 0. Therefore, any non-zero real number is a regular value of f . According to the Theorem 1.57, the n-dimensional unit sphere S n = f −1 (1) is an n-dimensional regular submanifold on Rn+1 .  Example 1.60. Let f : R3 → R, and f (u1 , u2 , u3 ) = (a − (u1 )2 + (u2 )2 )2 + (u3 )2 , a > 0. The assumption tells us that any non-zero real number is a regular point of f . Then, 0 < b2 < a2 is a regular value of f . Therefore, by Theorem 1.57, T 2 = f −1 (b2 ) is a 2-dimensional regular submanifold on R2 . If M  is a regular submanifold of M , then dim M − dim M  =codim M  is called the M -codimension of M  . Denote M  = {p ∈ M | fi (p) = 0 (i = 1, · · · , k)} and consider the mapping F : M −→ Rk ,

F (p) = (f1 (p), · · · , fk (p)).

If fi : M → R is smooth, then F is smooth too, and M  = F −1 (O). Proposition 1.61. Suppose M  is a subset of M . Then, M  is a k-codimensional regular submanifold of M if and only if for all q ∈ M  , there exists a neighborhood U of q ∈ M and a smooth mapping F : U → Rk , s.t. 1◦ U ∩ M  = F −1 (O). 2◦ F : U → Rk is a submersion.

54

1. Preliminaries of Differentiable Manifolds

Proof. Necessity. By the definition of the regular submanifold, if M  is a k-codimensional regular submanifold of M , then ∀p ∈ M  , there exists a submanifold chart (U, ϕ) of M that contains p s.t. ϕ(p) = O ∈ Rm , and ϕ(U ∩ M  ) = {(u1 , · · · , um ) ∈ ϕ(U ) | um−k+1 = · · · = um = 0}. Let us denotes the projection by π : Rm = Rm−k × Rk → Rk , let F = π ◦ ϕ : U → Rk . Then, F is smooth, and F −1 (O) = (π ◦ ϕ)−1 (O) = U ∩ M  , F∗q = π∗ϕ(q) ◦ ϕ∗q . Since ϕ∗q is an isomorphism and π∗ϕ(q) is surjective, F submerses at q. Sufficiency. If ∀ q ∈ U ∩ M  , F submerses at q, then O ∈ Rk is a regular value of F . By Theorem 1.57, F −1 (O) is a k-codimensional regular submanifold of U , i.e., M  is a k-codimensional regular submanifold of M .  We know that if q ∈ N is a regular value of f : M → N , and f −1 (q) = Ø, then f −1 (q) is a regular submanifold of M . Assume that Z is a regular submanifold of N . Then, under what condition would f −1 (Z) be a regular submanifold of M ? For this question, we have the following definition. Definition 1.62. Suppose Z is a regular submanifold of N , f : M → N is smooth, p ∈ M . Then, we say f is transversal to Z at p, if f (p) ∈ / Z or when f (p) ∈ Z has f∗p Tp M + Tf (p) Z = Tf (p) N, denoted by f p Z. If ∀p ∈ M , f p Z, then f is transversal to Z, denoted by f Z. Remark 1.63. If dim M + dim Z < dim N , then f Z ⇔ f (M ) ∩ Z = Ø; if q ∈ N is a regular value of f , then ∀p ∈ f −1 (q), f p Z; if f : M → N is a submersion, then for any regular submanifold Z of N , f Z. For transversality, we focus on its geometric property. Example 1.64. M = R, N = R2 , Z is x-axis in R2 , f : M → N, f (t) = (t, t2 ). When t = 0, as a result of f (t) ∈ / Z, f t Z;      d  ∂ ∂  d  , 2 = , When t = 0, Jf (0) = (1, 0) , note that f∗0  =(1, 0) 1 dt 0 ∂u ∂u d u1 f∗0 T0 M = T(0,0) Z. Therefore, f∗0 T0 M + T(0,0) Z = T(0,0) N is impossible to establish. Thus, f is not transversal to Z at 0. / However, if we change f to f (t) = (t, t2 − 1), we obtain:   t = ±1, f (t) ∈  when d ∂   + Z, so f t Z; when t = 1, Jf (1) = (1, 2) , therefore f∗1 =   dt 1 ∂u1 (1,0)  ∂  , i.e., f∗1 T1 M + T(1,0) Z = T(1,0) N , and hence f 1 Z. Similarly, we have 2 2 ∂u

(1,0)

f −1 Z. Thus, f Z. Submanifold transverse: Let Z, Z  be two regular submanifolds of N , i : Z  → N is an inclusion mapping. If iZ, then submanifold Z  is transversal to Z, denoted as Z  Z. If Z  Z, ∀p ∈ Z ∩ Z  , by definition, we have i∗p (Tp Z  ) + Tp Z = Tp N, i.e.,

1.1 Differentiable Manifolds

55

Tp Z  + Tp Z = Tp N. We assume that f : M → N is smooth, and Z is a k-codimensional regular submanifold of N , p ∈ M, f (p) = q ∈ Z. According to the Proposition 1.61, there exists a submanifold chart (V, ψ) of N that contains q, s.t. π ◦ ψ : V → Rk is a submersion, and Z ∩ V = (π ◦ ψ)−1 (O). Now, take a neighborhood of p in M , s.t., f (U ) ⊂ V , then π ◦ ψ ◦ f : U → Rk . Proposition 1.65. f p Z ⇔ π ◦ ψ ◦ f : U → Rk submerses at p. Proof. Since π ◦ ψ submerses at f (p), O ∈ Rk is a regular value of π ◦ ψ. From Z ∩ V = (π ◦ ψ)−1 (O), we know for every q ∈ Z ∩ V , there exists a (π ◦ ψ)∗q Tq N = To Rk . By Theorem 1.57, ker(π ◦ ψ)∗q = Tq Z. Therefore, f∗p Tp M + Tq Z = Tq N ↔ (π◦ψ)∗q (f∗p Tp M ) = To Rk ↔ (π◦ψ◦f )∗p (Tp M ) = To Rk , i.e., π◦ψ◦f submerses at p.  Remark 1.66. Extending from the conclusion of Proposition 1.65, we have f Z ↔ O ∈ Rk are regular values of π ◦ ψ ◦ f : U → Rk . Remark 1.67. Since f p Z, i.e., π ◦ ψ ◦ f : U → Rk submerses at p. By Proposition 1.52, we can choose a coordinate chart s.t. π ◦ ψ ◦ f ◦ ϕ−1 : ϕ(U ) → Rk has the form (π ◦ ψ ◦ f ◦ ϕ−1 )(u1 , · · · , um ) = (um−k+1 , · · · , um ). Then, f = ψ ◦ f ◦ ϕ−1 can be represented by f = (η1 (u1 , · · · , um ), · · · , ηn−k (u1 , · · · , um ), um−k+1 , · · · , um ). Theorem 1.68 (Extension of Theorem 1.57). Suppose f : M → N is smooth, Z is a k-codimensional regular submanifold of N . If f Z and f −1 (Z) = Ø, then f −1 (Z) is a k-codimensional regular submanifold of M , and ∀p ∈ f −1 (Z), −1 {Tf (p) Z}. Tp {f −1 (Z)} = f∗p

Proof. ∀ p ∈ f −1 (Z), there exists q ∈ Z, denoted by q = f (p). Since Z is a kcodimensional regular submanifold of N , there exists a submanifold chart (V, ψ) of N that contains p. Let U = f −1 (V ). From f Z, we know that O ∈ Rk is a regular value of π ◦ ψ ◦ f , and U ∩ f −1 (Z) = (π ◦ ψ ◦ f )−1 (O). By Theorem 1.57, U ∩ f −1 (Z) is a k-codimensional regular submanifold of U , and Tp {f −1 (Z)}

= ker(π ◦ ψ ◦ f )∗p −1 = f∗p {(π ◦ ψ)−1 ∗q (O)} −1 {Tf (p) Z}. = f∗p

The theorem is proved.



56

1. Preliminaries of Differentiable Manifolds

1.2 Tangent Bundle The tangent bundle of a differentiable manifold M is the disjoint union of the tangent spaces of M . It is useful, in distinguishing between the tangent space and bundle, to consider their dimensions, n and 2n respectively. In other words, the tangent bundle accounts for dimensions in the positions in the manifold as well as directions tangent to it. Since we can define a projection map, for each element of the tangent bundle giving the element in the manifold whose tangent space the first element lies in, tangent bundles are also fiber bundles.

1.2.1 Tangent Bundle and Orientation In this section, we will discuss two invariable properties under diffeomorphism– tangent bundle and orientation. 1. Tangent Bundle Definition 2.1. The Triple (T M, M, π) is called tangent bundle # of differentiable manTp M , projection map ifold M (sometimes simply called T M ), where T M = p∈M

π : T M → M satisfies π(Xp ) = p, ∀ Xp ∈ T M . For every p ∈ M, π −1 (p) = Tp M is called fiber at p of tangent bundle T M . Proposition 2.2. Let M be an m-dimensional differentiable manifold, then T M is a 2m-dimensional differentiable manifold, and π : T M → M is a submersion. Proof. Let (U, ϕ) be a chart on M , and its coordinate function be x1 , · · · , xm . Then,  ∂  ai i  . Define ϕU : π −1 (U ) → ϕ(U ) × Rm , s.t. ∀ Xp ∈ π −1 (U ), Xp = ∂x p i ϕU (Xp ) = (ϕ(p); a1 , · · · , am ), obviously ϕU is a 1 to 1 mapping. Note that as (U, ϕ) takes all the charts on M , all the corresponding π −1 (U ) constitutes a covering of T M . Hence, if the topology of π −1 (U ) is given, the subset of π −1 (U ) is open, iff the image of ϕU is an open set of ϕ(U ) × Rm . It is easy to prove that by the 1 to 1 correspondence of ϕU , the topology of ϕU on the Rm ×Rm = R2m subspaces can be lifted on π −1 (U ). The topology on T M can be defined as follows: W is called an open subset of T M , iff W ∩ π −1 (U ) is an open subset of π −1 (U ). It is easy to deduce that T M constitutes a topological space that satisfies the following conditions: 1◦ T M is a Hausdorff space that has countable bases. 2◦ π −1 (U ) is an open subset of T M , and ϕU is a homeomorphism from π −1 (U ) to an open subset of R2m . Furthermore, it can be proved that the manifold structure on T M can be naturally induced from the manifold structure on M . We say that {(π −1 (U ), ϕU )} = A is a smooth atlas of T M . For any chart (π −1 (U ), ϕU ), there exists a (π −1 (V ), ψV ) ∈

1.2 Tangent Bundle

57

A, and π −1 (U ) ∩ π −1 (V ) = Ø. Let x1 , · · · , xm and y 1 , · · · , y m be the coordinate functions of the charts (U, ϕ), (V, ψ). Then,  & ∂  ∂xi ϕ−1 (u) i   %% & & ∂  ∂y j  = ψV ai i  ∂x ϕ−1 (u) ∂y j ψ◦ϕ−1 (u) j i

1 m ψV ◦ ϕ−1 U (u; a , · · · , a ) = ψV

% =

%

ai

−1

ψ◦ϕ

(u);

 i

i ∂y

1 

 a ∂xi 

ϕ−1 (u)

,···,



a

i

m 

&  . ∂x ϕ−1 (u)

i ∂y

i

It is easy to conclude that T M is a 2m-dimensional manifold, A is a differentiable structure on T M . From the charts (U, ϕ) of M and (π −1 (U ), ϕU ) of T M , we know m π  = ϕ ◦ π ◦ ϕ−1 U : ϕ(U ) × R → ϕ(U ) has the form: π (u; a1 , · · · , am ) = u. 

By the definition of submersion, π is a submersion.

Given below are examples of two trivial tangent bundles (if there exists a diffeomorphism from its tangent bundle T M to M × Rm , and this diffeomorphism limited on each fiber of T M (Tp M ) is a linear isomorphism from Tp M to {p} × Rm ). Example 2.3. Let U be an open subset of Rm and T U  U × Rm .     ∂  ∂  (i = 1, · · · , m) is the basis of ∀ Xu ∈ T U, Xu = ai i  , where ∂ui u ∂u u i Tu U . Then, it is easy to prove that Xu −→ (u; a1 , · · · , am ) is a diffeomorphism from T U to U × Rm . Moreover, since each fiber Tu U of T U is a linear space, maps limited on Tu U is a linear isomorphism from Tu U to {u} × Rm , i.e., T U is a trivial tangent bundle. Example 2.4. T S 1 is a trivial tangent bundle, i.e., T S 1  S 1 × R. Let A={(U, ϕ), (V, ψ)} be a smooth atlas on S 1 , where U = {(cos θ, sin θ)|0 < θ < 2π}, ϕ(cos θ, sin θ) = θ, V = {(cos θ, sin θ)| − π < θ < π}, ψ(cos θ, sin θ) = θ, ' θ, 0 < θ < π, −1 ψ ◦ ϕ (θ) = θ − 2π, π < θ < 2π. Define f : T S 1 → S 1 × R, s.t. ⎧ ⎪ ⎪ ⎨ (p; a), p ∈ U, f (Xp ) = ⎪ ⎪ ⎩ (p; b), p ∈ V,



Xp = a

∂   , ∂x p

Xp = b

∂   , ∂y p



58

1. Preliminaries of Differentiable Manifolds

where x, y are the coordinate functions on (U, ϕ), (V, ψ) respectively. When p ∈ U ∩ V , we have ∂y  ∂  ∂  ∂   =   =  . ∂x p ∂x p ∂y p ∂y p Therefore, f has the definition and is a 1 to 1 correspondence. Moreover, f and f −1 are smooth. Hence, T S 1 is a trivial tangent bundle. Apart from trivial tangent bundles, there exists a broad class of nontrivial tangent bundles. For an example, T S 2 is a nontrivial tangent bundle. Definition 2.5. Let f : M → N be smooth. Define a mapping T f : T M → T N , s.t. T f |Tp M = f∗p ,

∀ p ∈ M,

then T f is called the tangent mapping of f . Remark 2.6. ∀ Xp ∈ Tp M , there exist charts (U, ϕ) on M about p and (V, ψ) on N about f (p), s.t. f (U ) ⊂ V . By π1 : T M → M, π2 : T N → N , it is naturally derived that (π1−1 (U ), ϕU ), (π2−1 (V ), ψV ) are charts on T M, T N , and T f (π1−1 (U )) ⊂ π2−1 (V ). Note that 1 m ψV ◦ T f ◦ ϕ−1 U (u; a , · · · , a )    ∂f1   ∂fn  ai i  ,···, ai i  , = ψ ◦ f ◦ ϕ−1 U ; ∂x ϕ−1 (u) ∂x ϕ−1 (u) i i

which may be simplified as    ˆ ψV ◦ T f ◦ ϕ−1 U (u; α) = f (u); Df (u)α , where α = (a1 , · · · , am ). Therefore, T f is a smooth mapping. Remark 2.7. Let M, N, L be the differentiable manifolds. By the definition of tangent mapping, if f : M → N and g : N → L are smooth, then T (g ◦ f ) = T g ◦ T f. Remark 2.8. If f : M → N is a diffeomorphism, then T f : T M → T N is also a diffeomorphism. 2. Orientation Next, we introduce the concept of orientation for differentiable manifolds. Given V as a m-dimensional vector space, {e1 , · · · , em }, {e1 , · · · , em } as V ’s two m  ordered bases, if ej = aij ei (j = 1, · · · , m), then i=1

(e1 , · · · , em ) = (e1 , · · · , em )A,

1.2 Tangent Bundle

59

where A = (aij )m×m . If det A > 0, we call {ei } and {ej } concurrent; otherwise, if det A < 0, we call {ei } and {ej } reverse. Then, a direction μ of V can be expressed by a concurrent class [{ej }] equivalent to {ej }. The other direction −μ can be expressed by an equivalent class to the reverse direction of {ej }. (V, μ) is called an orientable vector space. Let (V, μ), (W, ν) be two orientable vector spaces. A : V → W is a linear isomorphism from V to W . If the orientation of W , which is induced by A, is consistent with ν, i.e., Aμ = ν, then A preserves orientations. Otherwise, A reverses orientations. In the below section, we extend the orientation concept to differentiable manifolds. Definition 2.9. Let M be an m-dimensional differentiable manifold, ∀p ∈ M, μp is the orientation of Tp M , s.t. ϕ∗q : (Tq M, μq ) −→ (Tϕ(q) Rm , νϕ(q) ),

∀q ∈ U

are all linear isomorphisms that preserves orientations, where (U, ϕ) is a chart that contains p, and   ∂  ∂   νϕ(q) = . , · · · ,   ∂u1 ϕ(q) ∂um ϕ(q) Then, μ = {μp | p ∈ M } is the orientation on M , and (M, μ) is called an orientable differentiable manifold. Remark 2.10. The Definition 2.9 shows that if (M, μ) is an orientable differentiable manifold, W is an open subset of M , then ∀p ∈ M and there exists an orientation μp of Tp M . This gives an orientation on W , denoted by μ|W . Then, (W, μ|W ) is also an orientable differentiable manifold. Specifically, if (U, ϕ) is a chart on M , then (U, μp ) is an orientable differentiable manifold. Remark 2.10 shows that M may be locally orientable. Next, we discuss how to construct a global orientation. Proposition 2.11. Let M be an m-dimensional differentiable manifold, then M is orientable, iff there exists a smooth atlas, A = {(Uα , ϕα )}, on M , s.t. ∀ (Uα , ϕα ), (Uβ , ϕβ ) ∈ A, if Uα ∩ Uβ = Ø, then (ϕα (q)) > 0, det Jϕβ ◦ϕ−1 α

∀ q ∈ Uα ∩ Uβ ,

where Jϕβ ◦ϕ−1 (ϕα (q)) is the Jacobian matrix of ϕβ ◦ ϕ−1 α at ϕα (q). α Proof. Necessity. Since M is orientable, select one of the orientations of M , μ = {μp | p ∈ M }. According to Definition 2.9, ∀p ∈ M , there exists a chart (U, ϕ) on M about p, s.t. ∀q ∈ U ,  ∂   ∂   ϕ∗q μq = , · · · , .   ∂u1 ϕ(q) ∂um ϕ(q) Denote a set consisting of all such charts by A. Then, A is a smooth atlas of M , and the properties of A described in the proposition are easy to prove.

60

1. Preliminaries of Differentiable Manifolds

Sufficiency. Let A be an atlas that satisfies all the properties of the proposition. Choose (Uα , ϕα ), (Uβ , ϕβ ) ∈ A and Uα ∩ Uβ = Ø, and use x1 , · · · , xm and y 1 , · · · , y m to represent the coordinate functions of (Uα , ϕα ), (Uβ , ϕβ ) respectively. Note that  ∂  ∂    ∂  ∂    −1 (ϕα (q)), = , · · · , , · · · ,     J ∂x1 p ∂xm p ∂y 1 p ∂y m p ϕβ ◦ϕα and by supposition Jϕβ ◦ϕ−1 (ϕα (q)) > 0, we have α  ∂  ∂    ∂  ∂    = , · · · , , · · · ,     , ∂x1 p ∂xm p ∂y 1 p ∂y m p 

i.e., M is orientable.

Remark 2.12. If f : M → N is a diffeomorphism, f A = {f (Uα ), ϕα ◦ f −1 } is a smooth atlas. Pick two charts on N , (f (ϕα ), ϕα ◦ f −1 ), (f (ϕβ ), ϕβ ◦ f −1 ), we have det Jϕβ ◦f −1 ◦f ◦ϕ−1 (ϕα (q)) = det Jϕβ ◦ϕ−1 (ϕα (q)), ∀ q ∈ Uα ∩ Uβ . If M is α α orientable, then N is possible, which means orientation is an invariable property under diffeomorphism. Proposition 2.13. Let M be a connected differentiable manifold; if M is orientable, then M has only two orientations. Proof. If μ = {μp | p ∈ M } is an orientation of M , then −μ is also an orientation. Therefore, M has at least two orientations. Assume there exists another orientation, denoted as ν = {νp | p ∈ M }. Let S = {p ∈ M | μp = νp }. ∀p ∈ S, take charts (U, ϕ), (V, ψ) of M about p, s.t. μ, ν satisfy all the requirements of Definition 2.9. As a result of μp = νp , we have det Jψ◦ϕ−1 (ϕ(p)) > 0. By continuity, there exists a neighborhood of ϕ(p), W ⊂ ϕ(U ∩ V ), s.t. det Jψ◦ϕ−1 (ϕ(u)) > 0,

∀ u ∈ W.

Denote O = ϕ−1 (W ). Then, O is a neighborhood of p in M , and O ⊂ S, i.e., S is an open subset of M . Similarly, M \S is also an open subset of M . Since M is connected, we have either S = Ø or S = M . If S = Ø, then μ = −ν; if S = M , then μ = ν.  Remark 2.14. By the Proposition 2.13, any connected open set on an orientable differentiable manifold M has two and only two orientations. Remark 2.15. Let (U, ϕ), (V, ψ) be two charts on M , and U and V be connected. If U ∩ V = Ø, then det Jψ◦ϕ−1 preserves the orientation on ϕ(U ∩ V ). Example 2.16. S 1 is an orientable differential manifold. Let the smooth atlas of S 1 be A = {(U+ , ϕ+ ), (U− , ϕ− )}, where

1.2 Tangent Bundle

U+ = S 1 \{(0, −1)},

61

U− = S 1 \{(0, 1)},

ϕ± : U± → R, s.t. ϕ+ (u1 , u2 ) = Since

u1 , 1 + u2

ϕ− (u1 , u2 ) =

1 ϕ+ ◦ ϕ−1 − (u) = − , u

we have det Jϕ+ ◦ϕ−1 (u) = −

−u1 . u2 − 1

∀ u ∈ ϕ− (U+ ∩ U− ),

1 > 0, u2

∀ u ∈ ϕ− (U+ ∩ U− ).

Similarly ∀ u ∈ ϕ+ (U+ ∩ U− ),

det Jϕ− ◦ϕ−1 (u) > 0, +

i.e., S 1 is orientable. Example 2.17. M¨obius strip is a non-orientable surface. Define equivalent relation“∼” on [0, 1] × (0, 1): (u, v) ∼ (u, v),

0 < u < 1,

0 < v < 1,

(0, v) ∼ (1, 1 − v), 0 < v < 1, [0, 1] × (0, 1)\ ∼ is a M¨obius strip, A = {(U, ϕ), (V, ψ)} is its smooth atlas   1 × (0, 1), U = M \{0} × (0, 1), V = M\ 2   1 1 × (0, 1), ϕ : U −→ (0, 1) × (0, 1), ψ : V −→ − , 2 2

which satisfies: ϕ(u, v) = (u, v), ⎧ 1 ⎪ 0≤ , ⎨ (u, v), 2 ψ(u, v) = ⎪ 1 ⎩ (u − 1, 1 − v), < u ≤ 1, 2

  1 × (0, 1), (u, v) ∈ 0, 2 −1 ψ ◦ ϕ (u, v) =   ⎪ ⎩ (u − 1, 1 − v), (u, v) ∈ 1 , 1 × (0, 1), ⎧ ⎪ ⎨ (u, v),

2

i.e.,

  1 × (0, 1), (u, v) ∈ 0, 2 det Jψ◦ϕ−1 (u, v) =   ⎪ 1 ⎩ , 1 × (0, 1). −1, (u, v) ∈ ⎧ ⎪ ⎨ 1,

2

By the Remark 2.15, M¨obius strip is a nonorientable surface.

62

1. Preliminaries of Differentiable Manifolds

Definition 2.18. Let M, N be two orientable differential manifolds, and f : M → N be a local diffeomorphism ( diffeomorphism in the neighborhood of any p ∈ M ). If for every p ∈ M , there exists a f∗p : Tp M → Tf (p) N that preserves (or reverses) the orientation, then f is said to preserve the orientation (or reverse the orientation). Proposition 2.19. f : M → N is a diffeomorphism; if M is a connection, then f preserves the orientation or reverses the orientation. Proof. Let S = {p ∈ M | f∗p : Tp M → Tf (p) N preserves the orientation}. ∀ p ∈ S, because f∗p preserves orientation, det Jf (p) > 0. From the continuity, there exists U , s.t. det Jf (q) > 0, ∀ q ∈ U , i.e., U ⊂ S. Similarly, M \S is also an open subset of M . Since M is connected, S = Ø or S = M . When S = Ø, then f preserves the inverse orientation, otherwise (S = M ) f preserves the orientation. 

1.2.2 Vector Field and Flow Similar to Euclidean space, differentiable manifold also has the concept of vector field and curve of solution. Definition 2.20. Let M be a differentiable manifold. If map X : M → T M has the property π ◦ X = I : M → M , then X is said to be a vector field of M , and is also called a section in the tangent bundle T M , where π : T M → M is a projection. If the map X is smooth, then X is called a smooth vector field. Proposition 2.21. X is a smooth vector field on M , iff for every f ∈ C ∞ (M ) there exists a Xf ∈ C ∞ (M ), and C ∞ (M ) = {all smooth f unctions on M }, Xf : M → R, Xf (p) = Xp (f ), ∀ p ∈ M, f ∈ C ∞ (M ). Proof. Necessity. Suppose (U, ϕ) is a chart of M , and (π −1 (U ), ϕU ) is an induced natural chart of T M . Suppose X can be expressed as  ∂  ai (p) i  , ∀ p ∈ U, Xp = ∂x p i by

 = ϕU ◦ X ◦ ϕ−1 : ϕ(U ) −→ ϕ(U ) × Rm , X  X(u) = (u; a1 ◦ ϕ−1 (u), · · · , am ◦ ϕ−1 (u)),

we know, if X is smooth, then a1 , · · · , am are smooth too. Since  ∂f  ai (p) i  , ∀ p ∈ U, (Xf )(p) = Xp f = ∂x p i (Xf )|U is smooth. Sufficiency. ∀ p ∈ M , let (U, ϕ) be a chart on M about p, where its coordinate function x1 , · · · , xm may be expanded to smooth x "i on the entire M , and satisfy x "i = i x on some neighborhood of p on V ⊂ U . Then,

1.2 Tangent Bundle

Xq =



Xq (" xi )

i

∂   , ∂xi q

63

∀ q ∈ U, 

by the supposition Xq (" xi ) is smooth, i.e., X is also smooth.

Definition 2.22. Let X be a smooth vector field on the differentiable manifold M . The solution curve of X through p refers to a smooth mapping c : J → M s.t. c(0) = p, and   d c∗t  = Xc(t) , ∀ t ∈ J, dt

t

i.e., the velocity vector at t of a smooth curve c is exactly the value of the vector field at p on M . Proposition 2.23. Let f : M → N be a diffeomorphism, X be a smooth vector field on M . If we denote f∗ X = T f ◦ X ◦ f −1 : N → T N , then f∗ X is a smooth vector field on N , and c is a solution curve of X through p ∈ M , iff f ◦ c is a solution curve of f∗ X through f (p). Proof. By the definition of tangent mapping, we have π2 ◦ (f∗ X) = π2 ◦ (T f ◦ X ◦ f −1 ) = f ◦ (π1 ◦ X) ◦ f −1 = I. Since f −1 , X, T f are smooth, f∗ X is a smooth vector field on N . If c : J → M is a solution curve of X through p, then f ◦ c(0) = f (0) = f (p), and     d  d  (f ◦ c)∗t = f∗c(t) ◦ c∗t   dt

dt

t

t

= f∗c(t) (Xc (t)) = (f∗ X)f ◦c(t) ,

∀ t ∈ J. 

Therefore the proposition is completed.

Remark 2.24. Let X be a smooth vector field on the differentiable manifold M , and (U, ϕ) be a chart on M . By Proposition 2.23, we have ϕ∗ (X | U ) to be a smooth vector field of ϕ(U ). Remark 2.25. If ϕ∗ (X | U ) has an expression {ϕ∗ (X | U )}u =

m  i=1

then

ai (u)

∂   , ∂ui u

∀ u ∈ ϕ(U ),

m ∂  ∂(ϕ ◦ c)i  ∂   , (ϕ ◦ c)∗t  =   ∂t t ∂t t ∂ui ϕ◦c(t) i=1

∀ t ∈ J,

where (ϕ ◦ c)i is the i-th component of ϕ ◦ c. Therefore, according to Proposition 2.23, there exists a solution curve, c : J → U , of X through p, iff ϕ ◦ c is a solution of

64

1. Preliminaries of Differentiable Manifolds

⎧ i ⎨ d u = ai (u1 , · · · , um ), ⎩

dt

i = 1, · · · , m,

u(0) = ϕ(p).

Strictly speaking, a vector field on Rn is a mapping A : Rn → T (Rn ),i.e., ∀ x ∈ Rn ,

A(x) ∈ Tx Rn .

Since (e1 )x , · · · , (en )x form a basis on Tx Rn , we can write fore n  Ai (x)(ei )x . A(x) =



 ∂ , · · · , ∂ , there∂x1 ∂xn

i=1

If Ai (x) ∈ C ∞ , then A(x) is called a smooth vector field on Rn . The set of all smooth vector fields on Rn is denoted by X (Rn ). For any vector A(x), B(x) ∈ X (Rn ), define: (αA + βB)(x) = αA(x) + βB(x), α, β ∈ R, (f A)(x) = f (x)A(x),

f (x) ∈ C ∞ (Rn ).

Then, X (Rn ) is a C ∞ module of C ∞ -vector on Rn . If we denote A(x) ∈ X (Rn ) by (A1 (x), · · · , An (x)) , i.e., A(x) =

n 

Ai (x)(ei )x = (A1 (x), · · · , An (x)) ,

i=1

then A(x) is a C ∞ n-value function.

1.3 Exterior Product Exterior product is one of the algebraic operations. It has quite an interesting geometric background. In this section, we would like to construct a new linear space from an original linear space so that the new space has not only the linear space algebraic structure, but also a new algebraic operation — exterior product. This forms a basis for the differential form introduced later. These materials can be found in a series of books[AM78,Che53,Arn89,Ede85,Fla] . In R3 , let the vectors be a1 = a11 i + a12 j + a13 k, a2 = a21 i + a22 j + a23 k, a3 = a31 i + a32 j + a33 k, where a1 , a2 , a3 are linearly independent. Then,

1.3 Exterior Product

65

( ) V = x ∈ R3 | x = α1 a1 + α2 a2 + α3 a3 , 0 ≤ α1 , α2 , α3 ≤ 1 a spanned parallelepiped by vectors a1 , a2 , a3 . tween a1 , a2 , a3 as follows:   a11  a1 ∧ a2 ∧ a3 =  a21  a31

We introduce a new operation, ∧ bea12 a22 a32

a13 a23 a33

   .  

The geometric meaning of a1 ∧a2 ∧a3 is the orientable volume of V , where orientation means the sign of the volume is positive or negative. If the right hand law is followed, the volume has the plus sign, otherwise it has the minus sign . It is easy to see that operation ∧ satisfies the following laws: 1◦ Multilinear. Let a2 = βb + γc, b, c be vectors, β, γ be real numbers. Then, a1 ∧ (βb + γc) ∧ a3 = β(a1 ∧ b ∧ a3 ) + γ(a1 ∧ c ∧ a3 ). 2◦

Anti-commute a1 ∧ a2 ∧ a3 = −a2 ∧ a1 ∧ a3 , a1 ∧ a2 ∧ a3 = −a3 ∧ a2 ∧ a1 , a1 ∧ a2 ∧ a3 = −a1 ∧ a3 ∧ a2 .

From 2◦ we know that if a1 , a2 , a3 has two identical vectors, then a1 ∧a2 ∧a3 = 0. Example 3.1. Let e1 , e2 , e3 be a basis in R3 , which are not necessarily orthogonal, and let a1 , a2 , a3 be three vectors in R3 , which can be represented by a1 = a11 e1 + a12 e2 + a13 e3 , a2 = a21 e1 + a22 e2 + a23 e3 , a3 = a31 e1 + a32 e2 + a33 e3 . By multilinearity and anti-commutativity of ∧, after the computation, we have    a11 a12 a13    a1 ∧ a2 ∧ a3 =  a21 a22 a23  e1 ∧ e2 ∧ e3 .  a31 a32 a33  Example 3.2. Let e1 , e2 , e3 be a basis in R3 , and a1 = a11 e1 + a12 e2 + a13 e3 , a2 = a21 e1 + a22 e2 + a23 e3 . Then,

  a a1 ∧ a2 =  11 a21

   a a12  e ∧ e2 +  12 a22  1 a22

   a a13  e ∧ e3 +  13 a23  2 a23

 a11  e ∧ e1 . a21  3

The geometric significance of this formula is that the projection of the parallelepiped spanned by the pair of vectors a1 and a2 onto the coordinate plane e1 e2 , e2 e3 , e3 e1 is equal to A12 , A23 , and A31 respectively. Abstracting from the multilinearity and the anti-commutativity, which is satisfied by the operation wedge, we can obtain the concept of exterior product.

66

1. Preliminaries of Differentiable Manifolds

1.3.1 Exterior Form 1. 1- Form In this section, Rn is an n-dimensional real vector space, where the vectors are denoted by ξ, η, · · · ∈ Rn . Definition 3.3. A form of degree 1 (or a 1-form) on Rn is a linear function ω : Rn → R, i.e., ω(λ1 ξ1 + λ2 ξ2 ) = λ1 ω(ξ1 ) + λ2 ω(ξ2 ),

λ1 , λ2 ∈ R,

ξ1 , ξ2 ∈ Rn .

The set of all 1-forms on Rn is denoted by Λ1 (Rn ). For ω1 , ω2 ∈ Λ1 (Rn ), define (λ1 ω1 + λ2 ω2 )(ξ) = λ1 ω1 (ξ) + λ2 ω2 (ξ),

λ1 , λ2 ∈ R.

Then, Λ1 Rn becomes a vector space, i.e., the dual space (Rn )∗ of Rn . Let ⎡ ⎡ ⎤ ⎡ ⎤ ⎤ 1 0 0 ⎢ 0 ⎥ ⎢ 1 ⎥ ⎢ 0 ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎥ e1 = ⎢ . ⎥ , e2 = ⎢ . ⎥ , · · · , en = ⎢ . ⎥ ⎣ .. ⎦ ⎣ .. ⎦ ⎣ .. ⎦ 0 0 1 be the standard basis on Rn . x1 , x2 , · · · , xn forms the coordinate system on Rn , i.e., if ξ = a1 e1 + a2 e2 + · · · + an en , then xi (ξ) = ai , especially xi (ej ) = δij . Obviously, xi ∈ Λ1 (Rn ). For any ω ∈ Λ1 (Rn ), ω(ξ) = ω

n 

 ai ei =

i=1

n 

ai ω(ei ) =

i=1

i=1 

ω(ei )xi (ξ),

n

and so ω = ω(e1 )x1 + ω(e2 )x2 + · · · + ω(en )xn . Thus, x1 , · · · , xn is a basis on Λ1 (Rn ), Λ1 (Rn ) = {xi }i=1,···,n . Example 3.4. If F is a uniform force field on a Euclidean space R3 , then its work A on a displacement ξ is a 1-form acting on ξ, ωF (ξ) = (F, ξ) = F1 a1 + F2 a2 + F3 a3 ,

ξ = a1 e1 + a2 e2 + a3 e3

or ωF = F1 x1 + F2 x2 + F3 x3 . 2. 2-Forms Definition 3.5. An exterior form of degree 2 (or a 2-form) is a bilinear, skewsymmetric function ω 2 : Rn × Rn → R, i.e., ω 2 (λ1 ξ1 + λ2 ξ2 , ξ3 ) = λ1 ω 2 (ξ1 , ξ3 ) + λ2 ω 2 (ξ2 , ξ3 ), ω 2 (ξ1 , ξ2 ) = −ω 2 (ξ2 , ξ1 ),

ξ1 , ξ2 , ξ3 ∈ Rn ,

λ1 , λ2 ∈ R.

1.3 Exterior Product

67

The set of all 2-forms on Rn is denoted by Λ2 (Rn ) = Λ2 . Similarly, if we define the sum of two 2-forms ω12 , ω22 and scalar multiplication as follows (λ1 ω12 + λ2 ω22 )(ξ1 , ξ2 ) = λ1 ω12 (ξ1 , ξ2 ) + λ2 ω22 (ξ1 , ξ2 ), ω12 , ω22 ∈ Λ2 (Rn ),

λ1 , λ2 ∈ R,

then Λ2 (Rn ) becomes a vector space on R. Property 3.6. The skew-symmetric condition ω 2 (ξ1 , ξ2 ) − ω 2 (ξ2 , ξ1 ) is equivalent to ω 2 (ξ, ξ) = 0, ∀ ξ ∈ Rn since from the latter it follows that 0

= ω 2 (ξ1 + ξ2 , ξ1 + ξ2 ) = −ω 2 (ξ1 , ξ1 ) + ω 2 (ξ1 , ξ2 ) + ω 2 (ξ2 , ξ1 ) + ω 2 (ξ2 , ξ2 ) = ω 2 (ξ1 , ξ2 ) + ω 2 (ξ2 , ξ1 ).

i.e., ω 2 (ξ1 , ξ2 ) = ω 2 (ξ2 , ξ1 ). Example 3.7. Let S(ξ1 , ξ2 ) be the oriented area of the parallelogram constructed on the vector ξ1 and ξ2 of the oriented Euclidean plane R2 , i.e.,    ξ11 ξ12  ,  S(ξ1 , ξ2 ) =  ξ21 ξ22  where ξ1 = ξ11 e1 + ξ12 e2 ,

ξ2 = ξ21 e1 + ξ22 e2 .

Example 3.8. Let v be a given vector on the oriented Euclidean space R3 . The triple scalar product on other two vectors ξ1 and ξ2 is a 2-form:    v1 v2 v3    ω(ξ1 , ξ2 ) = (v, [ξ1 , ξ2 ]) =  ξ11 ξ12 ξ13  ,  ξ21 ξ22 ξ23  where v =

3  i=1

vi eji , ξj =

3 

ξji ei (j = 1, 2).

i=1

3. k-Forms We denote the set of all permutations of the set {1, 2, · · · , k} by Sk and its element by σ = {σ(1), σ(2), · · · , σ(k)} = {i1 , i2 , · · · , ik } ∈ νk ,  1, if σ ∈ νk is even, ε(σ) = −1, if σ ∈ νk is odd.

68

1. Preliminaries of Differentiable Manifolds

Definition 3.9. An exterior form of degree k (or a k-form) is a function of k vectors that is k-linear and skew-symmetric: ω(λ1 ξ1 + λ2 ξ1 , ξ2 , · · · , ξk ) = λ1 ω(ξ1 , ξ2 , · · · , ξk ) + λ2 ω(ξ1 , ξ2 , · · · , ξk ), ω(ξi1 , ξi2 , · · · , ξik ) = ε(σ)ω(ξ1 , ξ2 , · · · , ξk ), ξ1 , ξ1 , ξ2 , · · · , ξk ∈ Rn ,

λ1 , λ2 ∈ R,

σ ∈ νk ,

where σ = (i1 , i2 , · · · , ik ) ∈ Sk . Example 3.10. The oriented volume of the parallelepiped with edges ξ1 , ξ2 , · · · , ξn in the oriented Euclidean space Rn is an n-form.    ξ11 · · · ξ1n     ξ21 · · · ξ2n    V (ξ1 , ξ2 , · · · , ξn ) =  . ..  ,  .. .    ξn1 · · · ξnn  where ξi = ξi1 e1 + · · · + ξin en . The set of all k-forms in Rn is denoted by Λk (Rn ). It forms a real vector space if we introduce operations of addition. (λ1 ω1k + λ2 ω2k )(ξ1 , ξ2 , · · · , ξk ) = λ1 ω1k (ξ1 , ξ2 , · · · , ξk ) + λ2 ω2k (ξ1 , ξ2 , · · · , ξk ), ω1k , ω2k ∈ Λk (Rn ),

λ1 , λ2 ∈ R.

it Question 3.11. Show that if ηj =

k 

aji ξi (j = 1, · · · , k), then

i=1

ω k (η1 , η2 , · · · , ηk ) = det (aji )ω k (ξ1 , ξ2 , · · · , ξk ).

1.3.2 Exterior Algebra 1. The exterior product of two 1-forms In the previous section, we have defined various exterior forms. We now introduce one more operation: exterior multiplication of forms. As a matter of fact, these forms can be generated from the 1-forms by an operation called exterior product. Definition 3.12. For ω1 and ω2 ∈ Λ1 (Rn ), the exterior product of ω1 and ω2 denoted by ω1 ∧ ω2 is defined by the formula    ω1 (ξ1 ) ω2 (ξ1 )   , ξ1 , ξ2 ∈ Rn ,  (ω1 ∧ ω2 )(ξ1 , ξ2 ) =  ω1 (ξ2 ) ω2 (ξ2 ) 

1.3 Exterior Product

69

which denotes the oriented area of the image of the parallelogram with sides ω(ξ1 ) and ω(ξ2 ) on the ω1 , ω2 plane. It is not hard to verify that ω1 ∧ ω2 really is a 2-form and has properties ω1 ∧ ω2 = −ω2 ∧ ω1 , (λ1 ω1 + λ2 ω1 ) ∧ ω2 = λ1 ω1 ∧ ω2 + λ2 ω1 ∧ ω2 . Now suppose we have chosen a system of linear coordinates on Rn , i.e., we are given n independent 1-forms, x1 , x2 , · · · , xn . We will call these forms basic. The exterior products of the basic forms are the 2-forms xi ∧ xj . By skew-symmetry, xi ∈ Λ1 (Rn ), xi ∧ xi = 0, xi ∧ xj = −xj ∧ xi ,   xi (ξ1 ) xj (ξ1 )    (xi ∧ xj )(ξ1 , ξ2 ) =  xi (ξ2 ) xj (ξ2 )     ai aj   = ai bj − aj bi ,  = bi bj    ai ei , ξ2 = bi ei . It is the oriented area of the parallelogram with where ξ1 = i

i

sides (xi (ξ1 ), xi (ξ2 )) and (xj (ξ1 ), xj (ξ2 )) in the (xi , xj )-plane. For any ω ∈ Λ2 (Rn ), ω(ξ1 , ξ2 ) = =

n 

ω(ai ei , bj ej ) =

i,j=1 

n 

(ai bj − aj bi )ω(ei , ej ) =

i 1. By taking N = 2, we get

(1.43) (1.44)

14.1 Total Variation in Lagrangian Formalism

d S = θL− + Φ∗ θL+ .

589

(1.45)

By exterior differentiation of (1.45), we obtain Φ∗ (d θL+ ) = −d θL+ .

(1.46)

From the definition of θL− and θL+ , we know that θL− + θL+ = d (L(tk+1 − tk )).

(1.47)

By exterior differentiation of (1.47), we obtain d θL+ = −d θL− . Define ΩL ≡ d θL+ = −d θL− .

(1.48)

Finally, we have shown that the discrete flow Φ preserves the discrete extended Lagrange 2-form ΩL Φ∗ (ΩL ) = ΩL . (1.49) Now, the variational integrator (1.35), the discrete energy conservation law (1.36), and the discrete extended Lagrange 2-form ΩL converge to their continuous counterparts as tk+1 → tk , tk−1 → tk . Consider a conservative Lagrangian L(q, q). ˙ For simplicity, we choose the discrete Lagrangian as % & qk+1 − qk L(tk , qk , tk+1 , qk+1 ) = L qk , . (1.50) tk+1 − tk The variational integrator (1.35) becomes % ∂L 1 ∂L (qk , Δt qk ) − (qk , Δt qk ) − ∂ qk

tk+1 − tk

∂ Δ t qk

&

∂L (qk−1 , Δt qk−1 ) ∂ Δt qk−1

q − qk q − qk−1 , Δt qk−1 = k . where Δt qk = k+1 tk+1 − tk tk − tk−1

= 0, (1.51)

It is easy to see that, as tk+1 → tk , tk−1 → tk , the Equation (1.51) converges to d ∂L ∂L − = 0. ∂ qk d t ∂ q˙k

(1.52)

The discrete energy conservation law (1.36) becomes Ek+1 − Ek = 0, tk+1 − tk where

& % ∂L qk+1 − qk , Δt qk − L q k , Ek+1 = ∂Δt qk tk+1 − tk % & ∂L qk − qk−1 Δt qk−1 − L qk−1 , . Ek = ∂Δt qk−1 tk − tk−1

The Equation (1.53) converges to

(1.53)

590

14. Lee-Variational Integrator

d dt

%

& ∂L q˙k − L = 0 ∂ q˙k

(1.54)

as tk+1 → tk , tk−1 → tk . Now, we will consider the discrete extended Lagrange 2-form ΩL defined by (1.48). By discretization of (1.50), the discrete extended Lagrange 1-form θL+ defined in (1.42) becomes & % ∂L ∂L + Δt qk d tk+1 + d qk+1 . (1.55) θL = L(qk , Δt qk ) − ∂ Δ t qk ∂ Δt q k From (1.55), we can deduce that θL+ converges to the continuous Lagrangian 1-form ϑL defined by (1.29) as tk+1 → tk , tk−1 → tk . Thus, we obtain ΩL = dθL+ −→ dϑL = ΩL ,

tk+1 → tk ,

tk−1 → tk .

(1.56)

In general, the variational integrator (1.35) with fixed time steps does not exactly conserve the discrete energy, and the computed energy will not have secular variation[GM88,SSC94] . In some cases, such as in discrete mechanics proposed by Lee in [Lee82,Lee87] , the integrator (1.35) is required to conserve the discrete energy (1.36) by varying the time steps. In other words, the steps can be chosen according to (1.36) so that the integrator (1.35) conserves the discrete energy (1.36). The resulting integrator also conserves the discrete extended Lagrange 2-form dθL+ . This fact had not been discussed in Lee’s discrete mechanics. Example 1.3. Let us consider an example. For the classical Lagrangian 1 L(t, q, q) ˙ = q˙2 − V (q), 2 we choose the discrete Lagrangian L(tk , qk , tk+1 , qk+1 ) as % & % &2 1 qk+1 − qk qk+1 − qk L(tk , qk , tk+1 , qk+1 ) = . −V 2 tk+1 − tk 2

(1.57)

(1.58)

The discrete Euler–Lagrange equation (1.35) becomes % & V  (¯ qk − qk−1 qk )(tk+1 − tk ) + V  (¯ qk−1 )(tk − tk−1 ) qk+1 − qk = 0, + − tk+1 − tk tk − tk−1 2 (1.59) which preserves the Lagrange 2-form & % 1 t − tk  + k+1 V (¯ qk ) d qk+1 ∧ d qk , (1.60) tk+1 − tk

q +q

4

q

+q

k+1 k , q¯k−1 = k−1 . where q¯k = k 2 2 If we take fixed variables tk+1 − tk = tk − tk−1 = h, then (1.59) becomes

qk+1 − 2qk + qk−1 qk ) + V  (¯ qk−1 ) V  (¯ + = 0, 2 h 2 which preserves the Lagrange 2-form % & 1 h  + V (¯ qk ) d qk+1 ∧ d qk . h 4

14.2 Total Variation in Hamiltonian Formalism

591

14.1.4 Concluding Remarks We have presented the calculus of total variation problem for discrete mechanics with variable time steps referring to continuous mechanics in this section. Using the calculus for discrete total variations, we have proved that Lee’s discrete mechanics is symplectic and derived Kane–Marsden–Ortiz integrators. It is well known that an energy-preserving variational integrator is a more preferable and natural candidate of approximations for conservative Euler–Lagrange equation, since the solution of conservative Euler–Lagrange equation is not only symplectic but also energy-preserving. As is mentioned, Kane–Marsden–Ortiz integrators are related closely to the discrete mechanics proposed by Lee[Lee82,Lee87] . In Lee’s discrete mechanics, the difference equations are the same as Kane–Marsden–Ortiz integrators. However, Lee’s difference equations are solved as boundary value problems, while Kane–Marsden–Ortiz integrators are solved as initial value problems. Finally, it should be mentioned that in very recent works[GLW01a,GLWW01,GW03] , two of the authors (HYG and KW) and their collaborators have presented a difference discrete variational calculus and the discrete version of Euler–Lagrange cohomology for vertical variation problems in both Lagrangian and Hamiltonian formalism for discrete mechanics and field theory. In their approach, the difference operator with fixed step-length is regarded as an entire geometric object. The advantages of this approach have already been seen in the last subsection in the course of taking continuous limits although the difference operator Δt in (1.50) is of variable step-length. This approach may be generalized to the discrete total variation problems.

14.2 Total Variation in Hamiltonian Formalism We present a discrete total variation calculus in Hamiltonian formalism in this section. Using this discrete variation calculus and generating function for flows of Hamiltonian systems, we derive symplectic-energy integrators of any finite order for Hamiltonian systems from a variational perspective. The relationship between the symplectic integrators derived directly from the Hamiltonian systems and the variationally derived symplectic-energy integrators is explored.

14.2.1 Variational Principle in Hamiltonian Mechanics Let us begin by recalling the ordinary variational principle in Hamiltonian formalism. Suppose Q denotes the configuration space with coordinates q i , and T ∗ Q the phase space with coordinates (q i , pi ) (i = 1, 2, · · · , n). Consider a Hamiltonian H : T ∗ Q → R. The corresponding action functional is defined by -

b

(pi · q i − H(q i , pi )) d t,

S((q i (t), pi (t))) = a

where (q i (t), pi (t)) is a C 2 curve in phase space T ∗ Q.

(2.1)

592

14. Lee-Variational Integrator

The variational principle in Hamiltonian formalism seeks the curves (q i (t), pi (t)) for which the action functional S is stationary under variations of (q i (t), pi (t)) with fixed end points. We will first define the variation of (q i (t), pi (t)). Let n n   ∂ ∂ φi (qq , p ) i + ψ i (qq , p ) i , (2.2) V = ∂q

i=1

i=1

∂p

n ). For simplicity, we be a vector field on T ∗ Q, here q = (q 1 , · · · , q n ), p = (p1 , · · · , pC will use Einstein convention and omit the summation notation in the following. q˜, p˜), which is written in compoLet us denote the flow of V by F ε : F ε (qq , p ) = (˜ nents as

q˜i = f i (ε, q , p ), p˜i = g i (ε, q , p ), where (qq , p ) ∈ T ∗ Q and

(2.3) (2.4)



d  f i (ε, q , p ) = φi (qq , p ),  d ε ε=0  d g i (ε, q , p ) = ψ i (qq , p ).  dε ε=0

Let (q i (t), pi (t)) be a curve in T ∗ Q. The transformation (2.3) and (2.4) transforms (q i (t), pi (t)) into a family of curves     i q˜ (t), p˜i (t) = f i (ε, q (t), p (t)), g i (ε, q (t), p (t)) .   Next, we will define the variation of q i (t), pi (t) :        d  (2.5) q˜i (t), p˜i (t) = φi (qq , p ), ψ i (qq , p ) . δ q i (t), pi (t) =:  dε

ε=0

  Next, we will calculate the variation of S at q i (t), pi (t) as follows:    d  S (˜ q i (t), p˜i (t))  d ε ε=0    d  i i = S (f (ε, q (t), p (t)), g (ε, q (t), p (t))) dε ε=0  - b%   d   d  = g i ε, q (t), p (t) f i ε, q (t), p (t)  dε ε=0 a dt &  −H f i (ε, q (t), p (t)), g i (ε, q (t), p (t)) d t

δS =

-

b

%

% & & ! b ∂H ∂H  i i q˙ − ψ + − p˙ − φi d t + pi φi  . i i ∂p ∂q a i

= a

(2.6)

  i  If φi q (a), p (a)  i φ qi(b),p (b) = 0, the requirement of δS = 0 yields the Hamilton equation for q (t), p (t) :

14.2 Total Variation in Hamiltonian Formalism

q˙i =

∂H , ∂ pi

p˙i = −

∂H . ∂ qi

593

(2.7)

    If we drop the requirement of φi q (a), p (a) φi q (b), p (b) = 0, we can naturally ∗ i i obtain the canonical 1-form  from the second term in (2.6): θ = p dq . Fur i oni T Q thermore, restricting (˜ q (t), p˜ (t)) to the solution space of (2.7), we can prove that the solution of (2.7) preserves the canonical 2-form ω = d θL = d pi ∧ d q i . On the other hand, it is not necessary to restrict ((˜ q i (t), p˜i (t)) to the solution space of (2.7). Introducing the Euler–Lagrange 1-form     ∂H ∂H E(q i , pi ) = q˙i − (2.8) d pi + − p˙i − d qi , i i ∂p

∂q

the nilpotency of d leads to d E(q i , pi ) +

d ω = 0, dt

(2.9)

namely, the necessary and sufficient condition for symplectic structure preserving is that the Euler–Lagrange 1-form (2.8) is closed[GLW01a,GLWW01,GLW01b,GW03] . Based on the above-given variational principle in Hamiltonian formalism and using the ideas of discrete Lagrange mechanics[Ves88,Ves91b,MPS98,WM97] , we can develop a natural version of discrete Hamilton mechanics with fixed time steps and derive symplectic integrators for Hamilton canonical equations from a variational perspective[GLWW01] . However, the symplectic integrators obtained in this way are not energy-preserving, in general, because of its fixed time steps[GM88] . An energy-preserving symplectic integrator is a more preferable and natural candidate of approximations for conservative Hamilton equations since the solution of conservative Hamilton equations is not only symplectic but also energy-preserving. To attain this goal, we use variable time steps and a discrete total variation calculus developed in [Lee82,Lee87,KMO99,CGW03] . The basic idea is to construct a discrete action functional with variable time steps and then apply a discrete total variation calculus. In this way, we can derive symplectic integrators and their associated energy conservation laws. These variationally derived symplectic integrators are two-step integrators. If we take fixed time steps, the resulting integrators are equivalent to the symplectic integrators derived directly from the Hamiltonian systems in some special cases.

14.2.2 Total Variation in Hamiltonian Mechanics In order to discuss total variation in Hamiltonian formalism, we will work with extended phase space R × T ∗ Q with coordinates (t, q i , pi ). Here t denotes time. For details, see [Arn89,GPS02] . By total variation, we refer to variations of both (q i , pi ) and t. Consider a vector field on R × T ∗ Q, V = ξ(t, q , p )

∂ ∂ ∂ + φi (t, q , p ) i + ψ i (t, q , p ) i . ∂t ∂q ∂p

(2.10)

594

14. Lee-Variational Integrator

Let F ε be the flow of V . For (t, q i , pi ) ∈ R × T ∗ Q, we have F ε (t, q i , pi ) = (t˜, q˜i , p˜i ): t˜ = h(ε, t, q , p ), q˜i = f i (ε, t, q , p ), p˜i = g i (ε, t, q , p ), where

(2.11) (2.12) (2.13)



d   h(ε, t, q , p ) = ξ(t, q , p ), d ε ε=0  d   f i (ε, t, q , p) = φi (t, q , p), d ε ε=0  d   g i (ε, t, q , p ) = ψ i (t, q , p ). d ε ε=0

(2.14) (2.15) (2.16)

The transformation (2.11) – (2.13) transforms a curve (q i (t), pi (t)) into a family of curves (˜ q i (ε, t˜), p˜i (ε, t˜)) determined by   t˜ = h ε, t, q (t), p (t) , (2.17)   i i q˜ = f ε, t, q (t), p (t) , (2.18)   i i p˜ = g ε, t, q (t), p (t) . (2.19) Suppose we can solve (2.17) for t : t = h−1 (ε, t˜). Then, q˜i (ε, t˜) = f i (ε, h−1 (ε, t˜), q (h−1 (ε, t˜)), p (h−1 (ε, t˜))), q˜i (ε, t˜) = f i (ε, h−1 (ε, t˜), q (h−1 (ε, t˜)), p (h−1 (ε, t˜))).

(2.20) (2.21)

Before calculating the variation of S directly, we will first consider the first-order prolongation of V , 1

pr V = ξ(t, q , p )

∂ ∂ ∂ ∂ ∂ i i i i + φ (t, q , p ) i + ψ (t, q , p ) i + α (t, q , p , ·q ·q, ·p ·p) i + β (t, q , p , qq, ˙ p p) ˙ , ∂t ∂q ∂p ∂ q˙ ∂ p˙ i (2.22)

where pr1 V denotes the first-order prolongation of V and αi (t, q , p , qq, ˙ pp) ˙ = Dt φi (t, q , p ) − q˙i Dt ξφi (t, q , p ), ˙ pp) ˙ = Dt ψ i (t, q , p ) − p˙i Dt ξφi (t, q , p ), β i (t, q , p , qq, where Dt denotes the total derivative. For example, Dt φi (t, q , p ) = φit + φqq˙ + φp pp. ˙ For prolongation of vector field and formulae (2.23) and (2.24), refer to[Olv93] . Now, let us calculate the variation of S directly as follows:  i  d  δS = q (ε, t˜), p˜i (ε, t˜))  S (˜ d ε ε=0 - ˜b   i d i   d  = p˜ (ε, t˜) q˜ (ε, t˜) − H q˜i (ε, t˜), p˜i (ε, t˜) d t˜  d ε ε=0 a˜ dt˜

(2.23) (2.24)

14.2 Total Variation in Hamiltonian Formalism

=

d   d ε ε=0 -

b

= a

-

=



 i  d t˜ d dt p˜i (ε, t˜) q˜i (ε, t˜) − H (˜ q (ε, t˜), p˜i (ε, t˜) dt d t˜ a    t˜ = h ε, t, q (t), p (t)     d d  p˜i (ε, t˜) q˜i (ε, t˜) − H q˜i (ε, t˜), p˜i (ε, t˜) d t  d ε ε=0 d t˜ b

+ -

-

595

b

  pi (t)q˙i (t) − H q i (t), pi (t) Dt ξ d t



(2.25)

a

 d    ∂ H  i  i ∂ H  i H q i (t), pi (t) ξ + − p˙i − φ + q˙ − ψ dt dt ∂ qi ∂ pi a Ab @ (2.26) + pi φi − H(q i , pi )ξ  . b

a

In (2.25), we have used (2.14)and the fact d  d d  ˜ d t˜ =   t = Dt ξ. d ε ε=0 d t d t d ε ε=0 In (2.26),  we have used  theprolongation formula (2.23).     If ξ a, q (a), p (a) = ξ b, q (b), p (b) = 0 and φi a, q (a), p (a) = φi b, q (b), p (b) = 0, the requirement of δS = 0 yields the Hamilton canonical equation q˙i =

∂H , ∂pi

p˙i = −

∂H ∂q i

(2.27)

from the variation φi , ψ i and the energy conservation law d H(q i , pi ) = 0 dt

(2.28)

from the variation ξ. Since d ∂H i ∂H i H(q i , pi ) = ·q + ·p , dt ∂ qi ∂ pi we can very well see that the energy conservation law (2.28) is a natural consequence of the Hamilton canonical Equation (2.27). If we drop the requirement     ξ a, q (a), p (a) = ξ b, q (b), p (b) = 0,     φi a, q (a), p (a) = φi b, q (b), p (b) = 0, we can define the extended canonical 1-form on R × T ∗ Q from the second term in (2.26) (2.29) θ = pi d q i − H(q i , pi )d t.  i  i Furthermore, restricting q˜ (t), p˜ (t) to the solution space of (2.27), we can prove that the solution of (2.27) preserves the extended canonical 2-form ω = d θ = d pi ∧ d q i − d H(q i , pi ) ∧ d t by using the same method in[MPS98] .

(2.30)

596

14. Lee-Variational Integrator

14.2.3 Symplectic-Energy Integrators In this section, we will develop a discrete version of total variation in Hamiltonian formalism. Using this discrete total variation calculus, we will derive symplectic-energy integrators. Let L(q i , pi , q˙i , p˙i ) = pi q˙i − H(q i , pi ) be a function from R × T (T ∗ Q) to R. Here L does not depend on t explicitly. We use P × P for the discrete version R × T (T ∗ Q). Here P is the discrete version of R × T ∗ Q. A point (t0 , q 0 , p 0 ; t1 , q 1 , p 1 ) ∈ P × P corresponds to a tangent vector q − q p − p  1 0 1 0 , . t1 − t0 t0 − t0 For simplicity, the vector symbols q = (q 1 , · · · , q n ) and p = (p1 , · · · , pn ) are used throughout this section. A discrete L is defined to be L : P × P → R and the corresponding discrete action as S=

N −1 

L(tk , q k , p k , tk+1 , q k+1 , p k+1 )(tk+1 − tk ),

(2.31)

k=0

where t0 = a, tN = b. The discrete variational principle in total variation is to extremize S for variations of both q k , p k and tk holding the end points (t0 , q 0 , p 0 ) and (tN , q N , p N ) fixed. This discrete variational principle determines a discrete flow Φ : P × P → P × P by Φ(tk−1 , q k−1 , p k−1 , tk , q k , p k ) = (tk , q k , p k , tk+1 , q k+1 , p k+1 ).

(2.32)

Here, (tk+1 , q k+1 , p k+1 ) for all k ∈ (1, 2, · · · , N − 1) are found from the following discrete Hamilton canonical equation (tk+1 − tk )D2 L(tk , q k , p k , tk+1 , q k+1 , p k+1 ) + (tk − tk−1 )D5 L(tk−1 , q k−1 , p k−1 , tk , q k , p k ) = 0, (2.33) (tk+1 − tk )D3 L(tk , q k , p k , tk+1 , q k+1 , p k+1 ) + (tk − tk−1 )D6 L(tk−1 , q k−1 , p k−1 , tk , q k , p k ) = 0

and the discrete energy conservation law (tk+1 −tk )D1 L(tk , q k , p k , tk+1 , q k+1 , p k+1 )+(tk −tk−1 )D4 L(tk−1 , q k−1 , p k−1 , tk , q k , p k ) (2.34) −L(tk , q k , p k , tk+1 , q k+1 , p k+1 ) + L(tk−1 , q k−1 , p k−1 , tk , q k , p k ) = 0.

Di denotes the partial derivative of L with respect to the ith argument. Equation (2.33) is the discrete Hamilton canonical equation (variational integrator). Equation (2.34) is the discrete energy conservation law associated with (2.33). Unlike the continuous case, the variational integrator (2.33) does not satisfy (2.34) for arbitrarily given tk+1 in general. Therefore, we need to solve (2.33) and (2.34) simultaneously with qk+1 , pk+1 and tk+1 taken as unknowns. Now, we will prove that the discrete flow determined by (2.33) and (2.34) preserves a discrete version of the extended Lagrange 2-form ω defined in (2.30) so that

14.2 Total Variation in Hamiltonian Formalism

597

we call (2.33) and (2.34) a symplectic-energy integrator. We will do this directly from the variational point of view, consistent with the continuous case[MPS98] . As in the continuous case, we will calculate dS for variations with varied end points. dS(t0 , q 0 , p 0 , · · · , tN , q N , p N ) · (δt0 , δqq 0 , δpp0 , · · · , δtN , δqq N , δppN ) & N −1 %  = D2 L(vv k )δqq k + D5 L(vv k )δqq k+1 + D3 L(vv k )δppk + D6 L(vv k )δppk+1 (tk+1 − tk ) k=0

+

N −1  

N −1   L(vv k )(δtk+1 − δtk ) D1 L(vv k )δtk + D4 L(vv k )δtk+1 (tk+1 − tk ) +

k=0

=

N −1 % 

&

D2 L(vv k )(tk+1 − tk ) + D5 L(vv k−1 )(tk − tk−1 δqq k

k=1

+

k=0

N −1 % 

& D3 L(vv k )(tk+1 − tk ) + D6 L(vv k−1 )(tk − tk−1 δppk

k=1

+

N −1  

 D1 L(vv k )(tk+1 − tk ) + D4 L(vv k−1 )(tk − tk−1 ) + L(vv k−1 ) − L(vv k ) δtk

k=1

  +D2 L(vv 0 )(t1 − t0 )δqq 0 + D3 L(vv 0 )(t1 − t0 )δp0 + D1 Lvv 0 )(t1 − to ) − L(vv 0 ) δt0 +D5 L(vv N −1 )(tN − tN −1 )δqq N + D6 L(vv N −1 )(tN − tN −1 )δppN   + D4 L(vv N −1 )(tN − tN −1 ) − L(vv N −1 ) δtN , (2.35)

where v k = (tk , q k , p k , tk+1 , q k+1 , p k+1 ) (k = 0, 1, · · · , N − 1). We can see that the last six terms in (2.35) come from the boundary variations. Based on the boundary variations, we can define two 1-forms on P × P , θL− (vv k ) = D2 L(vv k )(tk+1 − tk )dqq k + D3 L(vv k )(tk+1 − tk )dppk +D1 L(vv k )(tk+1 − tk ) − L(vv k )dtk

(2.36)

and θL+ (vv k ) = D5 L(vv k )(tk+1 − tk )dqq k+1 + D6 L(vv k )(tk+1 − tk )dppk+1 +D4 L(vv k )(tk+1 − tk ) − L(vv k )dtk+1 . (2.37) Here, we have used the notation in [MPS98] . We regard the pair (θL− , θL+ ) as being the discrete version of the extended canonical 1-form θ defined in (2.29). Now, we will parametrize the solutions of the discrete variational principle by (t0 , q0 , t1 , q1 ), and restrict S to that solution space. Then, Equation (2.35) becomes d S(t0 , q 0 , p 0 , · · · , tN , q N , p N ) · (δt0 , δqq 0 , δpp0 , · · · , δtN , δqq N , δppN ) = θL− (t0 , q 0 , p 0 , t1 , q 1 , p 1 ) · (δt0 , δqq 0 , δpp0 , δt1 , δqq 1 , δpp1 ) +θL+ (tN −1 , q N −1 , p N −1 , tN , q N , p N ) · (δtN −1 , δqq N −1 , δppN −1 , δtN , δqq N , δppN ) = θL− (t0 , q 0 , p 0 , t1 , q 1 , p 1 ) · (δt0 , δqq 0 , δpp0 , δt1 , δqq 1 , δpp1 ) (2.38) +(ΦN −1 )∗ θL+ (t0 , q 0 , p 0 , t1 , q 1 , p 1 ) · (δt0 , δqq 0 , δpp0 , δt1 , δqq 1 , δpp1 ).

598

14. Lee-Variational Integrator

From (2.38), we can obtain d S = θL− + (ΦN −1 )∗ θL+ .

(2.39)

The Equation (2.39) holds for arbitrary N > 1. Taking N = 2, we obtain d S = θL− + Φ∗ θL+ .

(2.40)

By exterior differentiation of (2.40), we obtain Φ∗ (d θL+ ) = −d θL− .

(2.41)

From the definition of θL− and θL+ , we know that θL− + θL+ = d L.

(2.42)

By exterior differentiation of (2.42), we obtain dθL+ = −d θL− . Define ωL ≡ d θL+ = −d θL− .

(2.43)

Finally, we have shown that the discrete flow Φ preserves the discrete extended canonical 2-form ωL : (2.44) Φ∗ (ωL ) = ωL . We can now call the coupled difference system (2.33) and (2.34) a symplecticenergy integrator in the sense that it satisfies the discrete energy conservation law (2.34) and preserves the discrete extended canonical 2-form ωL . To illustrate the above-mentioned discrete total variation calculus, we present an example. We choose L in (2.31) as L(tk , q k , p k , tk+1 , q k+1 , p k+1 ) = p k+1/2

q k+1 − q k − H(qq k+1/2 , p k+1/2 ), tk+1 − tk

(2.45)

where

p k + p k+1 q + q k+1 , q k+1/2 = k . 2 2 Using (2.33), we can obtain the corresponding discrete Hamilton equation p k+1/2 =

q k+1 − q k−1 2 p k+1 − p k−1 2

+

*

− +

1 ∂H ∂H = 0, (tk+1 − tk ) (qq ,p ) + (tk − tk−1 ) (qq ,p 2 ∂pp k+1/2 k+1/2 ∂pp k+1/2 k+1/2 * + 1 ∂H ∂H (tk+1 − tk ) (qq k+1/2 , p k+1/2 ) + (tk − tk−1 ) (qq k+1/2 , p k+1/2 ) = 0, 2 ∂qq ∂qq p p +p

q +qq

(2.46)

where p k−1/2 = k 2 k−1 , q k−1/2 = k 2 k−1 . Using (2.34), we can obtain the corresponding discrete energy conservation law     H q k+1/2 , p k+1/2 = H q k+1/2 , p k+1/2 . (2.47) The symplectic-energy integrator (2.46) and (2.47) preserves the discrete 2-form:

14.2 Total Variation in Hamiltonian Formalism

599

    d tk + dtk+1  1 d p k ∧ dqq k+1 + dppk+1 ∧ dqq k − H q k+1/2 , p k+1/2 ∧ . (2.48) 2 2 If we take fixed time steps tk+1 − tk = h (h is a constant), then (2.46) becomes   ∂H  q k+1 − q k−1 1 ∂H q k+1/2 , p k+1/2 + q k−1/2 , p k−1/2 , = 2h 2 ∂p ∂p  (2.49)  p k+1 − p k−1 1 ∂H ∂H =− (qq k−1/2 , p k−1/2 . q k+1/2 , p k+1/2 ) + 2h

2

∂q

∂q

Now, we will explore the relationship between (2.49) and the midpoint integrator for the Hamiltonian system ∂H , q˙ = ∂pp (2.50) ∂H . p˙ = − ∂qq

The midpoint symplectic integrator for (2.50) is  q k+1 − q k ∂H = ,p q , h ∂ p k+1/2 k+1/2  p k+1 − p k ∂H  q . = − ,p h ∂qq k+1/2 k+1/2

(2.51)

Replacing k by k − 1 in (2.51), we obtain  q k − q k−1 ∂H  q = ,p , h ∂pp k−1/2 k−1/2   p k − p k−1 ∂H q k−1/2 , p k−1/2 . = − h ∂qq

(2.52)

Adding (2.52) to (2.51) results in (2.49). Therefore, if we use (2.51) to obtain p k , q k , the two-step integrator (2.49) is equivalent to the midpoint integrator (2.51). However, the equivalence does not hold in general. For example, choose L in (2.31) as     q − qk − H q k+1/2 , p k+1/2 , L tk , q k , p k , tk+1 , q k+1 , p k+1 = p k k+1 tk+1 − tk and take fixed time steps tk+1 − tk = h. Then (2.33) becomes   ∂H  q k+1 − q k 1 ∂H  = q k+1/2 , p k+1/2 + q k−1/2 , p k−1/2 , h 2 ∂p ∂pp   ∂H   p k − p k−1 1 ∂H = − q k+1/2 , p k+1/2 + q k−1/2 , p k−1/2 . h

2

∂q

(2.53)

(2.54)

∂qq

The integrator (2.54) is a two-step integrator which preserves dpk ∧dqk+1 . In this case, we cannot find an one-step integrator which is equivalent to (2.54). In conclusion, using discrete total variation calculus, we have derived two-step symplectic-energy integrators. When taking fixed time steps, some of them are equivalent to one-step integrators derived directly from the Hamiltonian system while the others do not have this equivalence.

600

14. Lee-Variational Integrator

14.2.4 High Order Symplectic-Energy Integrator In this subsection, we will develop high order symplectic-energy integrators by using the generating function of the flow of the Hamiltonian system z˙ = J∇H(zz ), where z = (pp, q )T ,

J=

 O I

(2.55) −I  . O

Let us first recall the generating function with normal Darboux matrix of a symplectic transformation. For details, see Chapters 5 and 6, or [Fen86,FWQW89] . Suppose α is a 4n × 4n nonsingular matrix with the form * + A B α= , C D where A, B, C and D are both 2n × 2n matrices. We denote the inverse of α by + * A1 B1 −1 , α = C1 D1 where A1 , B1 , C1 and D1 are both 2n × 2n matrices. We know that a 4n × 4n matrix α is a Darboux matrix if αT J4n α = J"4n , (2.56) where J4n =

*

O I2n

−I2n O

*

+ ,

J˜4n =

J2n O

O −J2n

*

+ ,

J2n =

O In

−In O

+ ,

where In is an n × n identity matrix and I2n is a 2n × 2n identity matrix. Every Darboux matrix induces a fractional transform between symplectic and symmetric matrices σα : Sp(2n) −→ Sm(2n), σα = (AS + B)(CS + D)−1 = M,

for

S ∈ Sp(2n), det (CS + D) = 0

with the inverse transform σα−1 = σα−1 σα−1 : Sm(2n) −→ Sp(2n), σα = (A1 M + B1 )(C1 M + D1 )−1 = S, where Sp(2n) is the group of symplectic matrices and Sm(2n) the set of symmetric matrices.

14.2 Total Variation in Hamiltonian Formalism

601

We can generalize the above discussions to nonlinear transformations on R2n . Let us denote the set of symplectic transformations on R2n by SpD2n and the set of symmetric transformations (i.e., transformations with symmetric Jacobian) on R2n by Symm(2n). Every f ∈ Symm(2n) corresponds, at least locally, to a real function φ (unique to a constant) such that f is the gradient of φ, w ) = ∇φ(w w ), f (w   w ) = φw1 (w w ), · · · , φw2n (w w ) and w = (w1 , w2 , · · · , w2n ). where ∇φ(w Then, we have

(2.57)

σα : SpD2n −→ Symm(2n), σα = (A ◦ g + B) ◦ (C ◦ g + D)−1 = ∇φ, for g ∈ SpD2n , det(Cgz + D) = 0 or alternatively Ag(zz ) + Bzz = (∇φ)(Cg(zz ) + Dzz ), where ◦ denotes the composition of transformation and the 2n × 2n constant matrices A, B, C and D are regarded as linear transformations. gz denotes the Jacobian of symplectic transformation g. Let φ be the generating function of Darboux type α for symplectic transformation g. Conversely, we have σα−1 : Symm (2n) −→ SpD2n , σα−1 (∇φ) = (A1 ◦ ∇φ + B1 ) ◦ (C1 ◦ ∇φ + D1 )−1 = g, for det(C1 φww + D1 ) = 0, or alternatively w ) + B1w = g(C1 ∇φ(w w ) + D1w ), A1 ∇φ(w where g is called the symplectic transformation of Darboux type α for the generating function φ. For the study of integrators, we will restrict ourselves to normal Darboux matrices, i.e., those satisfying A + B = 0, C + D = I2n . The normal Darboux matrices can be characterized as + * J2n −J2n 1 (2.58) , E = (I2n + J2n F ), F T = F, α= E I2n − E 2 *

and α−1 =

(E − I2n )J2n EJ2n

I2n I2n

+ .

(2.59)

The fractional transform induced by a normal Darboux matrix establishes a oneone correspondence between symplectic transformations near identity and symmetric transformations near nullity. 1 For simplicity, we will take F = 0, then E = I2n and 2

602

14. Lee-Variational Integrator

* α=

J2n

−J2n

1 I2n 2

1 I2n 2

+ .

(2.60)

Now, we will consider the generating function of the flow of (2.55) and denote it by w , t) for the flow etH of Darboux type (2.60) is given etH . The generating function φ(w by ∇φ = (J2n ◦ etH − J2n ) ◦

−1 1 etH + I2n , 2 2

1

|t|,

for small

(2.61)

w , t) satisfies the Hamilton–Jacobi equation where φ(w   1 ∂ w , t) w , t) = −H w + J2n ∇φ(w φ(w ∂t 2

(2.62)

and can be expressed by Taylor series in t, w , t) = φ(w

∞ 

φk (w)tk ,

for small

|t|.

(2.63)

k=1

The coefficients φk (w) can be determined recursively as w) φ1 (w

= −H(w), k −1  1 w) = φk+1 (w k + 1 m=1 m !

%

& 1 1 j1 jm D H , (2.64) J2n ∇φ , · · · , J2n ∇φ 2 2



m

j1 + · · · + jm = k jl ≥ 1

where k ≥ 1, and we use the notation of the m-linear form   1 1 Dm H J2n ∇φj1 , · · · , J2n ∇φjm 2 2 2n 1  1   w) · · · w) J2n ∇φjm (w := Hz i1 ···zz im (zz ) J2n ∇φj1 (w . 2 2 i1 im i ,···,i =1 1

m

From (2.61), we can see that the phase flow z := etH z satisfies J2n ( z − z ) = ∇φ

 z − z  2

=

∞  j=1

tj ∇φj

 z + z  2

.

(2.65)

Now, we will choose L in (2.31) as L(tk , q k , p k , tk+1 , q k+1 , p k+1 ) = p k+1/2 where ψ m (qq k+1/2 , p k+1/2 ) =

q k+1 − q k − ψ m (qq k+1/2 , p k+1/2 ), (2.66) tk+1 − tk

m  j=1

tj φj (qq k+1/2 , p k+1/2 ).

(2.67)

14.2 Total Variation in Hamiltonian Formalism

603

The corresponding symplectic-energy integrator (2.33) and (2.34) is q k+1 − q k−1 2 p k+1 − p k−1 2

−

+

1 2 1 2

*

+

(tk+1 − tk ) *

∂ψ m ∂ψ m (qq k+1/2 , p k+1/2 ) + (tk − tk−1 ) (qq ,p ) ∂pp ∂pp k−1/2 k−1/2

= 0, +

(tk+1 − tk )

∂ψ m ∂ψ m (qq ,p ) + (tk − tk−1 ) (qq ,p ) ∂qq k+1/2 k+1/2 ∂qq k−1/2 k−1/2

(2.68) = 0,

ψ m (qq k+1/2 , p k+1/2 ) = ψ m (qq k−1/2 , p k−1/2 ), which satisfies the discrete extended canonical 2-form   dt + dt 1 k k+1 (dppk ∧ dqq k+1 + dppk+1 ∧ dqq k ) − ψ m (qq k+1/2 , p k+1/2 ) ∧ . (2.69) 2 2 The integrator (2.68) is a two-step symplectic-energy integrator with 2m-th order of accuracy.

14.2.5 An Example and an Optimization Method In this subsection, we will see an example. We will take the Hamiltonian as H(q, p) =

1 2 1 4 p + (q − q 2 ), 2 2

(2.70)

where q and p are scalars. Corresponding to (2.70) the discrete Lagrangian (2.31) is chosen as L(tk , qk , pk , tk+1 , qk+1 , pk+1 ) = pk+1/2

qk+1 − qk 1 4 2 − (qk+1/2 − qk+1/2 ). (2.71) tk+1 − tk 2

The corresponding symplectic-energy integrator (2.33) and (2.34) become qk+1 − qk−1 1 − ((tk+1 − tk )pk+1/2 + (tk − tk−1 )pk−1/2 ) = 0, (2.72) 2 2 pk+1 − pk−1 1 3 3 + ((tk+1 − tk )(2qk+1/2 − qk+1/2 ) + (tk − tk−1 )(2qk−1/2 − qk−1/2 )) = 0, 2 2 1 2 1 4 1 2 1 4 2 2 p + (q − qk+1/2 ) = pk−1/2 + (qk−1/2 − qk−1/2 ), 2 2 2 k+1/2 2 k+1/2

where tk−1 , qk−1 , pk−1 and tk , qk , pk are given and tk+1 , qk+1 , pk+1 are unknowns. In the following numerical experiment, we will use a robust optimization method suggested in [KMO99] to solve (2.72). Concretely, let  A=

qk+1 −qk−1 2

B=

pk+1 −pk−1 2

−

1 2

(tk+1 − tk )pk+1/2 + (tk − tk−1 )pk−1/2 ,

 3 3 + 12 (tk+1 − tk )(2qk+1/2 − qk+1/2 ) + (tk − tk−1 )(2qk−1/2 − qk−1/2 ) , 

4 2 4 2 − qk+1/2 ) − 12 p2k−1/2 − 12 (qk−1/2 − qk−1/2 ). C = 12 p2k+1/2 + 12 (qk+1/2

Then, we will minimize the quantity F = A2 + B 2 + C 2

(2.73)

604

14. Lee-Variational Integrator

Fig. 2.1.

The orbits calculated by (2.72), (2.74) left plot q0 = 0.77, p0 = 0 and right plot q0 = 0.99, p0 = 0

Fig. 2.2.

The energy evaluation by (2.72), (2.74) left plot q0 = 0.77, p0 = 0 and right plot q0 = 0.99, p0 = 0

over qk+1 , pk+1 and tk+1 under the constraint tk+1 > tk . This constraint guarantees that no singularities occur in choosing time steps. We will compare (2.72) with the following integrator with fixed time steps: qk+1 − qk−1 1 − (pk+1/2 + pk−1/2 ) = 0, 2h 2  pk+1 − pk−1 1 3 3 + (2qk+1/2 − qk+1/2 ) + (2qk−1/2 − qk−1/2 ) = 0. 2h 2

(2.74)

In our numerical experiment, we use two initial conditions q0 = 0.77, p0 = 0, t = 0 and q0 = 0.99, p0 = 0, t = 0. To obtain q1 and p1 , we apply the midpoint integrator with t1 = 0.1. In Fig. 2.1, the orbits calculated by (2.72) and (2.74) are shown for the two initial conditions. The two orbits in each initial condition are almost indistinguishable. In Fig.2.2, we plot the evolution of the energy H(qk+1/2 , pk+1/2 ) for both (2.72) and (2.74). The oscillating curve is for (2.74) and the lower line for (2.72). For more numerical examples, see [KMO99] in the Lagrangian setting. In principle, the results in[KMO99] apply to the Hamiltonian setting in the present method as well taking qk+1 − qk = pk+1/2 . The purpose is to develop a discrete total variation calculus in h

14.2 Total Variation in Hamiltonian Formalism

605

the Hamiltonian setting and obtain the symplectic-energy integrators. The comprehensive implementation of the obtained integrators is not the subject of present and will be a topic for future research.

14.2.6 Concluding Remarks We will develop a discrete total variation calculus in Hamiltonian formalism in this subsection. This calculus provides a new method for constructing structure-preserving integrators for Hamiltonian system from a variational point of view. Using this calculus, we will derive the energy conservation laws associated with integrators. The coupled integrators are two-step integrators and preserve a discrete version of the extended canonical 2-form. If we take fixed time steps, the resulting integrators are equivalent to the symplectic integrators derived directly from the Hamiltonian systems only in special cases. Thus, new two-step symplectic integrators are variationally obtained. Using generating function method, we will also obtain higher order symplectic-energy integrators. In principle, our discussions can be generalized to multisymplectic Hamiltonian system (2.75) Mzz t + Kzz x = ∇x H(zz ), z ∈ Rn , where M and K are skew-symmetric matrices on Rn (n ≥ 3) and S : R n → R is a smooth function [Bri97,BD01] . We call the above-mentioned system a multisymplectic Hamiltonian system, since it possesses a multisymplectic conservation law ∂ ∂ ω+ κ = 0, ∂t ∂x

(2.76)

where ω and κ are the presymplectic forms ω=

1 d z ∧ Md z, 2

κ=

1 d z ∧ Kd z . 2

The constructed action functional is -   1 T S= z (Mzz t + Kzz x ) − H(zz ) d x ∧ d t. 2

(2.77)

Performing total variation on (2.77), we can obtain the multisymplectic Hamiltonian system (2.75), the corresponding local energy conservation law   ∂ 1 T 1 ∂ S(z) − z T Kzx + z Kzt = 0, (2.78) ∂t 2 ∂x 2 and the local momentum conservation law   1 ∂ 1 T ∂  z M zx + S(z) − z T M zt = 0. ∂t 2 ∂x 2

(2.79)

In the same way, we can develop a discrete total variation in the multisymplectic form and obtain multisymplectic-energy-momentum integrators. This will be discussed in detail in Chapter 16.

606

14. Lee-Variational Integrator

14.3 Discrete Mechanics Based on Finite Element Methods Now, we will consider mechanics based on finite element methods. Let us go back to the variation problem of the action factional (1.1). The finite element method is an approximate method for solving the variation problem. Instead of solving the variation problem in the space C 2 ([a, b]), the finite element method solves the problem in a subspace Vhm ([a, b]) of C 2 ([a, b]). Vhm ([a, b]) consists of piecewise m-degree polynomials interpolating the curves q(t) ∈ C 2 ([a, b]).

14.3.1 Discrete Mechanics Based on Linear Finite Element First, let us consider the piecewise linear interpolation. Given a partition of [a, b] a = t0 < t1 < · · · < tk < · · · < tN −1 < tn = b, the intervals Ik = [tk , tk+1 ] are called elements. hk = tk+1 − tk .Vh ([a, b]) consists of piecewise linear function interpolating q(t) at (tk , qk )(k = 0, 1, · · · , N ). Now, we will derive the expressions of qh (t) ∈ Vh ([a, b]). First, we will construct the basis functions ϕk (t), which are piecewise linear functions on [a, b] satisfying ϕk (ti ) = δki (i, k = 0, 1, · · · , N ).  ϕ0 (t) =  ϕN (t) =

1− 0,

t − t0 , h0

t0 ≤ t ≤ t1 ; otherwise;

t − tN , 1+ hN −1

0,

tN −1 ≤ t ≤ tN ;

(3.1)

otherwise;

and for k = 1, 2, · · · , N − 1, ⎧ ⎪ ⎪ ⎨ 1+ ϕk (t) = 1− ⎪ ⎪ ⎩ 0,

t − tk , hk−1 t − tk , hk

tk−1 ≤ t ≤ tk ; tk ≤ t ≤ tk+1 ;

(3.2)

otherwise.

Using these basis functions, we obtain the expression qh ∈ Vh ([a, b]): qh (t) =

N 

qk ϕk (t).

k=0

In the space Vh ([a, b]), the action functional (1.1) becomes   S (t, qh (t)) =

-

b

L(t, qh (t), q˙h (t))dt a

(3.3)

14.3 Discrete Mechanics Based on Finite Element Methods

=

N −1 - tk+1  tk

k=0

=

N −1 

*

607

+ N d  L t, (qi ϕi (t), (qi ϕi (t)) d t d t i=0 i=0 N 

L(tk , qk , tk+1 , qk+1 )(tk+1 − tk ),

(3.4)

k=0

where L(tk , qk , tk+1 , qk+1 ) = =

+ N d  L t, (qi ϕi (t), (qi ϕi (t)) d t d t i=0 tk i=0 + - tk+1 * k+1 k+1  d  1 L t, (qi ϕi (t), (qi ϕi (t)) d t. tk+1 − tk tk dt 1 tk+1 − tk

-

tk+1

*

N 

i=k

i=k

(3.5)

Therefore, restricting to the subspace Vh ([a, b]) of C 2 ([a, b]), the original variational problem reduces to the extremum problem of the function (3.4) in qk (k = 0, 1, · · · , N ). Note that (3.4) is one of the discrete actions (1.33). Thus, what remains to be done is just to perform the same calculation on (3.4) as on (1.33). We can then obtain the discrete Euler–Lagrange equation (1.35) which preserves the discrete Lagrange 2-form (1.48). Therefore, discrete mechanics based on finite element methods consists of two steps: first, use finite element methods to obtain a kind of discrete Lagrangian, second, use the method of Veselov mechanics to obtain the variational integrators. Let us consider the previous example again. For the classical Lagrangian (1.57), we choose the discrete Lagrangian L(tk , qk , tk+1 , qk+1 ) as L(tk , qk , tk+1 , qk+1 ) ⎛ * +2 * N +⎞ - tk+1 N   1 d 1 ⎝ (qi ϕi (t)) −V (qi ϕi (t)) ⎠ d t = tk+1 − tk tk 2 dt i=0 i=0 % &2 &+ - tk+1 * % tk+1 − t t − tk 1 qk+1 − qk 1 −V qk + qk+1 dt = tk+1 − tk tk 2 tk+1 − tk tk+1 − tk tk+1 − tk &2 % 1 qk+1 − qk = − F (qk , qk+1 ), (3.6) 2 tk+1 − tk

where 1 F (qk , qk+1 ) = tk+1 − tk

-

%

tk+1

V tk

& tk+1 − t t − tk qk + qk+1 d t. tk+1 − tk tk+1 − tk

The discrete Euler–Lagrange equation (1.35) becomes % & qk+1 − qk qk − qk−1 ∂F (qk , qk+1 ) − (tk+1 − tk ) + tk+1 − tk tk − tk−1 ∂qk ∂F (qk−1 , qk ) + (tk − tk−1 ) = 0, ∂qk

(3.7)

(3.8)

608

14. Lee-Variational Integrator

which preserves the Lagrange 2-form & % ∂ 2 F (qk , qk+1 ) 1 d qk+1 ∧ d qk . + (tk+1 − tk ) tk+1 − tk ∂qk ∂qk+1

(3.9)

Again, if we take fixed time steps tk+1 − tk = tk − tk−1 = h, (3.8) becomes ∂F (qk , qk+1 ) ∂F (qk−1 , qk ) qk+1 − 2qk + qk−1 + + = 0, 2 h ∂qk ∂qk which preserves the Lagrange 2-form % & ∂ 2 F (qk , qk+1 ) 1 +h d qk+1 ∧ d qk . h ∂ qk ∂ qk+1 Suppose qk is the solution of (3.8) and q(t) is the solution of d2 q ∂ V (q) + = 0, d t2 ∂q

(3.10)

then from the convergence theory of finite element methods[Cia78,Fen65] , we have q(t) − qh (t) ≤ Ch2 , where  ·  is the L2 norm. qh (t) =

N 

(3.11)

qk , h = max{hk } and C is a constant

k=0

k

independent of h. If we use midpoint numerical integration formula in (3.7), we obtain & %  tk+1 1 tk+1 − t t − tk F (qk , qk+1 ) = V q + q k k+1 d t tk+1 − tk tk tk+1 − tk tk+1 − tk   q + qk+1 ≈v k . 2

In this case, (3.8) is the same as (1.59). We can also use trapezoid formula or Simpson formula and so on to integrate (3.7) numerically and obtain another kind of discrete Lagrangian.

14.3.2 Discrete Mechanics with Lagrangian of High Order Now, we will consider piecewise quadratic polynomial interpolation, which will result in a kind of discrete Lagrangian of high order. To this aim, we add an auxiliary node tk+ 12 to each element Ik = [tk , tk+1 ]. There are two kinds of quadratic basis functions: φk (t) for nodes tk and φk+ 12 (t) for tk+ 12 that satisfy φk (ti ) = δik ,   φk+ 12 ti+ 12 = δik ,

  φk ti+ 12 = 0, φk+ 12 (ti ) = 0,

i, k = 0, 1, · · · , N.

14.3 Discrete Mechanics Based on Finite Element Methods

We have the basis functions as follows: ⎧    ⎨ 2(t − t0 ) − 1 t − t0 − 1 , t0 ≤ t ≤ t 1 ; h0 h0 φ0 (t) = ⎩ 0, otherwise; ⎧    2(t − t) t − t N N ⎨ −1 − 1 , tN −1 ≤ t ≤ tN ; hN −1 hN −1 φN (t) = ⎩ 0, otherwise;

609

(3.12)

(3.13)

and for k = 1, 2, · · · , N − 1, &% & ⎧ % 2(tk − t) tk − t ⎪ ⎪ −1 − 1 , tk−1 ≤ t ≤ tk ; ⎪ ⎪ hk−1 ⎪ ⎨ % hk−1 &  φk (t) = 2(t − tk ) t − tk ⎪ − 1 − 1 , tk ≤ t ≤ tk+1 ; ⎪ ⎪ hk hk ⎪ ⎪ ⎩ 0, otherwise;

(3.14)

and for k = 0, 1, · · · , N − 1,  4

φk+ 12 (t) =

t − tk hk



1−

t − tk hk

 ,

0,

tk ≤ t ≤ tk+1 ; otherwise.

(3.15)

Using these basis functions, we will construct subspace Vh2 ([a, b]) of C 2 ([a, b]): qh2 (t) =

N 

qk φk (t) +

k=0

N −1 

qk+ 12 φk+ 12 (t),

qh2 (t) ∈ Vh2 ([a, b]).

(3.16)

k=0

In the space Vh2 ([a, b]), the action functional (1.1) becomes   S (t, qh2 (t)) = =

-

b

  L t, qh2 (t), q˙h2 (t) d t

a N −1 - tk+1  k=0

=

N −1 

  L t, qh2 (t), q˙h2 (t) d t

tk

L(tk , qk , qk+ 12 , tk+1 , qk+1 )(tk+1 − tk ),

(3.17)

k=0

where 1 L(tk , qk , qk+ 12 , tk+1 , qk+1 ) = tk+1 − tk For the discrete action (3.17), we have

-

tk+1

tk

  L t, qh2 (t), q˙h2 (t) d t.

(3.18)

610

14. Lee-Variational Integrator

d S(q0 , q 12 , q1 , · · · , qN −1+ 12 , qN ) · (δq0 , δq 12 , δq1 , · · · , δqN −1+ 12 , δqN ) =

N −1 

(D2 L(wk )δqk + D3 L(wk )δqk+ 12 + D5 L(wk )δqk+1 )(tk+1 − tk )

k=0

=

N −1 

(D2 L(wk )δqk + D3 L(wk )δqk+ 12 )(tk+1 − tk )

k=0

+

N 

D5 L(wk−1 )(tk − tk−1 )δqk

k=1

=

N −1 

(D2 L(wk )(tk+1 − tk ) + D5 L(wk−1 )(tk − tk−1 )δqk

k=1

+

N −1 

D3 L(wk )δqk+ 21 (tk+1 − tk ) + D2 L(w0 )(t1 − t0 )δq0

k=0

+D5 L(wN −1 )(tN − tN −1 )δqN ,

(3.19)

where wk = (tk , qk , qk+ 12 , tk+1 , qk+1 ) (k = 0, 1, · · · , N − 1). From (3.19), we obtain the discrete Euler–Lagrange equation D2 L(wk )(tk+1 − tk ) + D5 L(wk−1 )(tk − tk−1 ) = 0, D3 L(tk , qk , qk+ 12 , tk+1 , qk+1 ) = 0,

(3.20) (3.21)

D3 L(tk−1 , qk−1 , qk−1+ 12 , tk , qk ) = 0.

(3.22)

From (3.21) and (3.22), we can solve for qk+ 12 and qk−1+ 12 respectively, then substitute them into (3.20) and finally solve for qk+1 . Therefore, the discrete Euler– Lagrange equation (3.20) – (3.22) determines a discrete flow Ψ : M × M −→ M × M, Ψ(tk−1 , qk1 , tk , qk ) = (tk , qk , tk+1 , qk+1 ). From (3.19), we can define two 1-forms ΘLv− (tk , qk , qk+ 12 , tk+1 , qk+1 ) = D2 L(tk , qk , qk+ 12 , tk+1 , qk+1 )(tk+1 − tk )dqk , and ΘLv+ (tk , qk , qk+ 12 , tk+1 , qk+1 ) = D5 L(tk , qk , qk+ 12 , tk+1 , qk+1 )(tk+1 − tk )dqk+1 . Using the same method as before, we can prove that Ψ∗ (dΘLv+ ) = −dΘLv− .

(3.23)

From the definition of ΘLv− and ΘLv+ , we have ΘLv− + ΘLv+ = d((tk+1 − tk )L) − D3 L(tk , qk , qk+ 12 , tk+1 , qk+1 )dqk+ 12 .

(3.24)

14.3 Discrete Mechanics Based on Finite Element Methods

611

From (3.21), we obtain D3 L(tk , qk , qk+ 12 , tk+1 , qk+1 ) = 0. Thus ΘLv− + ΘLv+ = d((tk+1 − tk )L), which means dΘLv+ = −dΘLv− .

(3.25)

From (3.23) and (3.25), we arrive at Ψ∗ (ΩvL ) = ΩvL ,

(3.26)

where ΩvL = dΘLv+ . For the classical Lagrangian (1.57), from (3.16) and (3.18), we obtain L(tk , qk , qk+ 12 , tk+1 , qk+1 ) & - tk+1 % 2   1 1 q˙h2 (t) − V qh2 (t) d t = tk+1 − tk tk 2 % & 1 1 2 2 2 2 = a (tk+1 + tk tk+1 + tk ) + ab(tk + tK+1 ) + b 2 3 −G(qk , qk+ 12 , qk+1 ),

(3.27)

where  4  qk + qk+1 − 2qk+ 12 , 2 hk  1  b = 2 4(tk + tk+1 )qk+ 12 − (3tk + tk+1 )qk+1 − (tk + 3tk+1 )qk , hk

a=

and G(qk , qk+ 12 , qk+1 ) =

1 tk+1 − tk

%

where fk (t) =

-

tk+1

tk

  V qk fk (t)+qk+1 fk+1 (t)+qk+ 12 fk+ 12 (t) d t, &

2(t − tk ) −1 hk

%



t − tk −1 , hk

&



2(tk+1 − t) tk+1 − t −1 −1 , hk hk   t − tk t − tk fk+ 12 (t) = 4 1− . hk hk

fk+1 (t) =

For the discrete Lagrangian (3.27), the discrete Euler–Lagrange equations (3.20) – (3.22) become a1 qk−1 + a2 qk + a3 qk+1 + a4 qk− 12 + a5 qk+ 12 − d1 hk − d2 hk−1 = 0, (3.28) −

 ∂G(qk , qk+ 12 , qk+1 ) 8  = 0, qk + qk+1 − 2qk+ 12 − 2 3hk ∂qk+ 1

(3.29)

 ∂G(qk−1 , qk−1+ 12 , qk ) 8  = 0, qk−1 + qk − 2qk−1+ 12 − 2 3hk−1 ∂qk−1+ 1

(3.30)

2

−

2

612

14. Lee-Variational Integrator

where 1 1 a1 = , 3 hk−1

a4 = −

7 a2 = 3

%

1 hk−1

1 + hk

& ,

a3 =

1 1 , 3 hk

∂G(qk , qk+ 1 , qk+1 ) ∂G(qk−1 , qk−1+ 1 , qk ) 8 1 8 1 2 2 , a5 = − , d1 = , d2 = . 3 hk−1 3 hk ∂qk ∂qk

The solution of (3.28) – (3.30) preserves the Lagrange 2-form * + 2 ∂ G(qk , qk+ 1 , qk+1 ) 1 2 − hk −M 3hk ∂ qk ∂ qk+1

where

*

M=

d qk ∧ d qk+1 ,

(3.31)

+* + ∂ 2 G(qk , qk+ 1 , qk+1 ) ∂ 2 G(qk , qk+ 1 , qk+1 ) 16 16 2 2 + hk + hk 3hk ∂qk+ 1 ∂qk 3hk ∂qk+ 1 ∂qk 2 2 . * + 2 ∂ G(qk , qk+ 1 , qk+1 ) 32 2 − hk 2 3hk ∂qk+ 1 2

If we take the fixed time steps hk−1 = hk = h, then (3.28) – (3.30) become qk−1 − 8qk− 12 + 14qk − 8qk+ 12 + qk+1 3h2

−

 ∂G(qk , qk+ 12 , qk+1 ) 8  1 + q − 2q = 0, q − k k+1 k+ 2 3h2 ∂qk+ 12

(3.33)

−

 ∂G(qk−1 , qk−1+ 12 , qk ) 8  = 0, qk−1 + qk − 2qk−1+ 12 − 2 3h ∂qk−1+ 12

(3.34)

which preserve

where

*

M=

− d1 hk − d2 hk−1 = 0, (3.32)

*

∂ 2 G(qk , qk+ 12 , qk+1 ) 1 −M −h 3h ∂qk ∂qk+1

+ d qk ∧ d qk+1 ,

(3.35)

+* + ∂ 2 G(qk , qk+ 1 , qk+1 ) ∂ 2 G(qk , qk+ 1 , qk+1 ) 16 16 2 2 +h +h 3hk ∂qk+ 1 ∂qk 3hk ∂qk+ 1 ∂qk 2 . * 2 + 2 ∂ G(qk , qk+ 1 , qk+1 ) 32 2 −h 2 3h ∂qk+ 1 2

Suppose qk is the solution of (3.28) – (3.30) and q(t) is the solution of (3.10), then from the convergence theory of finite element methods [Cia78,Fen65] , we have q(t) − qh2 (t) ≤ Ch3 , where qh2 (t) =

N  k=0

qk φk (t) +

N −1 

qk+ 12 φk+ 12 (t),

k=0

h = maxk {hk } and C is a constant independent of h.

(3.36)

14.3 Discrete Mechanics Based on Finite Element Methods

613

14.3.3 Time Steps as Variables In the above section, the time steps tk play the role of parameters. They are determined beforehand according to some requirements. In fact, we can also regard tk as variables and the variation of the discrete action with respect to tk yields the discrete energy conservation law. This fact was first observed by Lee[Lee82,Lee87] . The symplecticity of the resulting integrators was investigated in [CGW03,KMO99] . These results are also applied to the discrete mechanics based on finite element methods. We regard tk as variables and calculate the variation of the discrete action (1.33) as follows: d S(t0 , q0 , · · · , tN , qN ) · (δt0 , δq0 , · · · , δtN , δqN ) d  =  S(t0 + εδt0 , q0 + εδq0 , · · · , tN + εδtN , qN + εδqN ) d ε ε=0 N −1  = [D2 L(wk )(tk+1 − tk ) + D4 L(wk−1 )(tk − tk−1 )]δqk k=1

+

N −1 

[D1 L(wk )(tk+1 − tk ) + D3 L(wk−1 )(tk − tk−1 ) + L(wk−1 ) − L(wk )]δtk

k=1

+D2 L(w0 )(t1 − t0 )δq0 + D4 L(wN −1 )(tN − tN −1 )δqN +[D1 L(w0 )(t1 − t0 ) − L(w0 )]δt0 +[D3 L(wN −1 )(tN − tN −1 ) + L(wN −1 )]δtN ,

(3.37)

where wk = (tk , qk , tk+1 , qk+1 ) (k = 0, 1, · · · , N − 1), so that we have the discrete energy evolution equation from the variation δqk D2 L(wk )(tk+1 − tk ) + D4 L(wk−1 )(tk − tk−1 ) = 0,

(3.38)

and the discrete energy evolution equation from the variation δtk D1 L(wk )(tk+1 − tk ) + D3 L(wk−1 )(tk − tk−1 ) + L(wk−1 ) − L(wk ) = 0, (3.39) which is a discrete version of (1.23). For a conservative L, (3.39) becomes the discrete energy conservation law. From the boundary terms in (3.37), we can define two 1-forms θL− (wk ) = (D1 L(wk )(tk+1 − tk ) − L(wk ))dtk + D2 L(wk )(tk+1 − tk )dqk , (3.40) and θL+ (wk ) = (D3 L(wk )(tk+1 − tk ) + L(wk ))dtk+1 + D4 L(wk )(tk+1 − tk )dqk+1 . (3.41) These two 1-forms are the discrete version of the extended Lagrange 1-form (1.29). Unlike the continuous case, the solution of (3.38) does not satisfy (3.39) in general. Therefore, we must solve (3.38) and (3.39) simultaneously. Using the same method in the above section, we can show that the coupled integrator

614

14. Lee-Variational Integrator

D2 L(wk )(tk+1 − tk ) + D4 L(wk−1 )(tk − tk−1 ) = 0, D1 L(wk )(tk+1 − tk ) + D3 L(wk−1 )(tk − tk−1 ) + L(wk−1 ) − L(wk ) = 0, (3.42) preserves the extended Lagrange 2-form ωL = dθL+ . For the discrete Lagrangian (3.6), (3.42) becomes % & ∂ F (wk ) ∂ F (wk−1 ) qk+1 − qk qk − qk−1 − hk + hk−1 = 0, + tk+1 − tk

1 2

%

qk+1 − qk tk+1 − tk

tk − tk−1

&2

∂ qk

∂ qk

∂ F (w )

k + F (wk ) − hk ∂ tk % &2 ∂ F (wk−1 ) 1 qk − qk−1 = + F (wk−1 ) + hk−1 .

2

tk − tk−1

∂ tk

For the kind of high order discrete Lagrangian, we can obtain similar formulae.

14.3.4 Conclusions Recently, it has been proved [GLWW01] that the symplectic structure is preserved not only on the phase flow but also on the flow with respect to symplectic vector fields as long as certain cohomological condition is satisfied in both continuous and discrete cases. This should be able to be extended to the cases in this chapter.

Bibliography

[Arn89] V. I. Arnold: Mathematical Methods of Classical Mechanics. Springer-Verlag, GTM 60, Berlin Heidelberg, Second edition, (1989). [BD01] T. J. Bridges and G. Derks: The symplectic evans matrix, and the instability of solitary waves and fonts. Arch. Rat. Mech. Anal, 156:1–87, (2001). [Bri97] T. J. Bridges: Multi-symplectic structures and wave propagation. Math. Proc. Cam. Phil. Soc., 121:147–190, (1997). [CGW03] J. B. Chen, H.Y. Guo, and K. Wu: Total variation in Hamiltonian formalism and symplectic-energy integrators. J. of Math. Phys., 44:1688–1702, (2003). [CH53] R. Courant and D. Hilbert: Methods of Mathematical Physics. Interscience, New York, Second edition, (1953). [Cia78] D. G. Ciarlet: The Finite Element for Elliptic Problem. North-Holland, Amsterdam, First edition, (1978). [Fen65] K. Feng: Difference schemes based on variational principle. J. of appl. and comput. math.in chinese, 2(4):238–262, (1965). [Fen86] K. Feng: Difference schemes for Hamiltonian formalism and symplectic geometry. J. Comput. Math., 4:279–289, (1986). [FWQW89] K. Feng, H. M. Wu, M.Z. Qin, and D.L. Wang: Construction of canonical difference schemes for Hamiltonian formalism via generating functions. J. Comput. Math., 7:71–96, (1989). [GLW01a] H. Y. Guo, Y. Q. Li, and K. Wu: A note on symplectic algorithms. Commun.Theor. Phys., 36:11–18, (2001). [GLW01b] H. Y. Guo, Y. Q. Li, and K. Wu: On symplectic and multisymplectic structures and their discrete version in Lagrange formalism. Commun.Theor. Phys., 35:703–710, (2001). [GLWW01] H. Y. Guo, Y. Q. Li, K. Wu, and S. K. Wang: Difference discrete variational principle, Euler-Lagrange cohomology and symplectic, multisymplectic structures. arXiv: math-ph/0106001, (2001). [GM88] Z. Ge and J. E. Marsden: Lie–Poisson–Hamilton–Jacobi theory and Lie–Poisson integrators. Physics Letters A, pages 134–139, (1988). [GPS02] H. Goldstein, C. Pole, and J. Safko: Classical Mechanics. Addison Wesley, New York, Third edition, (2002). [GW03] H. Y. Guo and K. Wu: On variations in discrete mechanics and field theory. J. of Math. Phys., 44:5978–6044, (2003). [KMO99] C. Kane, J. E. Marsden, and M. Ortiz: Symplectic-energy-momentum preserving variational integrators. J. of Math. Phys., 40:3353–3371, (1999). [Lag88] J. L. Lagrange: M´ecanique Analytique, 2 vols. Gauthier-Villars et fils, Paris, 4-th edition, 1888-89, (1781) [Lee82] T. D. Lee: Can time be a discrete dynamical variable? Phys.Lett.B, 122:217–220, (1982). [Lee87] T. D. Lee: Difference equations and conservation laws. J. Stat. Phys., 46:843–860, (1987).

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[MPS98] J. E. Marsden, G.P. Patrick, and S. Shloller: Multi-symplectic geometry, variational integrators, and nonlinear PDEs. Communications in Mathematical Physics, 199:351–395, (1998). [MV91] J. Moser and A. P. Veselov: Discrete versions of some classical integrable systems and factorization of matrix polynomials. Communications in Mathematical Physics, 139:217– 243, (1991). [Olv93] P. J. Olver: Applications of Lie Groups to Differential Equations. GTM 107. SpringerVerlag, Berlin, Second edition, (1993). [SSC94] J. M. Sanz-Serna and M. P. Calvo: Numerical Hamiltonian Problems. AMMC 7. Chapman & Hall, London, (1994). [Ves88] A. P. Veselov: Integrable discrete-time systems and difference operators. Funkts. Anal. Prilozhen, 22:1–33, (1988). [Ves91a] A. P. Veselov: Integrable Lagrangian correspondences and the factorization of matrix polynomials. Funkts. Anal. Prilozhen, 25:38–49, (1991). [Ves91b] A. P. Veselov: Integrable maps. Russian Math. Surveys, 46:1–51, (1991). [WM97] J. Wendlandt and J. Marsden: Mechanical integrators derived from a discrete variational principle. Physica D, 106:223–246, (1997).

Chapter 15. Structure Preserving Schemes for Birkhoff Systems

A universal symplectic structure for a Newtonian system including nonconservative cases can be constructed in the framework of Birkhoffian generalization of Hamiltonian mechanics. In this chapter, the symplectic geometry structure of Birkhoffian system is discussed, and the symplecticity of Birkhoffian phase flow is presented. Based on these properties, a way to construct symplectic schemes for Birkhoffian systems by the generating function method is explained[SSQS07],[SQ03] .

15.1 Introduction Birkhoffian representation is a generalization of Hamiltonian representation, which can be applied to hadron physics, statistical mechanics, space mechanics, engineering, biophysics, etc. Santilli[San83a,San83b] . All conservative or nonconservative, self-adjoint or non self-adjoint, unconstrained or nonholonomic constrained systems always admit a Birkhoffian representation (Guo[GLSM01] and Santilli[San83b] ). In last 20 years, many researchers have studied Birkhoffian mechanics and obtained a series of results in integral theory, stability of motion, inverse problem, and algebraic and geometric description, etc. Birkhoff’s equations are more complex than Hamilton’s equations, and the study of the computational methods of the former is also more complicated. There are no result on computational methods for Birkhoffian system before. In general, the known difference methods are not generally applicable to Birkhoffian system. A difference scheme used to solve Hamiltonian system should be Hamiltonian scheme (Hairer, Lubich and Wanner[HLW02] and Sanz-Serna and Calvo[SSC94] ), so a difference scheme to simulate Birkhoffian system should be a Birkhoffian scheme. However, the conventional difference schemes such as Euler center scheme, leap-frog scheme, etc., are not Birkhoffian schemes. So, a way to systematically construct a Birkhoffian scheme is necessary, and this is the main context in this chapter. Both the Birkhoffian and Hamiltonian systems are usually of finite dimensional (Arnold[Arn89] and Marsden and Ratiu[MR99] ), infinite dimension system has not been proposed before. The algebraic and geometric profiles of finite dimensional Birkhoffian systems are described in local coordinates, and general nonautonomous Hamiltonian systems are considered as autonomous Birkhoffian systems (Santilli[San83b] ). Symplectic schemes are systematically developed for standard Hamiltonian systems

618

15. Structure Preserving Schemes for Birkhoff Systems

and for general Hamiltonian systems on the Poisson manifold, which belong to autonomous and semi-autonomous Birkhoffian systems (Feng and Wang[FW91bandFW91a] and Feng and Qin[FQ87] ). So, in this chapter, we just discuss the nonautonomous Birkhoffian system in detail. Thereby, Einstein’s summation convention is used. In Section 15.2, Birkhoffian systems are sketched out via variational self-adjointness, with which we shows the relationship between Birkhoffian and Hamiltonian systems more essentially and directly. Then the basic geometrical properties of Birkhoffian sys" tems are presented. In Section 15.3, the definitions of K(z)-Lagrangian submanifolds " is extended to K(z, t)-Lagrangian submanifolds with parameter t. Then the relationship between symplectic mappings and gradient mappings are discussed. In Section 15.4, the generating functions for the phase flow of the Birkhoffian systems are constructed and the method to simulate Birkhoffian systems by symplectic schemes of any order is given. Section 15.5 contains an illustrating example. Schemes of order one, two, and four are derived for the linear damped oscillator. In the last Section 15.6, numerical experiments are given.

15.2 Birkhoffian Systems The generalization of Hamilton’s equation is given by % % & & ∂ B(z, t) ∂ F (z, t) ∂ Fj ∂ Fi d zi − − + = 0, ∂ zi

∂ zj

dt

∂ zi

∂t

i, j = 1, 2, · · · , 2n,

(2.1)

where the following abbreviations Kij =

∂ Fj ∂ Fi − , ∂ zi ∂ zj

K = (Kij )i,j=1,···,2n

are further used. Following the terminology suggested by Santilli[San83b] , this is called Birkhoff’s equation or Birkhoffian system under some additional assumptions. The function B(z, t) is called the Birkhoffian function because of certain physical difference with Hamiltonian. Also, the Fi (i = 1, 2, · · · , 2n) are Birkhoffian functions. A representation of Newton’s equations via Birkhoff’s equation is called a Birkhoffian representation. Definition 2.1. Birkhoff’s equations (2.1) are called autonomous when the functions Fi and B are independent of the time variable. In this case, the equations are of the simple form Kij (z)

∂ B(z) d zj − = 0. dt ∂ zi

(2.2)

They are called semi-autonomous when the functions Fi do not depend explicitly on time. In this case, the equations have the more general form Kij (z)

∂ B(z, t) d zj − = 0. dt ∂ zi

15.2 Birkhoffian Systems

619

They are called nonautonomous when both the functions Fi and B explicitly dependent on time. Then, the equations read as follow: Kij (z, t)

∂ B(z, t) ∂ Fi (z, t) d zj − − = 0. dt ∂ zi ∂t

(2.3)

They are called regular when the functional determinant is unequal to zero in the region considered, i.e., $ ) = 0, det (Kij )(Re otherwise, degenerate. Given an arbitrary analytic and regular first-order system Kij (z, t)

d zi + Di (z, t) = 0, dt

i = 1, 2, · · · , 2n,

(2.4) ∗

$ for which is self-adjoint if and only if it satisfies the following conditions in Re [AH75] : i, j = 1, 2, · · · , 2n Kij + Kji = 0, ∂ Kij ∂ Kjk ∂ Kki + + = 0, ∂ zk ∂ zi ∂ zj

(2.5)

∂ Kij ∂ Di ∂ Dj = − . ∂t ∂ zj ∂ zi

We now simply introduce the geometric significance of the condition of variational ∗ $ self-adjointness [MP91,SVC95] . Here the region considered is a star-shaped region Re of points of R×T ∗ M , T ∗ M the cotangent space of the M , M a 2n-dimensional manifold. The geometric significance of self-adjointness condition (2.5) is the integrability condition for a 2-form to be an exact symplectic form. Consider first the case for which Kij = Kij (z). Given a symplectic structure written as the 2-form in local coordinates Ω=

2n 

Kij (z, t) d zi ∧ d zj .

i,j=1

One of the fundamental properties of symplectic form is that dΩ = 0. Because the exact character of 2-form implies that Ω = d (Fi d zi ),

(2.6)

this geometric property is fully characterized by the first two equations of the condition (2.5); i.e., the 2-form (2.6) describes the geometrical structure of the autonomous case (2.2) of the Birkhoff’s equations, it even sketches out the geometric structure of the semi-autonomous case.

620

15. Structure Preserving Schemes for Birkhoff Systems

For the case Kij = Kij (z, t), the full set of condition (2.5) must be considered. The corresponding geometric structure can be better expressed by transition of the symplectic geometry on the cotangent bundle T ∗ M with local coordinates zi to the contact geometry on the manifold R × T ∗ M with local coordinates z"i (i = 0, 1, 2, · · · , 2n), z"0 = t[San83b] . More general formulations of an exact contact 2-form exist, although it is now referred to as a (2n+1)-dimensional space, = Ω

2n 

 ij d z"i ∧ d z"j = Ω + 2 Di d zi ∧ d t, K

i,j=0

where

5  = K

0

−DT

D

K

6 D = (D1 , · · · , D2n )T .

,

If the contact form is also of the exact type,  F"i =

 = d (F"i d z"i ), Ω

−B,

(2.7)

Fi ,

the geometric meaning of the condition of the self-adjointness is then the integrability condition for the exact contact structure (2.7). Here B can be calculated from −

∂B ∂ Fi = Di + ∂ zi ∂t

for 

∂ ∂ Fi Di + ∂ zj ∂t





=



∂ ∂ Fj Dj + . ∂ zi ∂t

All the above discussion can be expressed via the following property. Theorem 2.2 (Self-Adjointness of Birkhoffian System). For a general nonautonomous first-order system (2.4), a necessary and sufficient condition for self-adjointness ∗ $ of points of R × T∗ R2n is that it is of the Birkhoffian type, i.e., the following in Re representation holds for i, j = 1, 2, · · · , 2n,     ∂ F (z, t) dz ∂ Fj ∂ Fi d zi Kij (z, t) i + Di (z, t) = − − ∇ B(z, t) + . (2.8) dt

∂ zi

∂ zj

dt

∂t

Remark 2.3. The functions Fi and B can be calculated according to the rules [AH75] Fi =

1 2

B=

0

-

1

0 1

zj · Kji (λz, t) d λ,

  ∂ Fi zi · D i + (λz, t) d λ. ∂t

15.3 Generating Functions for K(z, t)-Symplectic Mappings

621

Due to the self-adjointness of Birkhoff’s equations, the phase flow of the system (2.8) conserves the symplecticity d d Ω = (Kij d zi ∧ d zj ) = 0. dt dt

So denoting the phase flow of the Equation (2.8) with ( z,  t) yields Kij ( z,  t) d zi ∧ d zj = Kij (z, t) d zi ∧ d zj , respectively the algebraic representation ∂ z T ∂ z K( z,  t) = K(z, t). ∂z ∂z

In the latter, the algorithm preserving this geometric property of the phase flow in discrete space will be constructed.

15.3 Generating Functions for K(z, t)-Symplectic Mappings In this section, general K(z, t)-symplectic mappings and their relationships with the gradient mappings and their generating functions are considered [FW91b,FW91a,FQ87] . Definition 3.1. Let denote 5 5 6 6 O In O I2n J2n = , J4n = , −In O −I2n O 5 6 K( z , t) O " z , z, t, t0 ) = K( . O −K(z, t0 )

5 J"4n =

J2n

O

O

−J2n

Then a 2n-dimensional submanifold L ⊂ R4n ' / z  4n 2n L= ∈ R | z = z(x, t0 ), z = z(x, t), x ∈ U ⊂ R , open set z " z , z, t, t0 )-Lagrangian submanifold if it holds is called a J4n - or J"4n - or K( (Tx L)T J4n (Tx L) = 0,

(Tx L)T J"4n (Tx L) = 0

or " z , z, t, t0 )(Tx L) = 0, (Tx L)T K( where Tx L is the tangent space to L at x.

6 ,

622

15. Structure Preserving Schemes for Birkhoff Systems

Definition 3.2. The mapping with parameters t and t0 is z → z = g(z, t, t0 ) : R2n → R2n is called a canonical map or a gradient map or a K(z, t)-symplectic map if its graph ' /  z 4n 2n ∈ R | z = g(z, t, t0 ), z = z ∈ R Γg = z " z , z, t, t0 )-Lagrangian submanifold. is a J4n - or J"4n - or K( For differentiable mappings, there exists an equivalent definition for the Ksymplecticness, which is also useful for difference schemes. Definition 3.3. A differentiable mapping g : M → M is K(z, t)-symplectic if ∂ g ∂ gT  K g(z, t, t0 ), t = K(z, t0 ). ∂z ∂z A difference scheme approximating the Birkhoffian system (2.8) with step size τ z k+1 = g k (z k , tk + τ, tk ),

k0

is called a K-symplectic scheme, if g k is K-symplectic for every k, i.e., T

∂ gk ∂ gk K(z k+1 , tk+1 ) k = K(z k , tk ). k ∂z ∂z

The graph of the phase flow of the Birkhoffian system (2.8) is g t (z, t0 ) = " z , z, t, t0 )-Lagrangian submanifold for g(z, t, t0 ) which is a K(   gzt (z, t0 )T K g t (z, t0 ), t gzt (z, t0 ) = K(z, t0 ). Similarly, the graph of the phase flow of standard Hamiltonian system is a J"4n Lagrangian submanifold. Consider the nonlinear transformation with two parameters t and t0 from R4n to itself, 5 6 6 5 6 5 z , z, t, t0 ) z w  α1 ( α(t, t0 ) : , (3.1) −→ = z , z, t, t0 ) α2 ( z w 5 6 6 5 6 5 1  w, t, t0 ) w  z α (w, −1 . α (t, t0 ) : −→ =  w, t, t0 ) α2 (w, w z Let denote the Jacobian of α and its inverse by 5 z , z, t, t0 ) = α∗ (

Aα

Bα

Cα

Dα

6

5 ,

α∗−1 (w,  w, t, t0 )

=

Aα

Bα

Cα

Dα

6 .

15.3 Generating Functions for K(z, t)-Symplectic Mappings

623

Let α be a diffeomorphism from R4n to itself, then it follows that α carries ev" ery K-Lagrangian submanifold into a J4n -Lagrangian submanifold, if and only if " i.e., α∗T J4n α∗ = K, 6T 5 65 6 5 5 6 O J2n Aα Bα K( z , t) O Aα Bα = . Cα Dα Cα Dα O −J2n O −K(z, t0 ) " subConversely, α−1 carries every J4n -Lagrangian submanifold into a K-Lagrangian manifold. Theorem 3.4. Let M ∈ R2n×2n , α given as in (3.1), and define a fractional transformation σα : M −→ M,

M −→ N = σα (M ) = (Aα M + Bα )(Cα M + Dα )−1

under the transversality condition |Cα M + Dα | = 0. Then the following four conditions are mutually equivalent: |Cα M + Dα | = 0,

|M C α − Aα | = 0,

|C α N + Dα | = 0,

|N Cα − Aα | = 0.

The proof is direct and simple, so it is omitted here. Theorem 3.5. Let α be defined as in (3.1). Let z → z = g(z, t, t0 ) be a K(z, t)" of R2n with Jacobian gz (z, t, t0 ) = symplectic mapping in some neighborhood R " M (z, t, t0 ). If M satisfies the transversality condition in R   Cα (g(z, t, t0 ), z, t, t0 )M (z, t, t0 ) + Dα (g(z, t, t0 ), z, t, t0 ) = 0, (3.2) " a gradient mapping w → w then there uniquely exists in R  = f (w, t, t0 ) with Jacobian fw (w, t, t0 ) = N (w, t, t0 ) and a uniquely defined scalar generating function φ(w, t, t0 ), such that f (w, t, t0 ) = φw (w, t, t0 ),     α1 (g(z, t, t0 ), z, t, t0 ) = f α2 (g z, t, t0 ), z, t, t0 , t, t0     = φw α2 g(z, t, t0 ), z, t, t0 , t, t0 , and

N = (Aα M + Bα )(Cα M + Dα )−1 , M = (Aα N + B α )(C α N + Dα )−1 .

Proof. Under the transformation α, the image of the graph Γg is   67 6 5 5 w  = α1 g(z, t, t0 ), z, t, t0 w  ∈ R4n | α(Γg ) = .   w w = α2 g(z, t, t0 ), z, t, t0

(3.3)

624

15. Structure Preserving Schemes for Birkhoff Systems

Inequality (3.2) implies       ∂ w   ∂ α2 ∂ z ∂ α2   · +  =  = Cα M + Dα  = 0, ∂z



∂ z

∂z

∂z



so w = α2 g(z, t, t0 ), z, t, t0 is invertible, the inverse function is denoted by z = z(w, t, t0 ). Set   (3.4) w  = f (w, t, t0 ) = α1 g(z, t, t0 ), z, t, t0 |z=z(w,t,t0 ) , then N=

∂f = ∂w



∂ α1 ∂ g ∂ α1 + ∂ z ∂ z ∂z



∂z ∂w



= (Aα M + Bα )(Cα M + Dα )−1 .

Notice that the tangent space to α(Γg ) at z is ⎡

⎤

∂w  5 6 Aα M + Bα  ⎢ ∂z ⎥  . Tz α(Γg ) = ⎣ ⎦= ∂w Cα M + Dα ∂z

It can be concluded that α (Γg ) is a J4n -Lagrangian submanifold for T    Tz α(Γg ) J4n Tz α(Γg ) 

T

T

= (Aα M + Bα ) , (Cα M + Dα )

5



J4n

Aα M + Bα

6

Cα M + Dα  M   M  " = 0. = (M T , I) K = (M T , I)α∗T J4n α∗ I I So,

(Aα M + Bα )T (Cα M + Dα ) − (Cα M + Dα )T (Aα M + Bα ) = 0,

 = i.e., N = (Aα M + Bα )(Cα M + Dα )−1 is symmetric. This implies that w f (w, t, t0 ) is a gradient mapping. By the Poincar´e lemma, there is a scalar function φ(w, t, t0 ) such that f (w, t, t0 ) = φw (w, t, t0 ).

(3.5)

Consider the construction of f (w, t, t0 ) and z(w, t, t0 ). Since z(w, t, t0 )◦α2 (g(z, t, t0 ), z, t, t0 ) ≡ z, substituting w = α2 (g(z, t, t0 ), z, t, t0 ) in (3.4) and (3.5) yields Equation (3.3).  Theorem 3.6. f (w, t, t0 ) obtained in Theorem 3.5 is also the solution of the following implicit equation:     α1 f (w, t, t0 ), w, t, t0 = g α2 (f (w, t, t0 ), w, t, t0 ), t, t0 .

15.4 Symplectic Difference Schemes for Birkhoffian Systems

625

Theorem 3.7. Let α be defined as in Theorem 3.5, let w → w  = f (w, t, t0 ) be 2n " a gradient mapping in some neighborhood R of R with Jacobian fw (w, t, t0 ) = " the condition N (w, t, t0 ). If N satisfies in R  α    C f (w, t, t0 ), w, t, t0 N (w, t, t0 ) + Dα f (w, t, t0 ), w, t, t0  = 0, " there uniquely exists a K(z,t)-symplectic mapping z → z = g(z, t, t0 ) with then in R Jacobian g(z, t, t0 ) = M (z, t, t0 ) such that α1 (f (w, t, t0 ), w, t, t0 ) = g(α2 (f (w, t, t0 ), w, t, t0 ), t, t0 ), M = (Aα N + B α )(C α N + Dα )−1 , N = (Aα M + Bα )(Cα M + Dα )−1 .

Remark 3.8. The proofs of Theorems 3.6 and 3.7 are similar to that of Theorem 3.5 and are omitted here. Similar to Theorem 3.6, the function g(z, t, t0 ) is the solution of the implicit equation α1 (g(z, t, t0 ), z, t, t0 ) = f (α2 (g(z, t, t0 ), z, t, t0 ), z, t, t0 ).

15.4 Symplectic Difference Schemes for Birkhoffian Systems In Section 15.2, it is indicated that for a general Birkhoffian system, there exists the common property that its phase flow is symplectic. With the result in the last section, symplectic schemes for Birkhoffian systems are constructed by approximating the generating functions. Birkhoff’s phase flow is denoted by g t (z, t0 ) and it is a one-parameter group of K(z, t)-symplectic mappings at least local in z and t, i.e., g t0 = identity, g t1 +t2 = g t1 ◦ g t2 . Here z is taken as an initial value when t = t0 , and z(z, t, t0 ) = g t (z, t0 ) = g(t; z, t0 ) is the solution of the Birkhoffian system (2.8). Theorem 4.1. Let α be defined as in Theorem 3.5. Let z → z = g t (z, t0 ) be the phase flow of the Birkhoffian system (2.8), M (t; z, t0 ) = gz (t; z, t0 ) is its Jacobian. At some initial point z, i.e., t = t0 , z = z, if |Cα (z, z, t0 , t0 ) + Dα (z, z, t0 , t0 )| = 0,

(4.1)

then for sufficiently small |t − t0 | and in some neighborhood of z ∈ R2n there exists a gradient mapping w → w  = f (w, t, t0 ) with symmetric Jacobian fw (w, t, t0 ) = N (w, t, t0 ) and a uniquely determined scalar generating function φ(w, t, t0 ) such that

626

15. Structure Preserving Schemes for Birkhoff Systems

f (w, t, t0 ) = φw (w, t, t0 ), (4.2)   ∂ φw (w, t, t0 ) = A φw (w, t, t0 ), w, φww (w, t, t0 ), t, t0 , (4.3) ∂t     ∂w  ∂w  A w,  w, , t, t0 = A¯ z(w,  w, t, t0 ), , t, t0 ,  w, t, t0 ), z(w, (4.4) ∂w ∂w   ∂w  d ∂w  d , t, t0 = w(  z , z, t, t0 ) − w( z , z, t, t0 ) A¯ z, z, ∂w dt ∂w d t   ∂w  ∂ α1 ∂w  ∂ α2 = Aα − Cα K −1 D( z , t) + − ,(4.5) ∂w ∂t ∂w ∂t     α1 (g(t; z, t0 ), z, t, t0 ) = f α2 g(t; z, t0 ), z, t, t0 , t, t0     = φw α2 g(t; z, t0 ), z, t, t0 , t, t0 , and N = σα (M ) = (Aα M + Bα )(Cα M + Dα )−1 , M = σα−1 = (Aα N + B α )(C α N + Dα )−1 . Proof. M (t; z, t0 ) is differentiable with respect to z and t. Condition (4.1) guarantees that for sufficiently small |t − t0 | and for z in some neighborhood of z ∈ R2n , there is   Cα ( z , z, t, t0 )M (t; z, t0 ) + Dα ( z , z, t, t0 ) = 0. Additionally, the Birkhoffian phase flow is a symplectic mapping; therefore, by Theorem 3.5, there exists a time-dependent gradient map w  = f (w, t, t0 ) and a scalar function φ(w, t, t0 ), such that f (w, t, t0 ) = φw (w, t, t0 ),

∂ f (w, t, t0 ) ∂ φw (w, t, t0 ) = . ∂t ∂t

(4.6)

Notice that z = g(t; z, t0 ) is the solution of the following initial-value problem; ⎧   d z ∂F ⎪ ⎨ = K −1 ( z , t) ∇B + ( z , t), dt

∂t

⎪ ⎩ z|t=t0 = z, therefore, from the equation in (3.2), it follows that



dw  ∂w  d z ∂ ∂F = · + α1 ( z , z, t, t0 ) = Aα K −1 ∇ B + dt ∂ z d t ∂t ∂t   dw ∂ F ∂ α 2 = Cα K −1 ∇ B + , + dt ∂t ∂t

so







∂w  dw  ∂w  dw ∂w  ∂F = − = Aα − Cα K −1 ∇ B + ∂t dt ∂ t dt ∂t ∂t

Since

 +

 +

∂ α1 , ∂t

∂ α1 ∂w  ∂ α2 − . ∂t ∂w ∂t

∂w  = 0, so w = w(w,  t) exists and is solvable in (w),  but it cannot be solved ∂w

explicitly from the transformation α and α−1 , we have   ∂w  ∂w  , t, t0 = , A¯ z, z, ∂w

∂t

and the Equations (4.4) and (4.5). Then, from (4.6), the Equation (4.3) follows.



15.4 Symplectic Difference Schemes for Birkhoffian Systems

627

According to[FW91b,FW91a,FQ87] , we can easily construct symplectic difference schemes of any order for the autonomous or semi-autonomous Birkhoffian systems. Because of the simplicity of the ordinary geometry structure, the transformation α in (3.1) needed in these cases is independent of the parameter t, accordingly   ∂w  dw  ∂w  dw ∂w  = − = Aα − Cα K −1 ∇ B ∂t

dt



∂ t dt



∂w 

T



∂t

= − B αT + AαT ∇z B ∂w     = −Bw z(w, z (w,  w), t) .  w) or = −Bw ( Therefore, the corresponding Birkhoffian system is completely a Hamiltonian system   ∂ φ(w, t) = −B z(φw , w) , ∂t

  ∂ φ(w, t, t0 ) = −B z(φw , w), t ∂t

(4.7)

in the autonomous and semi-autonomous case, respectively. Remark 4.2. Because of the forcing term in (2.1), the Hamilton–Jacobi equation for the generating function φ(w, t, t0 ) cannot directly be derived, but instead the Hamilton–Jacobi equation (4.3) for φw (w, t, t0 ) can be easily derived. Assume the generating function φw (w, t, t0 ) can be expanded as a convergent power series in t φw (w, t, t0 ) =

∞ 

(t − t0 )k φ(k) w (w, t0 ).

(4.8)

k=0

Lemma 4.3. The k-th order total derivative of A defined as in Theorem 4.1 with respect to t can be described as ∞ 

Dtk A = ∂φw A

∞    (t − t0 )i φ(k+i) A (t − t0 )i φ(k+i) + ∂ φww w ww

i=0

i=0

∞ 

+∂t ∂φw A

(t − t0 )

i

φ(k−1+i) w



∞ 

+ ∂t ∂φww A

i=0

+

k 

Cm k

k−m 

m=0

·

∞ 

n=1

(t −

Cnk−m

i=0 k−m−n 

(h1 +i) t0 )i φw ,·

· ·,

·

i=0

1 +i) (t − t0 )i φ(j , · · ·, ww



∂φnw ∂φl ww ∂tm A

h1 +···+hn

l=1

i=0 ∞ 

(t − t0 )i φ(k−1+i) ww

+j1 +···+jl =k−m ∞ 

(hn +i) (t − t0 )i φw ,

i=0 ∞ 

 (jl +i) (t − t0 )i φw ,

i=0

then at the point of t = t0 , the total derivative of A is as



628

15. Structure Preserving Schemes for Birkhoff Systems (k)

(k)

Dtk At0 = ∂φw At0 φw + ∂φww At0 φww (k−1)

+∂t ∂φw At0 φw +

k  m=0

Cm k

k−m 

(k−1)

+ ∂t ∂φw w At0 φww

Cnk−m

k−m−n 

n=1

l=1



∂φnw ∂φl ww ∂tm At0

h1 +···+hn

+j1 +···+jl =k−m

 (h ) (h ) (j1 ) (jl )  , · φw 1 , · · · , φw n , φww , · · · , φww (0)

(0)

where At0 = A(φw , w, φww , t0 , t0 ). By means of the representations of the total derivative of A, the following results are proved. Theorem 4.4. Let A and α be analytic. Then the generating function φwα,A (w, t, t0 ) = φw (w, t, t0 ) can be expanded as a convergent power series in t for sufficiently small |t − t0 | ∞  φw (w, t, t0 ) = (t − t0 )k φ(k) (4.9) w (w, t0 ), k=0

and

(k) φw

(k ≥ 0), can be recursively determined by the following equations φ(0) w (w, t0 ) = f (w, t0 , t0 ),  (0)  (0) φ(1) w (w, t0 ) = A φw , w, φww , t0 , t0 ,   1 (0) Dtk A φ(0) φk+1 w (w, t0 ) = w , w, φww , t0 , t0 . (k + 1) !

(4.10) (4.11) (4.12)

Proof. Differentiating Equation (4.9) with respect to w and t, we derive φww (w, t, t0 ) = ∂ φw (w, t, t0 ) = ∂t

∞  k=0 ∞ 

(t − t0 )k φ(k) ww (w, t0 ),

(4.13)

(k + 1)(t − t0 )k φ(k+1) (w, t0 ). w

(4.14)

k=0

By Equation (4.2), φ0w (w, t0 ) = φw (w, t0 , t0 ) = f (w, t0 , t0 ).   ∂w  , t, t0 , and expanding A in Substituting Equations (4.9) and (4.13) in A w,  w, ∂w t = t0 , we get A(φw , w, φww , t, t0 ) = A(f (w, t0 , t0 ), w, fw (w, t0 , t0 ), t0 , t0 ) ∞  1 (0) + (t − t0 )k Dtk A(φ(0) w , w, φww , t0 , t0 ). (4.15) k! k=1

Using Equation (4.3) and comparing (4.15) with (4.14), we get (4.11) and (4.12).



15.5 Example

629

In the autonomous and semi-autonomous case, A is replaced by the Birkhoffian B, which makes it much easier to expand the generating functions φ. With Theorems 3.5 and 3.7, the relationship between the Birkhoffian phase flow and the generating function φ(w, t, t0 ) is established. With this result, K(z, t)-symplectic difference schemes can be directly constructed. Theorem 4.5. Let A and α be analytic. For sufficiently small time-step τ > 0, take (m) ψw (w, t0 + τ, t0 ) =

m 

τ i φ(i) w (w, t0 ),

m = 1, 2, · · · ,

i=0 (i)

where φw are determined by Equations (4.10) – (4.12). (m) Then, ψw (w, t0 + τ, t0 ) defines a K(z,t)-symplectic difference scheme z = z k → z k+1 = z,   (m) α1 (z k+1 , z k , tk+1 , tk ) = ψw α2 (z k+1 , z k , tk+1 , tk ), tk+1 , tk (4.16) of m-th order of accuracy. (m)

Proof. Let be N = φww (w0 , t0 , t0 ) = ψww (w0 , t0 , t0 ) and w0 = α(z, z, t0 , t0 ), then Theorem 3.7 yields |C α N + Dα |, because of |Cα (z, z, t0 , t0 ) + Dα (z, z, t0 , t0 )| = 0. Thus, for sufficiently small τ and in some neighborhood of w0 , there exists |C α N (m) (w, t0 + τ, t0 ) + Dα | = 0, where (m) N (m) (w, t0 + τ, t0 ) = ψww (w, t0 + τ, t0 ). (m)

By Theorem 3.7, ψw (w, t0 + τ, t0 ) defines a K(z, t)-symplectic mapping which is expressed in (3.3). Therefore, Equation (4.16) determines a m-th order K(z, t)symplectic difference scheme for the Birkhoffian system (2.8). 

15.5 Example In this section, an example illustrates how to obtain schemes preserving the symplectic structure for a nonconservative system expressed in Birkhoffian representation. Consider the linear damped oscillator r¨ + ν r˙ + r = 0.

(5.1)

We introduce a gradient function p satisfying p = r, ˙ then a Birkhoffian representation of (5.1) is given by

630

15. Structure Preserving Schemes for Birkhoff Systems

5

−eνt 0

0 eνt

65

6

r˙ p˙

5 =

The structure and functions are 6 5 0 −eνt , K= eνt 0 ⎤ ⎡ 1 eνt p ⎦, F =⎣ 2 1 − eνt r

6

νeνt p + eνt r

.

eνt p 5

K

−1

=

0

e−νt

−e−νt

0

(5.2)

6 ,

1 2

B = eνt (r2 + rp + p2 ),

2

and the energy function reads as follows: 1 2

H(q, p) = (q 2 + p2 ) − νp2 .

(5.3)

The Euler midpoint scheme (or one-step Gauss–Runge–Kutta method) for the system (5.2), which can be derived via the discrete Lagrange–d’Alembert principle[MW01] , reads as follows: qk+1 − qk p + pk = k+1 , τ 2 pk+1 − pk pk+1 + pk q + qk = −ν − k+1 , τ 2 2

and hence,

5

qk+1 pk+1

6

1 = Δ

5

65

−τ 2 + 2ντ

4τ

−4τ

−τ 2 − 2ντ + 4

qk pk

6 ,

(5.4)

where Δ = τ 2 + 2ντ + 4, is not a K(z, t)-symplectic scheme. Now, let the transformation α in (3.1) be  = eνt p − eνt0 p, Q

P = q − q,

1 2

1 2

P = − (eνt p + eνt0 p),

Q = ( q + q), where the Jacobian of α is

⎡

0

⎢ 1 ⎢ ⎢ α∗ = ⎢ 1 ⎢ 2 ⎣ 0

eνt

0

−eνt0

0

−1

0

0

1 2

0

1 2

0

− eνt0

− eνt

1 2

(5.5)

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎦

The inverse transformation is 1 q = P + Q,

 − e−νt P, p = e−νt Q

1 q = − P + Q,

1  − e−νt0 P, p = − eνt0 Q

2

2

1 2

2

(5.6)

15.5 Example

and

⎡

α∗−1

⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣

0

1 2

1 −νt e 2

0

0 1 2

− e−νt0

−

1 2

0

1

0

631

⎤

⎥ ⎥ −e−νt ⎥ ⎥ ⎥. ⎥ 1 0 ⎥ ⎦ 0 −e−νt0 0

Consequently, using (5.5), (5.6) and (5.2), we derive ⎤ ⎡ 1 6 5 5 6 νt νt ˙ νt − eνt P − eνt Q p  −e νe p  + e q  dw  2 ⎥ ⎢ = = =⎣ ⎦, dt 1 ˙q p  − e−νt P e−νt Q 2 ⎤ ⎡ 1 −νt  1 −νt e Q− e P 2 dw ⎥ ⎢ 4 =⎣ ⎦. dt 1 νt  1 e P + eνt Q 4

2

Simple calculations (for m = 0,1) yields, 5 6  0    Q (0) φw = = ,  0 t=t0 P (1) φw

(0)





dw  (0) d w  = − φww =   d t t=t0 d t t=t0

5

−eνt0 Q

6 .

−e−νt0 P

(1)

Set w  = φw + φw τ , so the first order scheme for the system (5.2) reads as follows: qk+1 − qk eνtk+1 pk∗1 + eνtk pk = e−νtk+1 , τ 2 eνtk+1 pk+1 − eνtk pk q + qk = −eνtk k+1 , τ 2

and hence

5

qk+1 pk+1

6

5 1 = Δ

4 − τ2

4τ

−4τ e−ντ

(4 − τ 2 )e−ντ

65

qk pk

where Δ = 4 + τ 2 . The transition matrix denoted by A satisfies 6 5 6 5 νtk νtk+1 0 −e 0 −e . AT A= eνtk+1 0 eνtk 0 Then, consider the transformation α in (3.1) to be

6 ,

(5.7)

632

15. Structure Preserving Schemes for Birkhoff Systems

P = −eνt/2 q + eνt0 /2 q,

 = eνt/2 p − eνt0 /2 p, Q 1 2

1 2

Q = (eνt/2 q + eνt0 /2 q),

P = − (eνt p + eνt0 p).

The Jacobian of α is ⎡

0

⎢ −eνt/2 ⎢ ⎢ α∗ = ⎢ 1 νt/2 ⎢ e ⎣ 2 0

eνt/2

0

−eνt0 /2

0

eνt0 /2

0

0

1 νt0 /2 e 2

0

1 2

0

− eνt0

− eνt

⎡

and the inverse

α∗−1

⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣

1 −νt e 2

0 −

0 1 − e−νt0 2

⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤

1 2

0

1 2

⎤

1 2

0

1

0

0

−νt

−e

1

0

0

−e−νt0

⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦

Direct calculation yields the scheme of second order eνtk+1 /2 qk+1 − eνtk /2 qk eνtk+1 /2 pk+1 + eνtk /2 pk eνtk+1 /2 qk+1 + eνtk /2 qk = +ν , τ 2 4 eνtk+1 /2 pk+1 − eνtk /2 pk eνtk+1 /2 qk+1 + eνtk /2 qk eνtk+1 /2 pk+1 + eνtk /2 pk = −ν , τ 2 4

and hence,

5

qk+1 pk+1

6

e−ντ /2 = Δ

5

w1

−16τ

16τ

w−2

65

qk pk

6 ,

(5.8)

where Δ = ν 2 τ 2 − 4τ 2 − 16, w1 = −16 − 8ντ − ν 2 τ 2 + 4τ 4 ,

w2 = −16 + 8ντ − ν 2 τ 2 + 4τ 2 .

e−ντ /2

(∗) in (5.8) by M (τ ), then by composition[Yos90,QZ92] Abbreviating the matrix Δ we have the scheme of order four 5 5 6 6 qk+1 qk = M (c1 τ )M (c2 τ )M (c1 τ ) , (5.9) pk+1 pk where c1 = If take m = 2, we have

1 , 2 − 21/3

c2 =

φ(2) w = 0.

−21/3 . 2 − 21/3

15.5 Example

Now take m = 3, = φ(3) w













633



1 ∂ ∂ dw " ∂ dw " ∂w " ∂w " ∂ dw ∂ w " + − 3! ∂ t ∂ t dt ∂w " dt ∂ t ∂w∂w " dt ∂ t      ∂w " ∂ dw ∂ ∂w " dw − − . ∂ w ∂ t dt ∂ t ∂ w dt

(5.10)

For equation q¨ + ν q˙ + q = 0, 3rd derivatives of φ in time t = t0 , only one term to appear, i.e.,     −

Simple calculation yields (3)  φw 

t=t0

∂ ∂w " ∂ dw ∂ w " . ∂t ∂w ∂w " dt ∂ t

   1 ν   −   −ν P − Q  8 8   2  1   ν ν   − Q−P  − 4   8 2 2     1 1 νP  − + ν2 Q +  4 16 2      1 1 2 νQ +P  − + ν      −1 

  −1 1  = −  ν 6 −    1 = −  6 

ν − 2

4

16

2

  2   2   ν ν ν  − 2 Q + − 2 P  2 2 2 =     2  ν ν  ν2 −2 P + −2 Q  2

2

2

     

    ,    (1)

(3)

we get 4-th order symmetrical symplectic scheme: w " = φw Δt + φw Δt2 , i.e., eνtk+1 /2 qk+1 − eνtk /2 qk eνtk+1 /2 pk+1 + eνtk /2 pk eνtk+1 /2 qk+1 − eνtk /2 qk = +ν τ 2 4

+τ 2

1 24 × 4



  ν2 − 2 eνtk+1 /2 pk+1 + eνtk /2 pk 2

 2    ν ν νtk+1 /2 + −2 qk+1 + eνtk /2 qk , e 2

2

eνtk+1 /2 pk+1 − eνtk /2 pk eνtk+1 /2 qk+1 + eνtk /2 qk eνtk+1 /2 pk+1 + eνtk /2 pk = −ν τ 2 4

−τ 2  +

1 24 × 4



  ν2 − 2 eνtk+1 /2 qk+1 + eνtk /2 qk 2





 ν2 ν  νtk+1 /2 −2 pk+1 + eνtk /2 pk . e 2 2 (5.11)

This method is easily extended to more general ODEs such as  p˙ + β  (t)p + V (q, t) = 0, q˙ − G(p, t) = 0.

(5.12)

634

15. Structure Preserving Schemes for Birkhoff Systems

Remark 5.1. The derived schemes (5.7), (5.8), and (5.9) are K(z, t)-symplectic, i.e., for τ > 0 and k ≤ 0 they satisfy the Birkhoffian condition eνtk+1 d qk+1 ∧ d pk+1 = eνtk d qk ∧ d pk .

15.6 Numerical Experiments In this section, we present numerical results for the linear damped oscillator (5.1), resp., (5.2) using the derived K(z, t)-symplectic schemes (5.7), (5.8), and (5.9) of order one, two, and four, respectively. Further, we use Euler’s midpoint scheme (5.4), which is not K(z, t)-symplectic but shows convenient numerical results[MW01] , and further Euler’s explicit scheme for comparison. In the presented figures, the initial values are always chosen as q(0) = 1, p(0) = q(0) ˙ = −1, and the time interval is from 0 to 25. There are only small differences in the behavior of the different schemes choosing other initial values. The actual error, err = |approximate solution - true solution|, is computed with step size τ = 0.2. Using different step sizes, the schemes always show the same quality, which is emphasized by representing the results in a double logarithmic scale using step sizes τ = 0.01, 0.02, 0.05, 0.1, 0.2, 0.5. The orbits are computed with step size τ = 0.05. The first comparison is given between scheme (5.7) and Euler’s explicit scheme both are of order one. For smaller ν, i.e., 0 ≤ ν ≤ 1.3 scheme (5.7) is better, and for ν > 1.3 Euler’s explicit scheme is better. The second comparison is given between scheme (5.8) and Euler’s midpoint scheme (5.4) both are of order two. For 0 ≤ ν ≤ 0.5 both schemes show the same behavior, for 0.5 < ν < 2.8 scheme (5.8) is better, where the most advantage is around ν = 2, and for 2.8 ≤ ν Euler’s midpoint scheme behaves better. The third comparison is given between scheme (5.9) of order four and scheme (5.8) of order two. Both schemes have the same structure preserving property, and therefore the higher order scheme (5.9) shows a clear superiority over the twoorder scheme. These differences between the discussed schemes are illustrated by the error curves (Figs. 6.1 and 6.4). For the energy function (5.3), the comparisons of the energy error H, between the different schemes are also done in double logarithmic scales (Figs. 6.5 and 6.8). The result shows that the dominance is not clear between scheme (5.7) and Euler’s explicit scheme while scheme (5.8) is always better than Euler’s midpoint scheme for growing ν, even for ν ≥ 2.8. Scheme (5.9) keeps its superiority in the comparisons. The comparisons also show that it is possible for different schemes obtained from different transformation α, that different quantities are preserved. This point is proved to be true in the generating function method for Hamiltonian systems (see Feng et al[FW91b,FW91a] ). The extension to application in Birkhoffian systems will also be studied in a prospective paper.

15.6 Numerical Experiments

q’’ + 0.6 * q’ + q = 0

0

635

( tau = 0.01, 0.02, 0.05, 0.1, 0.2, 0.5 )

10

expl Euler midpoint (5.4) scheme (5.7) scheme (5.8) scheme (5.9)

−2

10

−4

max−err

10

−6

10

−8

10

−10

10

−2

−1

10

0

10

10

tau

Fig. 6.1.

Error comparison between the different schemes for ν = 0.6

q’’ + 1.3 * q’ + q = 0

0

( tau = 0.01, 0.02, 0.05, 0.1, 0.2, 0.5 )

10

expl Euler midpoint (5.4) scheme (5.7) scheme (5.8) scheme (5.9)

−2

10

−4

max−err

10

−6

10

−8

10

−10

10

−2

10

−1

10

tau

Fig. 6.2.

Error comparison between the different schemes for ν = 1.3

0

10

636

15. Structure Preserving Schemes for Birkhoff Systems

q’’ + 1.9 * q’ + q = 0

0

( tau = 0.01, 0.02, 0.05, 0.1, 0.2, 0.5 )

10

expl Euler midpoint (5.4) scheme (5.7) scheme (5.8) scheme (5.9)

−2

10

−4

max−err

10

−6

10

−8

10

−10

10

−12

10

−14

10

−2

−1

10

0

10

10

tau

Fig. 6.3.

Error comparison between the different schemes for ν = 1.9

q’’ + 2.8 * q’ + q = 0

0

( tau = 0.01, 0.02, 0.05, 0.1, 0.2, 0.5 )

10

expl Euler midpoint (5.4) scheme (5.7) scheme (5.8) scheme (5.9)

−2

10

−4

max−err

10

−6

10

−8

10

−10

10

−2

10

−1

10

tau

Fig. 6.4.

Error comparison between the different schemes for ν = 2.8

0

10

15.6 Numerical Experiments

q’’ + 2.8 * q’ + q = 0

0

637

( tau = 0.01, 0.02, 0.05, 0.1, 0.2, 0.5 )

10

expl Euler midpoint (5.4) scheme (5.7) scheme (5.8) scheme (5.9)

−2

10

max−err−H

−4

10

−6

10

−8

10

−10

10

−2

−1

10

0

10

10

tau

Fig. 6.5.

Energy error comparison between the different schemes for ν = 0.6

q’’ + 1.3 * q’ + q = 0

0

( tau = 0.01, 0.02, 0.05, 0.1, 0.2, 0.5 )

10

expl Euler midpoint (5.4) scheme (5.7) scheme (5.8) scheme (5.9)

−2

10

−4

max−err−H

10

−6

10

−8

10

−10

10

−12

10

−2

10

−1

10

tau

Fig. 6.6.

Energy error comparison between the different schemes for ν = 1.3

0

10

638

15. Structure Preserving Schemes for Birkhoff Systems

q’’ + 1.9 * q’ + q = 0

0

( tau = 0.01, 0.02, 0.05, 0.1, 0.2, 0.5 )

10

expl Euler midpoint (5.4) scheme (5.7) scheme (5.8) scheme (5.9)

−2

10

−4

max−err−H

10

−6

10

−8

10

−10

10

−12

10

−14

10

−2

−1

10

0

10

10

tau

Fig. 6.7.

Energy error comparison between the different schemes for ν = 1.9

q’’ + 2.8 * q’ + q = 0

0

( tau = 0.01, 0.02, 0.05, 0.1, 0.2, 0.5 )

10

expl Euler midpoint (5.4) scheme (5.7) scheme (5.8) scheme (5.9)

−2

10

max−err−H

−4

10

−6

10

−8

10

−10

10

−2

10

−1

10

tau

Fig. 6.8.

Energy error comparison between the different schemes for ν = 2.8

0

10

Bibliography

[AH75] R.W. Atherton and G.M. Homsy: On the existence and formulation of variational principles for nonlinear differential equations. Studies in Applied Mathematics, LIV(1):1531– 1551, (1975). [Arn89] V. I. Arnold: Mathematical Methods of Classical Mechanics. Springer-Verlag, GTM 60, Berlin Heidelberg, Second edition, (1989). [FQ87] K. Feng and M.Z. Qin: The symplectic methods for the computation of Hamiltonian equations. In Y. L. Zhu and B. Y. Guo, editors, Numerical Methods for Partial Differential Equations, Lecture Notes in Mathematics 1297, pages 1–37. Springer, Berlin, (1987). [FW91a] K. Feng and D.L. Wang: A Note on conservation laws of symplectic difference schemes for Hamiltonian systems. J. Comput. Math., 9(3):229–237, (1991). [FW91b] K. Feng and D.L. Wang: Symplectic difference schemes for Hamiltonian systems in general symplectic structure. J. Comput. Math., 9(1):86–96, (1991). [GLSM01] Y.X. Guo, S.K. Luo, M. Shang, and F.X. Mei: Birkhoffian formulations of nonholonomic constrained systems. Reports on Mathematical Physics, 47:313–322, (2001). [HLW02] E. Hairer, Ch. Lubich, and G. Wanner: Geometric Numerical Integration. Number 31 in Springer Series in Computational Mathematics. Springer-Verlag, Berlin, (2002). [MP91] E. Massa and E. Pagani: Classical dynamics of non-holonomic systems : a geometric approach. Annales de l’institut Henri Poincar (A) Physique thorique, 55(1):511–544, (1991). [MR99] J. E. Marsden and T. S. Ratiu: Introduction to Mechanics and Symmetry. Number 17 in Texts in Applied Mathematics. Springer-Verlag, Berlin, second edition, (1999). [MW01] J. E. Marsden and M. West: Discrete mechanics and variational integrators. Acta Numerica, 10:357–514, (2001). [QZ92] M. Z. Qin and W. J. Zhu: Construction of higher order symplectic schemes by composition. Computing, 47:309–321, (1992). [San83a] R.M. Santilli; Foundations of Theoretical Mechanics I. Springer-Verlag, New York, Second edition, (1983). [San83b] R.M. Santilli: Foundations of Theoretical Mechanics II. Springer-Verlag, New York, Second edition, (1983). [SQ03] H. L. Su and M. Z. Qin: Symplectic schemes for Birkhoffian system. Technical Report arXiv: math-ph/0301001, (2003). [SSC94] J. M. Sanz-Serna and M. P. Calvo: Numerical Hamiltonian Problems. AMMC 7. Chapman & Hall, London, (1994). [SSQS07] H. L. Su, Y.J. Sun, M. Z. Qin, and R. Scherer: Symplectic schemes for Birkhoffian system. Inter J of Pure and Applied Math, 40(3):341–366, (2007). [SVC95] W. Sarlet, A. Vandecasteele, and F. Cantrijn: Derivations of forms along a map: The framework for time-dependent second-order equations. Diff. Geom. Appl., 5:171–203, (1995). [Yos90] H. Yoshida: Construction of higher order symplectic integrators. Physics Letters A, 150:262–268, (1990).

Chapter 16. Multisymplectic and Variational Integrators

Recently, multisymplectic discretizations have been drawing much attention and, therefore, have become the vigorous component of the structure-preserving algorithms. In this chapter, we systematically develop what our research group has achieved in the field of multisymplectic discretizations. Some very interesting new issues arising in this field are also given. Multisymplectic and variational integrators are studied from a comparative point of view. The implementation issues of multisymplectic integrators are discussed, and composition methods to construct higher order multisymplectic integrators are presented. The equivalence of variational integrators to multisymplectic integrators is proved. Several generalizations are also described.

16.1 Introduction The introduction of symplectic integrators is a milestone in the development of numerical analysis[Fen85] . It has led to the establishment of structure-preserving algorithms, a very promising subject. Due to its high accuracy, good stability and, in particular, the capability for long-term computation, the structure-preserving algorithms have proved to be very powerful in numerical simulations. The applications of structure-preserving algorithms can be found on diverse branches of physics, such as celestial mechanics, quantum mechanics, fluid dynamics, geophysics[LQHD07,MPSM01,WHT96] , etc. Symplectic algorithms for finite dimensional Hamiltonian systems have been well established. They not only bring new insights into existing methods but also lead to many powerful new numerical methods. The structure-preserving algorithms for infinite dimensional Hamiltonian systems are comparatively less explored. Symplectic integrators for infinite dimensional Hamiltonian systems were also considered [Qin90,LQ88,Qin87,Qin97a] . The basic idea is, first to discretize the space variables appropriately so that the resulting semi-discrete system is a Hamiltonian system in time; and second, to apply symplectic methods to this semi-discrete system. The symplectic integrator obtained in this way preserves a symplectic form which is a sum over the discrete space variables. In spite of its success, a problem remains: the change of the symplectic structure over the spatial domain is not reflected in such methods. This problem was solved by introducing the concept of multisymplectic integrators (Bridges and Reich[BR01a,BR06] ). In general, an infinite dimensional Hamiltonian system can be reformulated as a multisymplectic Hamiltonian system in which associated to every time and space direction, there exists a symplectic structure and a

642

16. Multisymplectic and Variational Integrators

multisymplectic conservation law is satisfied. The multisymplectic conservation law is completely local and reflects the change of the symplecticity over the space domain. A multisymplectic integrator is a numerical scheme for the multisymplectic Hamiltonian system which preserves a discrete multisymplectic conservation law, characterizing the spatial change of the discrete symplectic structure. The multisymplectic integrator is the direct generalization of the symplectic integrator and has good performance in conserving local conservation laws. A disadvantage of the multisymplectic integrator is the introduction of many new variables which usually are not needed in numerical experiments. To solve this problem, we can eliminate the additional variables from some multisymplectic integrators and obtain a series of new schemes for the equations considered. On the construction of multisymplectic integrators, it was proved that using symplectic Runge–Kutta integrators in both directions lead to multisymplectic integrators[Rei00] . In this chapter, another approach, namely the composition method will be presented. The multisymplectic integrator is based on the Hamiltonian formalism. In the Lagrangian formalism, a geometric-variational approach to continuous and discrete mechanics and field theories is known by Marsden, Patrik, and Shkoller[MPS98] . The multisymplectic form is obtained directly from the variational principle, staying entirely on the Lagrangian side, but the local energy and momentum conservation laws are not particularly addressed. By disretizing the Lagrangian and using a discrete variational principle, variational integrators are obtained, which satisfy a discrete multisymplectic form[MPS98] . Taking the sine-Gordon equation and the nonlinear Schr¨odinger equation as examples, we will show that some variational integrators are equivalent to multisymplectic integrators. In addition to the standard multisymplectic and variational integrators, we have more ambitious goal of presenting some generalizations, including multisymplectic Fourier pseudospectral methods on real space, nonconservative multisymplectic Hamiltonian systems, constructions of multisymplectic integrators for modified equations and multisymplectic Birkhoffian systems[SQ01,SQWR08,SQS07] . This chapter is organized as follows. In the next Section 16.2, the basic theory of multisymplectic geometry and multisymplectic Hamiltonian systems is presented. Section 16.3 is devoted to developing multisymplectic integrators. In Section 16.4, the variational integrators are discussed. In Section 16.5, some generalizations are given.

16.2 Multisymplectic Geometry and Multisymplectic Hamiltonian Systems In this section, the basic theory needed for multisymplectic and variational integrators is discussed. The basic theory includes multisymplectic geometry and multisymplectic Hamiltonian system. We will present the theory from the perspective of the total variation[Lee82,Lee87] , always named Lee variational integrator (see Chapter 14)[Che02,CGW03] .

16.2 Multisymplectic Geometry and Multisymplectic Hamiltonian Systems

643

1. Multisymplectic geometry Exclusively, local coordinates are used and the notion of prolongation spaces instead of jet bundles[Olv86,Che05c] is employed. The covariant configuration space is denoted by X × U and X represents the space of independent variables with coordinates xμ (μ = 1, 2, · · · , n, 0), and U the space of dependent variables with coordinates uA (A = 1, 2, · · · , N ). The first-order prolongation of X × U is defined to be (2.1) U (1) = X × U × U1 , where U1 represents the space consisting of first-order partial derivatives of uA with respect to xμ . Let φ : X → U be a smooth function, then its first prolongation is denoted by pr1 φ = (xμ , φA , φA μ ). A Lagrangian density, L is defined as follows: L : U (1) −→ Λn+1 (X),

(2.2)

n+1 x, L(pr1 φ) = L(xμ , φA , φA μ)d

where Λn+1 (X) is the space of n + 1 forms over X. Corresponding to the Lagrangian density (2.2), the action functional is defined by n+1 L(xμ , φA , φA x, M is an open set in X. (2.3) S(φ) = μ)d M

Let V be a vector field on X × U with the form V = ξ μ (x)

∂ ∂ + αA (x, u) , ∂ xμ ∂ uA

where x = (x1 , · · · , xn , x0 ), u = (u1 , · · · , uN ) and we use Einstein summation convention here. The flow exp(λV ) of the vector field V is a one-parameter transformation group of ˜ → U depending X × U and transforms a map φ : M → U to a family of maps φ˜ : M on the parameter λ. Now, we calculate the variation of the action functional (2.3). For simplicity , let n = 1, N = 1 and x1 = x, x0 = t, u1 = u, α1 = α, then it follows that   d  d  ˜ ˜ φ˜x˜ , φ˜˜) d x S(φ) = L(˜ x, t˜, φ, ˜ ∧ d t˜ = A + B, δS =   t dλ

dλ

λ=0

where

-



A = M



+

˜ M







∂L ∂L ∂L + Dt φt − L + Dx φt ∂t ∂ φt ∂ φx







∂L ∂L ∂L + Dx φx − L + Dt φx ∂x ∂ φx ∂ φt

 +

λ=0





ξ0

ξ1

 !

∂L ∂L ∂L − Dx − Dt α d x ∧ d t, ∂φ ∂ φx ∂ φt

(2.4)

644

16. Multisymplectic and Variational Integrators

and -



B = ∂M



+  +

L−





∂L ∂L φt − L d x − φt d t ξ 0 ∂ φt ∂ φx





∂L ∂L φx d t + φx d x ξ 1 ∂ φx ∂ φt

 !

∂L ∂L dt − dx α . ∂ φx ∂ φt

(2.5)

If ξ 1 (x), ξ 0 (x), and α(x, t, φ(x, t)) have compact support on M , then B = 0. In this case, with the requirement of δS = 0 and from (2.4), the variation ξ 0 yields the local energy evolution equation     ∂L ∂L ∂L (2.6) + Dt φt − L + Dx φt = 0, ∂t

∂ φt

∂ φx

and the variation ξ 1 the local momentum evolution equation     ∂L ∂L ∂L + Dx φx − L + Dt φx = 0. ∂x

∂ φx

∂ φt

(2.7)

For a conservative L, i.e., the one that does not depend on x, t explicitly, (2.6) and (2.7) become the local energy conservation law and the local momentum conservation law respectively. The variation α yields the Euler–Lagrange equation ∂L ∂L ∂L − Dx − Dt = 0. ∂φ ∂φx ∂φt

(2.8)

If the condition that ξ 1 (x, t), ξ 0 (x, t), α(x, t, φ(x, t)) have compact support on M is not imposed, then from the boundary integral B, we can define the Cartan form ΘL =

  ∂L ∂L ∂L ∂L dφ ∧ dt − dφ ∧ dx + L − φx − φt d x ∧ d t, ∂ φt ∂ φx ∂ φt ∂φx

which satisfies using the interior product and the pull-back mapping ()∗ ,  1 ∗  1  B= pr φ pr V ΘL .

(2.9)

(2.10)

∂M

The multisymplectic form is defined to be ΩL = d ΘL . Theorem 2.1. [MPS98,GAR73] Suppose φ is the solution of (2.8), and let η λ and ζ λ be two one-parameter symmetry groups of Equation (2.8), and V1 and V2 be the corresponding infinitesimal symmetries, then we have the multisymplectic form formula  1 ∗  1  pr φ pr V1 pr1 V2 ΩL = 0. (2.11) ∂M

16.2 Multisymplectic Geometry and Multisymplectic Hamiltonian Systems

645

2. Multisymplectic Hamiltonian systems A large class of partial differential equations can be represented as[BR06,Bri97] M z t + Kz x = (z S(z),

(2.12)

where z ∈ Rn , M and K are antisymmetric matrices in Rn×n , n ≥ 3 and S : Rn → R is a smooth function. Here for simplicity, we only consider one space dimension. We call (2.12) a multisymplectic Hamiltonian system, since it possesses a multisymplectic conservation law Dt ω + Dx κ = 0, where Dt =

(2.13)

d d , Dx = and ω and κ are the presymplectic form dt dx

ω=

1 d z ∧ M d z, 2

κ=

1 d z ∧ K d z, 2

which are associated to the time direction and the space direction, respectively. The system (2.12) satisfies a local energy conservation law Dt E + Dx F = 0,

(2.14)

with energy density 1 2

E = S(z) − z T Kz x and energy flux 1 2

F = z T Kz t . The system (2.12) also has a local momentum conservation law Dt I + Dx G = 0

(2.15)

with momentum density 1 2

I = zTM zx and momentum flux 1 2

G = S(z) − z T M z t . The multisymplectic Hamiltonian system can be obtained from the Lagrangian density and the covariant Legendre transform, or Legendre–Hodge transformation[Bri06] . The relationship between the Lagrangian and the Hamiltonian formalisms is explained in the following diagram, where in each line the corresponding equations are given[Che05c,Che02,LQ02] .

646

16. Multisymplectic and Variational Integrators

L = L(φ, φx , φt ) ⇐⇒ H = L −

∂L ∂L ∂L − Dx − Dt = 0 ⇐⇒ M z t + Kz x = (z S(z), ∂φ ∂ φx ∂ φt

-

(pr1 φ)∗ (pr1 V1

∂M

%

Dt % Dx

∂L ∂L φx − φt , ∂ φx ∂ φt

pr1 V2 %

&

∂L φt − L ∂ φt

+ Dx &

∂L φx − L ∂ φx

% + Dt

ΩL ) = 0 ⇐⇒ Dt ω + Dx κ = 0,

∂L φt ∂ φx ∂L φx ∂ φt

& = 0 ⇐⇒ Dt E + Dx F = 0, & = 0 ⇐⇒ Dt I + Dx G = 0.

16.3 Multisymplectic Integrators and Composition Methods The concept of the multisymplectic integrators for the system (2.12) was introduced by Bridges and Reich[BR01a] . A multisymplectic integrator is a numerical scheme for (2.12) which preserves a discrete multisymplectic conservation law. The multisymplectic integrator is the direct generalization of the symplectic integrator and has good performance in maintaining local conservation laws. Using symplectic Runge–Kutta integrators in both directions leads to multisymplectic integrators[Rei00] . A popular multisymplectic integrator is the multisymplectic Preissman integrator which is obtained by using the midpoint method in both directions. Discretizing (2.12) by the midpoint method in both directions with step-size Δt and Δτ yields M

z j+1 − z ji+ 1 i+ 1 2

2

Δt

j+ 1

j+ 1 2

z 2 − zi + K i+1 Δx

  j+ 1 = ∇z S z i+ 12 , 2

(3.1)

where Δ t and Δ x are the time step size and space step size, respectively, and   1 j+1 j+1 z ji ≈ z(iΔz, jΔt), z j+1 + z z , 1 = i i+1 i+ 2 2   j+ 12 1 j j j+1 j+1 z i+ 1 = z i + z i+1 + z i + z i+1 , etc. 2

4

The scheme (3.1) satisfies the discrete multisymplectic conservation law j+1 j ωi+ 1 − ωi+ 1 2

2

Δt

j+ 1

+

j+ 1 2

κi+12 − κi Δx

= 0,

(3.2)

which can be proved by direct calculations. Example 3.1. First, consider the sine-Gordon equation[Che06b,WM01] utt − uxx + sin u = 0.

(3.3)

16.3 Multisymplectic Integrators and Composition Methods

647

Introducing the new variables v = ut and w = ux , Equation (3.3) is equivalent to the system −vt + wx = sin u, ut = v, −ux = −w, (3.4) which can be represented as M1 z t + K1 z x = ∇z S1 (z), where

1 2

z = (u, v, w)T ,

and

⎛

S1 (z) = (v 2 − w2 ) − cos (u)

−1

0

⎜ M1 = ⎝ 1

0

⎞

⎟ 0 ⎠, 0

0

0

(3.5)

0

⎛

0

⎜ K1 = ⎝ 0

0 1

⎞

⎟ 0 0 ⎠. −1 0 0

Applying the multisymplectic integrator (3.1) to (3.3) yields −

j+1 j vi+ 1 − vi+ 1 2

2

Δt

j+ 1

j+ 1 2

w 2 − wi + i+1 Δx uj+1 i+ 1 2

−

uji+ 1 2

Δt j+ 1

j+ 1

= sin ui+ 12 , 2

j+ 1

= vi+ 12 ,

(3.6)

2

j+ 1 2

u 2 − ui − i+1 Δx

j+ 1

= −wi+ 12 . 2

Eliminating v and w from (3.6), a nine-point integrator for u is derived j j−1 uj+1 (i) − 2u(i) + u(i)

Δ t2

−

(j)

(j)

(j)

ui+1 − 2ui + ui−1 + sin (¯ uji ) = 0, Δ x2

(3.7)

where ul(i) = (j)

um =

uli−1 + 2uli + uli+1 , 4 j j+1 uj−1 m +2um +um , 4

sin(¯ uji ) = u ¯ji =

l = j − 1, j, j + 1, m = i − 1, i, i + 1,

 1 uji−1 ) + sin (¯ ui−1j−1 ) + sin (¯ uij−1 ) , sin (¯ uji ) + sin (¯ 4

1 j j+1  u + uji+1 + uj+1 , i+1 + ui 4 i

u ¯j−1 i−1 =

 1  j−1 + uj−1 + uji + uji−1 , u i 4 i−1

u ¯ji−1 =

 1 j + uji + uj+1 + uj+1 u i i−1 , 4 i−1

= u ¯j−1 i

1  j−1 j j + uj−1 u i+1 + ui+1 + ui . 4 i

Second, consider the nonlinear Schr¨odinger equation, written in the form   iψt + ψxx + V  |ψ|2 ψ = 0.

(3.8)

648

16. Multisymplectic and Variational Integrators

Using ψ = p + iq and introducing a pair of conjugate momenta v = px , w = qx , Equation (3.8) can be represented[Che06b,Che05b,CQ02,CQT02,SHQ06,SMM04,SQL06,Che04a] as a multisymplectic Hamiltonian system M2 zt + K2 zx = (z S2 (z), where z = (p, q, v, w)T , and

⎛

0

1

S2 (z) =

0

0

 1 2 v + w2 + V (p2 + q 2 ) 2

⎞

⎛

⎟ ⎜ ⎜ −1 0 0 0 ⎟ ⎟ M2 = ⎜ ⎜ 0 0 0 0 ⎟, ⎠ ⎝ 0 0 0 0

(3.9)

−1

0 0

⎜ ⎜ 0 0 K2 = ⎜ ⎜ 1 0 ⎝ 0 1

⎞

0

⎟ −1 ⎟ ⎟. 0 ⎟ ⎠ 0

0 0 0

From the multisymplectic Preissman integrator (3.1), we obtain a six-point integrator for (3.8) j+1 j j+ 1 j+ 1 j+ 1 ψ[i] − ψ[i] ψi+12 − 2ψi 2 + ψi−12 1 i + Gi,j = 0, (3.10) + 2 Δt Δx 2 where   1 r ψ[i] ψi−1,r + 2ψi,r + ψi+1,r , r = j, j + 1, = 4

       j+ 1 2 j+ 1  j+ 1 2 j+ 1 Gi,j = V  ψi− 12  ψi− 12 + V  ψi+ 12  ψi+ 12 . 2

2

2

2

Third, consider the KdV equation (Korteweg & de Vries) ut + 3(u2 )x + uxxx = 0.

(3.11)

Introducing the new variables φ, v and w, Equation (3.11) can be represented as M3 z t + K3 z x = (z S3 (z), where z = (φ, u, v, w)T , and

⎛ 0

⎜ ⎜ 1 ⎜ − M3 = ⎜ ⎜ 2 ⎜ 0 ⎝ 0

1 2

0 0 0

S3 (z) =

⎞ 0

0

⎟ ⎟ 0 0 ⎟ ⎟, ⎟ 0 0 ⎟ ⎠ 0 0

(3.12)

1 2 v + u2 − uw 2 ⎛

0

0

⎜ ⎜ 0 0 K3 = ⎜ ⎜ 0 1 ⎝ −1 0

0 −1 0 0

1

⎞

⎟ 0 ⎟ ⎟. 0 ⎟ ⎠ 0

From the multisymplectic Preissman integrator (3.1), we obtain an eight-point integrator

16.3 Multisymplectic Integrators and Composition Methods

j uj+1 (i) − u(i)

Δt

j+ 1

j+ 1

j+ 1

649

j+ 1

u 2 − 3ui 2 + 3ui−12 − ui−22 u ¯2 − u ¯2i−1 + 3 i+1 + i+1 = 0, 2Δ x Δ x3

(3.13)

where ul(i) =

uli−2 + 3uli−1 + 3uli + uli+1 , 8

l = j, j + 1,

u ¯2m =

j+ 1  1  j+ 12 2 (um+1 ) + (um 2 )2 , 2

m = i − 1, i + 1.

A twelve-point integrator for the KdV equation is known[ZQ00,AM04,MM05] , which can be reduced to the eight-point integrator (3.13). Numerical experiments with the integrators mentioned above are given in[WM01,CQT02,ZQ00] . For other soliton equations such as the ZK equation and the KP equation, similar results are obtained[Che03,LQ02] . The coupled Klein–Gordon–Schr¨odinger equation[KLX06] 1 2

i ψt + ψxx + ψϕ = 0, ϕtt − ϕxx + ϕ − |ψ|2 = 0,

i=

√ −1

(3.14)

is a classical model which describes interaction between conservative complex neutron field and neutral meson Yukawa in quantum field theory. KGS equation with initial boundary value conditions ψ(0, x) = ψ0 (x), ϕ(0, x) = ϕ0 (x), ϕt (0, x) = ϕ1 (x), ψ(t, xL ) = ψ(t, xR ) = ϕ(t, xL ) = ϕ(t, xR ) = 0,

(3.15) (3.16)

where ψ0 (x), ϕ0 (x) and ϕ1 (x) are known functions. The problems (3.14), (3.15) and (3.16) has conservative quantity 2

-

ψ =

xR

ψ ψ¯ d x = 1.

xL

Setting ψ = p + i q, ψx = px + i qx = f + i g, pt = v,

ϕx = w,

z = (p, q, f, ϕ, v, w)T .

The multisymplectic formation of KGS system (3.14) is

650

16. Multisymplectic and Variational Integrators

⎧ 1 ⎪ qt + fx = −ϕp, ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ pt + gx = −ϕq, ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ − px = f, ⎪ ⎪ 2 2 ⎪ ⎪ ⎨ 1 1 − qx = − g, 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 1 1 ⎪ − vt + wx = ϕ − (p2 + q 2 ), ⎪ ⎪ 2 2 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ϕt = v, ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ 1 1 ⎩ − ϕx = − w. 2

(3.17)

2

System (3.17) can be written in standard Bridge form M

∂z ∂z +K = ∇ S, ∂t ∂x

(3.18)

where matrices M and K (3.18) are ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ 1⎜ M= ⎜ 2⎜ ⎜ ⎜ ⎜ ⎝

−2

0

2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

−1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

⎞

⎛

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎠

⎜ ⎜ ⎜ ⎜ ⎜ 1⎜ K= ⎜ 2⎜ ⎜ ⎜ ⎜ ⎝

0

⎞

0

0

1

0

0

0

0

0

0

1

0

0

−1

0

0

0

0

0

0

−1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 1 ⎟ ⎟ ⎟ 0 ⎠

0

0

0

0

−1

0

0

respectively, and the Hamiltonian function is 1 2

1 4

S(z) = − ϕ(p2 + q 2 ) + (ϕ2 + v 2 − w2 − f 2 − g 2 ). For the three local conservation laws corresponding to (3.17), (3.18), we have ω(z) = −2 d p ∧ d q − d ϕ ∧ d v, κ(z) = d p ∧ d f + d q ∧ d q + d ϕ ∧ d w, 1 2

1 4

E(z) = − ϕ(p2 + q 2 ) + (ϕ2 + v 2 − pfx − qgx − ϕwx ), 1 4

(3.19)

F (z) = (pft + qgt + ϕwt − f pt − gqt − rw), 1 2

1 4

1 2

I(z) = − ϕ(p2 + q 2 ) + (ϕ2 − w2 − f 2 − g 2 + ϕvt ) + (pqt − qpt ), 1 4

G(z) = (−2pg + 2qf − qvx + vw).

16.3 Multisymplectic Integrators and Composition Methods

651

Recently, many math physical equations can be solved by Multisymlectic methods, such as Gross–Pitaevskii equation[TM03,TMZM08] , Maxwell’s equations[SQS07,SQ03,CYWB06,STMM07] ,

Camassa–Holm equation[Dah07] , Kadomtsev–Petviashvili equation[JYJ06] , Seismic wave equation[Che04b,Che04a,Che07a,Che07c,Che07b] , Dirac equation[HL04] , and nonlinear “good” Boussinesq equation[HZQ03,Che05a] , etc. Now, let us discuss the composition method for constructing high order multisymplectic integrators[Che05c,CQ03] . First, recall the definition of a composition method for ODEs[Yos90,QZ92,Suz92] : Suppose there are n integrators with corresponding operators s1 (τ ), s2 (τ ), · · ·, sn (τ ) of corresponding order p1 , p2 , · · · , pn , respectively, having maximal order μ =maxi (pi ). If there exists constants c1 , c2 , · · · , cn such that the order of the integrator whose operator is the composition s1 (c1 τ )s2 (c2 τ ) · · · sn (cn τ ) is m > μ, then the new integrator is called composition integrator of the original n integrators. This construction of higher order integrators from the lower order ones is called the composition method. While constructing higher order integrators, the main task is to determine constants c1 , c2 , · · ·, cn such that the scheme with the corresponding operator Gm (τ ) = s1 (c1 τ )s2 (c2 τ ) · · · sn (cn τ ) has order m > μ. Now, we will present the basic formula for determining the constants ci (i = 1, · · · , n). For this purpose, we introduce the symmetrization operator S S(xp z q ) =

p!q!  Pm (xp z q ), (p + q) ! Pm

where x, z are arbitrary noncommutable operators, Pm denotes the summation of all the operators obtained in all possible ways of permutation[Suz92] . We also introduce a time-ordering operator P :  xi xj , if i < j; P (xi xj ) = xj xi , if j < i, where xi , xj are noncommutable operators[Suz92] . Set Gm (τ ) = s1 (c1 τ ) · · · sn (cn τ ). The condition on which Gm has order m reads P S(xn1 1 xn2 2 xn3 3 · · ·) = 0,

n 

ci = 1,

(3.20)

i=1

where n1 + 2n2 + 3n3 + · · · ≤ m, excluding n2 = n3 = · · · = 0. Given a multisymplectic integrator for (2.12) with accuracy of order O(τ p + τq ),     zi,j ), (3.21) M s(τ )zi,j + K s( τ )zi,j = ∇z (˜ where s(τ ) and s( τ ) are discrete operators in t-direction and x-direction respectively, and τ and τ are time step and space step respectively. z˜i,j = fs,s (zi,j ) is a function of τ ). zi,j corresponding to the operators s(τ ) and s(

652

16. Multisymplectic and Variational Integrators

Suppose Gm (τ ) is the composition operator of s(τ ) with accuracy of order  n ( O(τ m ), and G τ ) is the composition operator of s( τ ) with accuracy of order O( τ n ), then the multisymplectic integrator      n ( τ )zi,j = ∇z S(˜ zi,j ) (3.22) M Gm (τ )zi,j + K G   m has accuracy of order O τ + τn .

16.4 Variational Integrators In this section, variational integrators are discussed. First, we present Veselov-type dicretizations of first-order multisymplectic field theory developed in [MPS98] . For sim = (xi , tj ) and U = (uij ) plicity, let n = 1, N = 1, X = (x, t), U = (u), and take X as the discrete versions of X and U . It is more suitable to use only the indices of the grid and set X = (i, j). A rectangle 2 of X is an ordered quadruple of the form   2 = (i, j), (i + 1, j), (i + 1, j + 1), (i, j + 1) . (4.1) The i − th component of 2 is the i − th vertex of the rectangle, denoted by 2i . A point (i, j) ∈ X is touched by a rectangle if it is a vertex of that rectangle. If M ⊆ X, then (i, j) is an interior point of M, if M contains all four rectangles that touch (i, j). We denote M as the union of all rectangles touching interior points of M. A boundary point of M is a point in M which is not an interior point. If M = M, we call M regular. int M is the set of the interior points of M, and ∂M is the set of boundary points. The discrete first-order prolongation of X × U is defined by U(1) ≡ (2; uij , ui+1j , ui+1j+1 , uij+1 ), and the first order prolongation of the discrete map ϕ : X → U; ϕ(i, j) := ϕi,j by pr1 ϕ ≡ (2; ϕij , ϕi+1j , ϕi+1j+1 , ϕij+1 ).

(4.2)

(1)

→ R, we define the discrete Corresponding to a discrete Lagrangian L : U functional   L(pr1 ϕ)ΔxΔt = L(2, ϕij , ϕi+1j , ϕi+1j+1 , ϕij+1 )ΔxΔt, (4.3) S(ϕ) = 2⊂M

2⊂M

where Δx and Δt are the grid sizes in direction x and t, and M is a subset of X. In this chapter, only an equally spaced grid is considered. Now for brevity of notations, let M = [a, b] × [c, d] be a rectangular domain and consider a uniform rectangular subdivision a = x0 < x1 < · · · < xM −1 < xM = b, c = t0 < t1 < · · · < tN −1 < tN = d, xi = a + i Δ x,

tj = c + j Δ t,

M Δ x = b − a,

N Δ t = d − c.

i = 0, 1, · · · , M, j = 0, 1, · · · , N,

(4.4)

16.4 Variational Integrators

653

For autonomous Lagrangian and uniform rectangular subdivisions, the discrete action functional takes the form S(ϕ) =

M −1 N −1  

  L ϕij , ϕi+1j , ϕi+1j+1 , ϕij+1 Δ x Δ t.

(4.5)

i=0 j=0

Using the discrete variational principle, we obtain the discrete Euler–Lagrange equation (variational integrator) D1 Lij + D2 Li−1j + D3 Li−1j−1 + D4 Lij−1 = 0, which satisfies the discrete multisymplectic form formula ⎛    ⎝ (pr1 ϕ)∗ pr1 V1 pr1 V2 2;2∩∂ M=∅

l;2l ∈∂

(4.6)

⎞

 l

ΩL ⎠ = 0,

(4.7)

M

where ΩlL = d ΘLl (l = 1, · · · , 4) and V1 and V2 are solutions of the linearized equation of (4.6). Now the discretizations of an autonomous Lagrangian L(ϕ, ϕx , ϕt ) is considered  ϕi+1,j+ 1 − ϕij+ 1 ϕi+ 1 j+1 − ϕi+ 1 j  2 2 2 2 , , L(ϕij , ϕi+1,j , ϕi+1,j+1 , ϕi,j+1 ) = L ϕ¯ij , Δx Δt (4.8) where ϕ¯ij =

1 (ϕij + ϕi+1j + ϕi+1j+1 + ϕij+1 ) , 4

ϕij+ 12 =

1 (ϕij + ϕij+1 ) , 2

ϕi+ 12 j+1 =

1 (ϕij+1 + ϕi+1j+1 ) 2

etc. For the discrete Lagrangian, the discrete Euler–Lagrange equation (4.6) is a ninepoint variational integrator. The following results demonstrate the equivalence of variational integrators and multisymplectic integrators. Consider the sine-Gordon equation (3.3), then the Lagrangian is given by 1 2

1 2

L(u, ux , ut ) = u2x − u2t − cos (u).

(4.9)

The discrete Euler–Lagrange equation (4.6) corresponding to (4.9) is just the ninepoint integrator (3.7). Consider the nonlinear Schr¨odinger equation (3.8), then the Lagrangian for (3.8) is given by L(p, q, px , qx , pt , qt ) =

A 1@ 2 px + qx2 + pqt − qpt − V (p2 + q 2 ) . 2

The discrete Euler–Lagrange equation (4.6) corresponding to (4.10) reads

(4.10)

654

16. Multisymplectic and Variational Integrators

i

j+1 j−1 ψ[i] − ψ[i]

j+ 1

+

j− 1

j+ 1 2

ψi+12 + ψi+12 − 2ψi

2Δ t 1 1 + Gi,j + Gi,j−1 = 0. 4 4

j− 1 2

− 2ψi Δ x2

j+ 1

j− 1

+ ψi−12 + ψi−12

(4.11)

The integrator (4.11) is equivalent to the integrator (3.10), since replacing j by j − 1 in (3.10) and adding the resulting equation to (3.10) leads to (4.11) (see [CQ03] ).

16.5 Some Generalizations In this section, some generalizations based on the multisymplectic geometry and multisymplectic Hamiltonian systems are presented. 1. Multisymplectic Fourier pseudospectral methods On Fourier space, multisymplectic Fourier pseudospectral methods were considered in [BR01b] . Now, we discuss these methods on real space [CQ01a] and take the nonlinear Schr¨odinger equation as an example. Applying the Fourier pseudospectral method to the multisymplectic system (3.9) and using the notations p = (p0 , · · · , pN −1 )T ,

q = (q0 , · · · .qN −1 )T ,

v = (v0 , · · · , vN −1 )T ,

w = (w0 , · · · .wN −1 )T ,

it follows d qj − (D1 v)j = 2(p2j + qj2 )pj , dt dp − j − (D1 w)j = 2(p2j + qj2 )qj , dt

(5.1)

(D1 p)j = vj , (D1 q)j = wj ,

where j = 0, 1, · · · , N − 1 and D1 is the first order spectral differentiation matrix. The Fourier pseudospectral semidiscretization (5.1) has N semidiscrete multisymplectic conservation laws N −1  d (D1 )j,k κjk = 0, ωj + dt

j = 0, 1, · · · , N − 1,

(5.2)

k=0

where

1 2

ωj = (d zj ∧ M d zj ),

κjk = d zj ∧ K d zk ,

and zj = (pj , qj , vj , wj )T (j = 0, 1, · · · , N − 1). 2. Nonconservative multisymplectic Hamiltonian systems Nonconservative multisymplectic Hamiltonian systems refer to those depending on

16.5 Some Generalizations

655

the independent variables explicitly. Such an example is the Schr¨odinger equation with variable coefficients[HLHKA06] . Another example is the three-dimensional scalar seismic wave equation[Che04b,Che06a,Che07a,Che07b,Che04a] ∇2 u −

1 utt = 0, c(x, y, z)2

(5.3)

where ∇2 u = uxx + uyy + uzz and c(x, y, z) is the wave velocity. Introducing the new variables v=

1 ut , c(x, y, z)

w = ux ,

p = uy ,

q = uz ,

Equation (5.3) can be rewritten as M (x, y, z)Zt + KZx + LZy + N Zz = ∇Z S(Z),

(5.4)

1 2

where Z = (u, v, w, p, q)T , S(Z) = (v 2 − w2 − p2 − q 2 ) and ⎛

0

⎜ ⎜ 1 ⎜ ⎜ c(x, y, z) ⎜ M (x, y, z) = ⎜ 0 ⎜ ⎜ ⎝ 0 0

−

⎛

K

⎛ ⎜ ⎜ ⎜ L=⎜ ⎜ ⎜ ⎝

0 0 0 −1 0

0 0 0 0 0

0 0 0 0 0

0 0 ⎜ ⎜ 0 0 ⎜ =⎜ ⎜ −1 0 ⎜ ⎝ 0 0 0 0 ⎞ 1 0 ⎟ 0 0 ⎟ ⎟ 0 0 ⎟ ⎟, ⎟ 0 0 ⎠ 0 0

1 c(x, y, z)

1 0 0 0 0

0

0

0

0

0

0 0 0

0 0 0

0 0 0

0 0 0 0 0 ⎛

⎜ ⎜ ⎜ N =⎜ ⎜ ⎜ ⎝

0 0 0 0 0

0

⎞

⎟ ⎟ ⎟ 0 ⎟ ⎟, 0 ⎟ ⎟ ⎟ 0 ⎠ 0

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

0 0 0 0 −1

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

1 0 0 0 0

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

The corresponding four presymplectic forms associated to the time direction and three space directions are respectively: ω=

1 d Z ∧ M (x, y, z) d Z, 2

κy =

1 d Z ∧ L d Z, 2

κx =

1 d Z ∧ K d Z, 2

κz =

1 d Z ∧ N d Z. 2

(5.5)

656

16. Multisymplectic and Variational Integrators

Note that the time direction presymplectic form ω depends on the space variables (x, y, z). We can also obtain the corresponding multisymplectic integrators[Che06a] . 3. Construction of multisymplectic integrators for modified equations Consider the linear wave equation utt = uxx .

(5.6)

Based on the two Hamiltonian formulations of (5.6) and using the hyperbolic functions, various symplectic integrators were constructed in[QZ93] . By deriving the corresponding Lagrangians and their discrete counterparts, these symplectic integrators were proved to be multisymplectic integrators for the modified versions of (5.6) in[SQ00] . Let us present an example. Using hyperbolic function tanh(x), we can obtain a symplectic integrator for (5.6) of accuracy O(Δ t2s + Δ x2m ):     Δt Δt uj+1 − 2uji + uj−1 = tanh 2s, Δ (2m) (uj+1 − 2uji + uij−1 ), tanh 2s, i i i 2 2 (5.7) where Δ (2m) = ∇+ ∇−

m−1 

(−1)j βj



j=0

where βj =

Δ x2 ∇+ ∇− 4

j ,

[(j !)2 22j ] [(2j + 1) ! (j + 1)]

and ∇+ and ∇− are forward and backward difference operators respectively. For m = 2 and s = 2, the integrator (5.7) is a multisymplectic integrator of the modified equation utt = uxx −

Δ t2 Δ t4 uxxxx − uxxxxxx . 6 144

(5.8)

For other hyperbolic functions, we can obtain similar results. 4. Multisymplectic Birkhoffian systems The multisymplectic Hamiltonian system can be generalized to include dissipation terms. This generalization leads to the following multisymplectic Birkhoffian system M (t, x, z)z t + K(t, x, z)z x = ∇z B(t, x, z) +

∂F ∂G + , ∂t ∂x

(5.9)

where z = (z1 , · · · , zn )T , F = (f1 , · · · , fn )T , G = (g1 , · · · , gn )T and M = (mij ) and K = (kij ) are two antisymmetric matrices with entries respectively: mij =

∂ fj ∂ fi − , ∂ zi ∂ zj

kij =

∂ gj ∂ gi − . ∂ zi ∂ zj

16.5 Some Generalizations

The system (5.9) satisfies the following multisymplectic dissipation law:     d 1 d 1 dz ∧ M dz + d z ∧ K d z = 0. dt 2

dx 2

657

(5.10)

Let us present an example[SQ03,SQWR08] . Consider the equation describing the linear damped string: utt − uxx + u + αut + βux = 0. (5.11) Introducing new variables p = ut and q = ux , the Equation (5.11) can be cast into the form of (5.9) with ⎞ ⎛ ⎞ ⎛ 0 0 −eαt−βx 0 eαt−βx 0 ⎟ ⎜ ⎟ ⎜ 0 0 0 0 0 ⎠, K = ⎝ M = ⎝ −eαt−βx ⎠, 0

0

eαt−βx

0

0

0

and 1

z = (u, p, q)T , B = − eαt−βx (u2 + p2 − q 2 + αup + βuq), 2  T T  1 αt−βx 1 αt−βx 1 αt−βx 1 F = − e p, e u, 0 , G = e q, 0, − eαt−βx . 2

2

2

2

Similarly, we can develop multisymplectic dissipation integrators for the system (5.9) which preserve a discrete version of the multisymplectic dissipation law (5.10). 5. Differential complex, methods and multisymplectic structure Differential complexes have come to play an increasingly important role in numerical analysis recently. In particular, discrete differential complexes are crucial in designing stable finite element schemes[Arn02] . With regard to discrete differential forms, a generic Hodge operator was introduced in[Hip02] . It was shown that most finite element schemes emerge as its specializations. The connection between Veselov discrete mechanics and finite element methods was first suggested in[MPS98] . Symplectic and multisymplectic structures in simple finite element methods are explored in[GjLK04] . It will be of particular significance to study the multisymplectic structure for the finite element methods by using discrete differential complexes and in particular, discrete Hodge operators[STMM07] . We will explore this issue in the future.

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Symbol

Symbol A, B  A∗ = A A , AT A A⊥ A = {Uλ , ϕλ } Ad Ad∗ adv ad∗v Br (a) Bk Bk bk , bk B(ρ), C(η), D(ζ) C Cn Ck C∞ C(z) C k (M ) i Cjk d dxi D det A div deg ω (deg f )( deg P (x)) Eτ e ex ei , {ei , fj } eta exp, Exp F (t)f

Description Matrix A = {aij ∈ M (n)} conjugate transpose of A transpose of A J-orthogonal complement of A orthogonal complement of A smooth atlas adjoint representation coadjoint representation adjoint vector field coadjoint vector field take a as the center of circle, r is the radius ball space consist of all exact k-form set of all k-boundaries Betti number order conditions of Butcher. the complex numbers complex vector space of complex n-vector space of k-times differentiable functions space of smooth functions Casimir function k-dimensional chain on M structure constant exterior derivative, exterior differential operator basis differential 1-form total differential determinant of matrix A divergence order of form (order of map) (order of polynomial)

Euler step-transient operator identity element of group exponential function of x basis, symplectic basis phase of flow with vector field a exponential map differential element of function f

664

Symbol

Symbol F Fn f∗p F(Rn ) g k (M ) G G2n,k g g∗ Gl(n), Gl(n, R), Gl(n, C) gl(n) grad H K (M, R)(HK (M, R)) H(p, q), H(z) i iX , I In , I2n id (Id) im L J I2n , J4n J"4n K ker L  L[u] = L dx LX Y, LX ω M, N M (n, m, R) M (n, m, C) M (n, R) M (n, C) M (n, F) O(n), o(n) O P p Q q p R Rn Rnp , Rnq RP n r(t) S h, s

Description a field (usually R or C) vector space (over F) of n-vectors differential of the map f in the p place a class of all differentiable function on Rn set of all k-differential form on M group, Lie group M (2n, k) nonsingular equivalent class Lie algebra dual to the Lie algebra linear group on Rn ,(Cn ) Lie algebra of n × n matrix gradient k-th cohomology (homology) group on M Hamiltonian function including map contraction, interior product identity map identity matrix, standard Euclidean structure identity image of map L momentum map symplectic structure J"4n -symplectic structure K-symplectic structure kernel of mapping L variation of L vector field Y , differential form ω of Lie derivative

manifold set of all real matrix with n-row and m-column set of all complex matrix with n-row and m-column

set of all real matrix of order n × n on Rn set of all complex matrix of order n × n on Cn set of all matrix of order n × n on Fn orthogonal group, orthogonal Lie algebra zero matrix coordinate in momentum space coordinate in configuration space order of p real number n-dimensional real vector space momentum space, configuration space in Rn real projection space order of t-tree symplectic transformation, S-transformation step of time

Symbol

Symbol SL(n), SL(n, R), SL(n, C) sl(n) SO(n) so(n) Sp(2n) sp(2n) CSp(2n) Sp(0) Sp(I) Sp(II) Sp(III) Sp-diff or Sp-Diff TM Tx M T ∗M Tx∗ M Sm u, v (U, ϕ), (V, ϕ) V Xp x, ˙ x ¨ x, xi y, y i X(M ) XH X (Rn )   A Bα α α= Cα Dα  Aα  Bα α−1 = α α C D δ σ(t) γ(t) δij α(t) Γf , Gf , gr (f ) Δt, τ, s Δx θ dθ π T Rn −→ Rn π −1 (x) = Tx Rn

665

Description special linear group, (real), (complex) Lie algebra of special linear group special orthogonal group Lie algebra of special orthogonal group symplectic group, symplectic matrix symplectic algebra, infinitesimal symplectic matrix

conformal symplectic group 0-class of symplectic matrix I-class of symplectic matrix II-class of symplectic matrix III-class of symplectic matrix symplectic diffeomorphism tangent bundle tangent space in place x cotangent bundle cotangent space in place of x symmetric group vector in Rn space local coordinate vector space vector field in place p on manifold first, second, order derivative at x x vector, coordinate component y vector, coordinate component set of all tangent vector on M Hamiltonian vector field set of all smooth vector field on Rn Darboux transformation inverse of Darboux transformation variational derivative, codifferential operator symmetry of t-tree density of t-tree Kronecker symbol essential different labelings graphic of f step size of time step size of apace differential 1-form exterior of differential 1-form projection T Rn to Rn fiber in point x

666

Symbol

Symbol ϕ∗ ω (ϕ∗ ω) ϕ∗ f (ϕ∗ f ) ϕ∗ Y (ϕ∗ Y ) × ∧ Λk (Rn ) Λn Λn (K) f Z f p Z Ω Ω# Ωb Ωk (Rn ), Ω0 (Rn ) = C ∞ (Rn )

Description pull back of differential form (push-forward) pull back of function(push-forward) pull back of vector field (push-forward) product exterior product k-th exterior bundle over Rn Lagrangian subspace K-Lagrangian subspace f transverse to Z f in the p transverse to Z standard symplectic structure lift of mapping Ω# (z1 )(z2 ) = Ω−1 (z1 , z2 ) down mapping Ωb (z1 )(z2 ) = Ω(z1 , z2 ) k-differential form on Rn

∂ ∂xI

partial derivative with respect to xi

or ∂xi

∂ (×f (  ·f  pds ω ∅ ⊗ ∩ ∪ ⊂ ∈   ◦ f ◦ g = f (g) /  ∈ ∀   ∼ = ≡ := ∼ c ∼ −→ =⇒ ⇐⇒  n  n! Cnk = = k!(n − k)! k

boundary operator rotation divergence boundary integral integral of differential form empty set tensor product set-theoretic intersection set-theoretic union inclusion element of element of composition division not element of for homomorphism approximate similarly identity definition corresponding, equivalent, congruent relation conjugate congruent mapping extrusion extrusion mutually binomial coefficient

Symbol

Description

Symbol *

+

n! n = k1 , k 2 , · · · , k r k1 !k2 ! · · · , kr ! where k1 + k2 + · · · + kr = n

(a, b) [a, b] [u, w] [A, B] [F, H] (u, v) [U, V ] B ◦

U +V P1 "P  2 ,  {ϕ, φ} a⊥b ab 1N (x) = x

multinomial coefficients. open interval closed interval Lie bracket matrix commutator Poisson bracket inner product, Euclidean inner product symplectic inner product norm of matrix direct sum symplectic direct sum inner product Poisson bracket vector a orthogonal to b (Euclidean) vector a symplectic orthogonal to b identity function

667

Index

A A(α)-stability, 550 a*–linear differential operator, 407 a∗ –Jacobian matrix, 407 A-stability, 550 ABC flow, 446 action functional of Lagrangian density, 643 Ad*-equivariant, 503 adjoint integrator, 374 adjoint method, 372 all polynomials is symplectically separability in R2n , 207 alternative canonical forms, 130 angular momentum in body description, 505 angular momentum in space description, 505 angular momentum-preserving schemes for rigid body, 525 angular velocity in body description, 505 angular velocity in space description, 505 anti-symmetric product, 117 atlas, 40 automorphism, 39 autonomous Birkhoff’s equations, 618

B B-series, 417 B-stability, 550 backward error analysis, 432 base of tangent space, 45 BCH formula, 380, 413 Betti numbers, 99 bijective, 39 bilinear antisymmetric form, 188 binary forms, 116 Birkhoffian system, 618 black (fat )vertex, 309 boundary of chains, 92 Butcher tableau, 278

C calculate the formal energy, 267 canonical equation, 170 canonical forms under orthogonal transformation, 134 canonical reductions of bilinear forms, 128 canonical transformation, 172, 188 Cartan form, 644 Cartan’s Magic formula, 106 Casimir function, 501 Cayley transformation, 193 centered Euler method, 416 centered Euler scheme, 192, 200, 231 chains, 91 characteristic equations, 477 chart, 40 Chebyshev spectral method, 508 classical Stokes theorem, 98 closed form, 84 closed nondegenerate differential 2-form, 165 coadjoint orbits, 505 coclosed form, 90 codifferential operator, 89 coefficient B-series for centered Euler scheme, 418 coefficient B-series for exact solution, 418 coefficient B-series for explicit Euler scheme, 418 coefficient B-series for implicit Euler scheme, 418 coefficient B-series for R–K method, 418 coefficient B-series for trapezoidal scheme, 418 coefficients can be determined recursively, 233 coexact form, 90 cohomology space, 98 coisotropic subspace, 138 commutativity of generator maps, 261 commutator, 124, 179

670

Index

commutator of two vector fields, 100 comparison order conditions between symplectic R–K (R–K–N) method, 302 comparison order conditions P–R–K method and symplectic P–R–K method, 318, 319, 333 compatible of two local coordinate systems, 40 complete non-integrability, 477 complexifiable, 124 complexification of real vector space and real linear transformation, 123 composition laws, 419 composition of centered Euler scheme, 372 composition of trapezoid scheme, 365 composition scheme is not A-stable, 389 compositional property of Lie series, 379 condition for centered Euler to be volumepreserving, 444 condition of symplectic P–R–K method, 303 condition of variational self-adjointness, 619 configuration space, 188 conformally K-symplectic group CSp(K, n, F), 120 conformally canonical transformation, 173, 182 conformally Hermitian, 117 conformally identical, 114 conformally orthogonal group CO(S, n, F), 120 conformally symmetric, 114 conformally symplectic group CSp(2n), 144 conformally unitary group CU (H, n, C), 120 congruence canonical forms of conformally symmetric, 130 congruence canonical forms of Hermitian matrices, 130 congruent reductions, 129 conic function, 484 conic Hamiltonian vector fields, 488 conic map, 484 conic symplectic, 484 conic symplectic map, 484 conic transformation, 488 conservation Laws, 234 conservation of spatial angular momentum theorem, 506 constrained Hamiltonian algorithm, 537 construction of the difference schemes via generating function, 213 construct volume-preserving difference schemes, 454 constructing s-scheme by 2nd kind g.f., 227

constructing s-scheme by Poincar´e type g.f., 229 constructing s-scheme via 1st kind g.f., 227 construction of volume-preserving schemes via g.f., 464 contact 1-form, 480 contact algorithm, 483 contact algorithm–C, 493 contact algorithm–P , 492 contact algorithm–Q, 492 contact difference schemes, 492 contact dynamical systems, 477 contact element, 482 contact generating function, 487 contact geometry, 477 contact Hamiltonian, 483, 492 contact map, 486 contact structure, 477, 481 contact transformation, 483 contactization of conic symplectic maps, 487 contraction, 105 convergence of symplectic difference schemes, 239 coordinate Lagrangian subspaces, 147 coordinate of tangent vector, 45 coordinate subspaces, 139 cotangent bundle, 76, 249 cotangent vector, 76 cycle, 93

D Darboux matrix, 231, 600 Darboux theorem, 168, 190 Darboux transformation, 249 De Rham theorem, 99 decomposed theorem of symplectic matrix, 155 decompositions of source-free vector fields, 452 definition of symplectic for LMM, 356 density of tree γ(t), 294 diagonal formal flow, 415 diagonal Pad´e approximant, 194 diagonally implicit method, 284 diagonally implicit symplectic R–K method, 284 diffeomorphism, 39, 102, 126, 188 diffeomorphism group, 102 differentiable manifold, 40 differentiable manifold structure, 40 differentiable mapping, 41 differentiable mapping, differential concept, 43

Index

differentiable structure, 40 differential, 45 differential k-form, 77 differential complex, 657 diophantine condition, 566, 572 diophantine frequency vectors, 552 diophantine step sizes, 569 direction field, 477 discrete energy conservation law, 587 discrete Euler–Lagrange equation, 587, 652 discrete extended Lagrange 2-form, 589 discrete Lagrange 2-form, 589 discrete Lagrangian, 652 discrete mechanics based on finite element methods, 606 discrete multisymplectic conservation law, 646 discrete multisymplectic form formula, 652 discrete total variation in the multisymplectic form, 605 discrete variational principle in total variation, 596 divergence-free system, 443, 449

E eigenvalues of infinitesimal symplectic matrix, 159 eigenvalues of symplectic matrix, 158 elementary divisor in real space, 136 elementary divisors in complex space, 136 embedded submanifold, 538 embedding submanifold, 51 endomorphism, 39 energy conservation law, 645 energy density, 645 energy equation, 644 energy flux, 645 energy-preserving schemes for rigid body, 525 epimorphism, 39 equivalent atlas, 40 Euclidean form, 118 Euclidian structure, 137 Euler centered scheme, 194 Euler equation, 506 Euler–Lagrange 1-form, 583 Euler–Lagrange equation, 644 Euler–Lagrange equation in FEM, 607 even polynomial, 159 exact form, 84 exact symplectic mapping, 551 exp maps, 412 explicit Euler method, 415

671

explicit Euler scheme, 204 explike function, 349 exponential matrix transform, 125 extended canonical two form, 595 extended configuration space, 581 extended Lagrangian 1-form, 585 extended phase space, 242 extended symplectic 2-form, 585 exterior algebra, 68 exterior differential operator, 82 exterior form, 66 exterior monomials, 70 exterior product, 64 exterior product of forms, 72

F fathers’ and sons’ relations, 297 fiber of tangent bundle, 56 first integrals, 234 first order prolongation, 594, 643 first order prolongation of V , 584 fixed point, 236 formal energy, 264 formal energy for symplectic R–K method, 333, 339 formal energy of centered Euler scheme, 344 formal power series, 265, 407 formal vector field, 432 fourth order with 3-stage scheme, 365 Frechet derivatives, 289 free rigid body, 529

G G-stability, 550 Gauss IA-IA, 472 Gauss theorem, 98 Gauss–Legendre polynomial, 279 Ge–Marsden theorem, 273 general Hamilton–Jacobi equation, 221 general linear group GL(n, F), 119 general vector field, 583 generalized Cayley transformation, 197, 198 generalized Euler schemes, 231 generalized Hamiltonian equation, 500 generalized Lagrangian subspaces, 162 generalized Noether theorem, 502 generating function, 182, 219, 233, 601 generating function and H.J. equation of the first kind, 223 generating function and H.J. equation of the second kind, 223

672

Index

generating function for Lie–Poisson system, 519 generating function for volume-preserving, 460 generating function method, 432 generating functions, 221, 255 generating functions for Lagrangian subspaces, 160 generator map, 255 generators of Sp(2n), 155 gradient map, 220 gradient mapping, 219 gradient transformation, 174 graph of gradient map, 219 graph of symplectic map, 219 Grassmann algebra, 75 Grassmann manifold, 143 Green theorem, 97 Gronwall inequality, 241 group homomorphism, 126 group of contact transformations, 483

H H-Stability, 401 H-stability interval of explicit scheme, 404 Hamilton–Jacobi equation, 182, 233, 462, 602 Hamilton–Jacobi equation for contact system, 494 Hamiltonian function, 187 Hamiltonian mechanics, 165, 168 Hamiltonian operator, 500 Hamiltonian phase flow, 171 Hamiltonian systems, 187 Hamiltonian vector fields, 167, 170 Hamiltonian–Jacobi equation, 627 heavy top, 534 Hermitian form, 117 Hermitian, anti-Hermitian, 116 high order symplectic-energy integrator, 600 Hodge operator, 88 homeomorphism, 39 homogeneous symplectic, 484 homology space, 99 homomorphism, 39 Hopf algebra, 433 horizontal variation of q i , 586 hyperplane, 478 hypersurface, 477

I immersed submanifold, 48

immersion, 47 implicit Euler method, 415 impossible to construct volume-preserving algorithms analytically depending on source-free vector fields, 452 infinitesimal generator vector field, 502 infinitesimal symplectic matrices, 190 injective, 39 integral invariant, 171 integral surface, 477 integrator S(τ ) has a formal Lie expression, 381 invariance of generating functions, 261 invariant groups for scalar products, 119 invariant integral, 192 invariant tori, 574 invariant under the group G of symplectic transformations, 234 invariant under the phase flow of any quadratic Hamiltonian, 235 invariants under congruences, 132 inverse mapping, 39 isomorphic mapping, 39 isotropic subspace, 138 isotropic, coisotropic, Lagrangian, 182

J Jacobi identity, 124, 177 J4n , J˜4n -Lagrangian submanifold, 219, 622 jet bundles, 643

K " K-Lagrangian submanifold, 623 K(z, t)-symplectic, 621 k-forms, 67 K-symplectic group, 120 K-symplectic scheme, 622 K-symplectic structure, 190 KAM iteration process, 556 KAM theorem, 551 KAM theorem of symplectic algorithms, 559 Kane–Marsden–Ortiz integrator, 587

L L-stability, 550 labeled n-tree λτ , 297 labeled P -tree, 309 labeled graph, 292 labeled trees, 298 Lagrange 2-form in FEM, 607 Lagrangian 2-form, 583 Lagrangian density, 643

Index

Lagrangian mechanics, 581 Lagrangian submanifold, 182, 250 Lagrangian subspace, 138 Lee-variational integrator, 581 left translation action, 503 Legendre transform, 645 Legendre–Hodge transformation, 645 Lie algebra, 125, 179, 190, 409 Lie algebra of conformally invariant groups, 128 Lie bracket, 409 Lie derivative, 103 Lie group, 125 Lie group action, 502 Lie series, 377 Lie–Poisson bracket, 501, 504 Lie–Poisson equation, 504 Lie–Poisson scheme, 519 Lie–Poisson systems, 501 Lie-Poisson-Hamilton-Jacobi equation, 514 lifted action, 502 linear damped oscillator, 629 linear fractional transformation, 213 linear Hamiltonian systems, 192 linear multistep method, 347 Liouville frequency vectors, 552 Liouville’s phase-volume conservation law, 189 Liouville’s theorem, 172, 443 Lobatto III A, 279, 280 Lobatto III B, 279, 280 Lobatto III C, 279, 280 Lobatto IIIC-IIIC, 472 Lobatto polynomial, 279 local coordinate systems, 40 log maps, 412 logarithmic map, 434 loglike function, 350

M M¨obius strip, 61 manifold, 40 matrix representation of subspaces, 143 maximum non-degeneracy, 477 modified centered Euler scheme of sixth order, 433 modified equation, 334, 432 modified equation for centered Euler scheme, 336, 433 modified integrators, 432 momentum, 502 momentum conservation law, 605, 645 momentum density, 645

673

momentum equation, 644 momentum flux, 645 momentum mapping, 502 monomial, 207 monomorphism, 39 monotonic rooted labeled trees, 298 Morse–Smale systems, 551 multi-stage P–R–K method, 473 multisymplectic Birkhoffian systems, 656 multisymplectic conservation law, 605, 645 multisymplectic dissipation law, 656 multisymplectic form, 644 multisymplectic form formula, 644 multisymplectic Fourier pseudospectral methods, 654 multisymplectic geometry, 643 multisymplectic Hamiltonian system, 605 multisymplectic Hamiltonian system for KdV equation, 648 multisymplectic Hamiltonian system for KGS equation, 649 multisymplectic Hamiltonian system for Schr¨odinger equation, 647 multisymplectic Hamiltonian system for sine-Gordon equation, 646 multisymplectic Hamiltonian systems, 645 multisymplectic integrators, 646 multisymplectic integrators for modified equations, 655 multisymplectic-energy-momentum integrators, 605

N natural product symplectic structure, 249 near-0 formal power series, 409 near-1 formal power series, 409 nilpotent of degree 2, 204 Noether theorem, 179 non-exceptional matrices, 197 non-existence of symplectic schemes preserving energy, 273 non-superfluous tree, 299 nonautonomous Birkhoff’s equation, 619 nonautonomous Hamiltonian System, 242 nonconservative multisymplectic Hamiltonian systems, 654 nonexistence of SLMM for nonlinear Hamiltonian systems, 356 nonresonant frequencies, 570 normal Darboux matrix, 232, 239, 494 normal Darboux matrix of a symplectic transformation, 600 normalization coefficient B-series, 418

674

Index

normalization Darboux transformation, 251 normalizing conditions, 453 null space of 1-form, 478 number of essential different labelings α(t), 294 numerical version of KAM theorem, 564

O obstruction, 450 one-form (1-form), 66 one-leg weighted Euler schemes, 231 one-parameter group of canonical maps, 221 operation ∧, 65 optimization Method, 603 orbit-preserving schemes, 527 order conditions for symplectic R–K–N method, 319 orientable differentiable manifold, 59 orientable vector spaces, 59 orthogonal group O(n, F), 119

P P–R–K method, 302 Pad´e approximation, 193 Pad´e approximation table, 196 partitions and skeletons, 418 Pfaffian theorem, 118 phase flow, 102, 221, 408 phase flow of contact system, 483 phase flow- etF , 235 phase space, 102 phase-area conservation law, 189 Poincar´e lemma, 85, 220, 222 Poincar´e transformation, 250 Poincar´e’s generating function and H.J. equation, 223 Poisson bracket, 177, 192, 499 Poisson manifold, 499 Poisson mapping, 500 Poisson scheme, 508 Poisson system, 500 Poisson theorem, 179 postprocessed vector field, 432 Preissman integrator, 646 preprocessed vector field integrators, 432 preserve all quadratic first integrals of system, 236 preserve angular momentum pT Bq, 236 preserving the contact structure, 483 presymplectic form, 645 presymplectic forms, 605 product of cotangent bundles, 249

product preservation property of Lie series, 379 prolongation spaces, 643 proper mapping, 51 properties of Lie series, 379 pull-back, 80 pull-back mapping, 374 push-forward mapping, 374

Q quadratic bilinear form, 115 quaternion form, 524

R Radau I A, 279 Radau IA-IA, 471 Radau II A, 280 Radau IIA-IIA, 472 Radau polynomial, 279 rational fraction, 200 real representation of complex vector space, 121 reduction method, 540 reflective polynomial, 158 regular submanifold, 51, 53 relationship between rooted tree and elementary differential, 293 resonant, 568 revertible approximations, 450 Riemann structure, 167 right translation, 503 rigid body in Euclidean space, 523 Rodrigue formula, 543 root isomorphism, 298 rooted n-tree, 299 rooted P -tree, 309 rooted S-tree, 321 rooted 3-tree, 298 rooted labeled n-tree ρλτ , 297 rooted labeled P -tree, 309 rooted labeled S-tree, 321 rooted labeled 3-tree, 298 rooted labeled trees, 298

S S-graph, 321 S-orthogonal group, 119 S-tree, 321 scalar product, 117 section of tangent bundle, 62 self-adjoint integrator, 376 self-adjoint method, 372

Index

semi-autonomous Birkhoff’s equation, 618 separable Hamiltonian system, 202 separable systems for source-free systems, 447 sesquilinear form, 116 simplify symplectic R–K conditions, 300 simplifying condition of R–K method, 279 Sm(2n)matrices, 600 small twist mappings, 558 some theorems about Sp(2n), 151 sons of the root, 297 source-free system, 443, 449, 467 Sp(2n) matrices, 600 SpD2n the totality of symplectic operators, 232 SpD2n the set of symplectic transformations, 601 special linear group SL(n, F ), 119 special separable source-free systems, 458 special type Sp2n (I), 150 special type Sp2n (II), 151 special type Sp2n (III), 151 special type Sp2n (I, II), 151 special types of Sp(2n), 148 stability analysis for composition scheme, 388 standard antisymmetric matrix, 192 standard symplectic structure, 169, 188, 249 star operators, 88 step size resonance, 568 step transition, 415 step-forward operator, 240 Stokes theorem, 93 structure-stability, 551 subalgebra of a Lie algebra, 179 submanifold, 46 submersion, 51 substitution law, 432 superfluous trees, 298 surjective, 39 Sylvester’s law of inertia, 132 Symm(2n) the set of symmetric transformations, 601 symm(2n) the totality of symmetric operators, 232 symmetric operators near nullity, 232 symmetric pair, 216 symmetric product, 117 symmetrical composition, 376 symmetry of tree σ(t), 294 symplectic algebra, 216 symplectic algorithms as small twist mappings, 560 symplectic basis, 145

675

symplectic conditions for R–K method, 281 symplectic explicit R–K–N method (non-redundant 5-stage fifth order), 331 symplectic form, 118 symplectic geometry, 165, 188 symplectic group, 188 symplectic group Sp(2n), 144 symplectic group Sp(2n, F ), 119 symplectic invariant algorithms, 235 symplectic leave, 505 symplectic LMM for linear Hamiltonian systems, 348 symplectic manifold, 165 symplectic map, 220 symplectic mapping, 215 symplectic matrix, 189 symplectic operators near identity, 232 symplectic pair, 217 symplectic R–K method, 277, 279 symplectic R–K–N method, 319 symplectic R–K–N method (3-stage and 4-th order), 323 symplectic schemes for Birkhoffian Systems, 625 symplectic schemes for nonautonomous system, 244 symplectic space, 137 symplectic structure, 137, 165, 215, 477 symplectic structure for trapezoidal scheme, 202 symplectic structure in product space, 215 symplectic subspace, 137 symplectic-energy integrator, 596, 602 symplectic-energy-momentum, 581 symplectically separable Hamiltonian systems, 205 symplectization of contact space, 487 symplified order conditions for symplectic R–K–N method, 327 symplified order conditions of explicit symplectic R–K method, 307

T table of coefficient ω(τ ) for trees of order  5, 435 table of coefficients σ(τ ), γ(τ ), ˘b(τ ), and b(τ ), 434 table of composition laws for the trees of order ≤ 4, 436 table of substitution law ∗ defined in for the trees of order ≤ 5, 437 tangent bundle, 56 tangent mapping, 58

676

Index

tangent space, 44 tangent vector, 43 the elementary differential, 291 the inverse function to exp, 126 the order of tree r(t), 294 time-dependent gradient map, 221 topological manifold, 40 total variation for Lagrangian mechanics, 583 total variation in Hamiltonian mechanics, 593 transversal, 54, 140, 143 transversal Lagrangian subspaces, 148 transversality condition, 181, 213, 221, 225, 227, 250, 251, 460, 623 trapezoidal method, 416 trapezoidal scheme, 201 tree, 298 trivial tangent bundle, 57 truncation, 233 two-forms (2-forms), 66

U Unitary group U (n, C), 119 Unitary product, 118

V variational integrators, 651 variational principle in Hamiltonian mechanics, 591 vector field, 62 vertical vector field, 582 Veselov–Moser algorithm, 539 volume-preserving 2-Stage P–R–K methods, 471 volume-preserving P-R–K method, 467 volume-preserving R–K method, 467 volume-preserving schemes, 444

W W -transformation, 304, 470 white (meagre) vertex, 309 Witt theorem, 132

X X-matrix, 305