Classical Mechanics: Pearson New International Edition [Third new international edition] 1292026553, 9781292026558, 9781306517928, 1306517923

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Classical Mechanics Goldstein Safko Poole Third Edition Classical Mechanics Goldstein Safko Poole Third Edition Classical Mechanics Goldstein Safko Poole Third Edition Classical Mechanics Goldstein Poole Third Edition Classical Mechanics Goldstein SafkoSafko Poole Third Edition Classical Mechanics Goldstein Safko Poole Third Edition

Classical Mechanics Classical Mechanics Classical Mechanics PEARSON NEW INTERNATIONAL EDITION PEARSON NEW INTERNATIONAL EDITION PEARSON NEW INTERNATIONAL EDITION Goldstein Goldstein Safko Poole Goldstein Safko Safko Poole Poole Third Edition Classical Mechanics Third Edition Third Edition Classical Mechanics Classical Mechanics Goldstein Goldstein Safko Poole Goldstein Safko Safko Poole Poole Third Edition Third Edition Third Edition

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CLASSICAL MECHANICS

Pearson New International Edition Classical Mechanics Goldstein Safko Poole Third Edition

PEARSON

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PEARSON

ISBN 1 0 : 1-292-02655-3 ISBN 1 3 : 978-1-292-02655-8

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library

Printed in the United States of America

Contents Survey of the Elementary Principles 1.1 1.2 1.3 1.4 1.5 1.6

Mechanics of a Particle 1 Mechanics of a System of Particles 5 Constraints 12 D' Alembert's Principle and Lagrange's Equations 16 Velocity-Dependent Potentials and the Dissipation Function Simple Applications of the Lagrangian Formulation 24

1

22

Variational Principles and Lagrange's Equations 2.1 2.2 2.3 2.4 2.5 2.6 2.7

The Central Force Problem 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12

34

Hamilton's Principle 34 Some Techniques of the Calculus of Variations 36 Derivation of Lagrange's Equations from Hamilton's Principle 44 Extending Hamilton's Principle to Systems with Constraints 45 Advantages of a Variational Principle Formulation 51 Conservation Theorems and Symmetry Properties 54 Energy Function and the Conservation of Energy 60

Reduction to the Equivalent One-Body Problem 70 The Equations of Motion and First Integrals 72 The Equivalent One-Dimensional Problem, and Classification of Orbits 76 The Virial Theorem 83 The Differential Equation for the Orbit, and Integrable Power-Law Potentials 86 Conditions for Closed Orbits (Bertrand's Theorem) 89 The Kepler Problem: Inverse-Square Law of Force 92 The Motion in Time in the Kepler Problem 98 The Laplace-Runge-Lenz Vector 102 Scattering in a Central Force Field 106 Transformation of the Scattering Problem to Laboratory Coordinates 114 The Three-Body Problem 121

70

Contents

The Kinematics of Rigid Body Motion 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10

The Rigid Body Equations of Motion 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

238

Formulation of the Problem 238 The Eigenvalue Equation and the Principal Axis Transformation 241 Frequencies of Free Vibration, and Normal Coordinates 250 Free Vibrations of a Linear Triatomic Molecule 253 Forced Vibrations and the Effect of Dissipative Forces 259 Beyond Small Oscillations: The Damped Driven Pendulum and the Josephson Junction 265

The Classical Mechanics of the Special Theory of Relativity 7.1 7.2 7.3 7.4

184

Angular Momentum and Kinetic Energy of Motion about a Point 184 Tensors 188 The Inertia Tensor and the Moment of Inertia 191 The Eigenvalues of the Inertia Tensor and the Principal Axis Transformation 195 Solving Rigid Body Problems and the Euler Equations of Motion 198 Torque-free Motion of a Rigid Body 200 The Heavy Symmetrical Top with One Point Fixed 208 Precession of the Equinoxes and of Satellite Orbits 223 Precession of Systems of Charges in a Magnetic Field 230

Oscillations 6.1 6.2 6.3 6.4 6.5 6.6

134

The Independent Coordinates of a Rigid Body 134 Orthogonal Transformations 139 Formal Properties of the Transformation Matrix 144 The Euler Angles 150 The Cayley-Klein Parameters and Related Quantities 154 Euler's Theorem on the Motion of a Rigid Body 155 Finite Rotations 161 Infinitesimal Rotations 163 Rate of Change of a Vector 171 The Coriolis Effect 174

Basic Postulates of the Special Theory 277 Lorentz Transformations 280 Velocity Addition and Thomas Precession 282 Vectors and the Metric Tensor 286

276

vii

Contents

7.5 7.6 7.7 7.8 7.9 7.10 7.11

1-Forms and Tensors 289 Forces in the Special Theory; Electromagnetism 297 Relativistic Kinematics of Collisions and Many-Particle Systems 300 Relativistic Angular Momentum 309 The Lagrangian Formulation of Relativistic Mechanics 312 Covariant Lagrangian Formulations 318 Introduction to the General Theory of Relativity 324

8 ■ The Hamilton Equations of Motion 8.1 8.2 8.3 8.4 8.5 8.6

Legendre Transformations and the Hamilton Equations of Motion 334 Cyclic Coordinates and Conservation Theorems 343 Routh's Procedure 347 The Hamiltonian Formulation of Relativistic Mechanics Derivation of Hamilton's Equations from a Variational Principle 353 The Principle of Least Action 356

334

349

9 ■ Canonical Transformations 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9

368

The Equations of Canonical Transformation 368 Examples of Canonical Transformations 375 The Harmonic Oscillator 377 The Symplectic Approach to Canonical Transformations 381 Poisson Brackets and Other Canonical Invariants 388 Equations of Motion, Infinitesimal Canonical Transformations, and Conservation Theorems in the Poisson Bracket Formulation 396 The Angular Momentum Poisson Bracket Relations 408 Symmetry Groups of Mechanical Systems 412 Liouville's Theorem 419

10 ■ Hamilton-Jacobi Theory and Action-Angle Variables 10.1 The Hamilton-Jacobi Equation for Hamilton's Principal Function 430 10.2 The Harmonic Oscillator Problem as an Example of the Hamilton-Jacobi Method 434 10.3 The Hamilton-Jacobi Equation for Hamilton's Characteristic Function 440 10.4 Separation of Variables in the Hamilton-Jacobi Equation 444 10.5 Ignorable Coordinates and the Kepler Problem 445 10.6 Action-angle Variables in Systems of One Degree of Freedom 452

430

viii

Contents

10.7 Action-Angle Variables for Completely Separable Systems 457 10.8 The Kepler Problem in Action-angle Variables 466 11 ■ Classical Chaos 11.1 Periodic Motion 484 11.2 Perturbations and the Kolmogorov-Arnold-Moser Theorem 487 11.3 Attractors 489 11.4 Chaotic Trajectories and Liapunov Exponents 491 11.5 Poincare Maps 494 11.6 Henon-Heiles Hamiltonian 496 11.7 Bifurcations, Driven-damped Harmonic Oscillator, and Parametric Resonance 505 11.8 The Logistic Equation 509 11.9 Fractals and Dimensionality 516

483

12 ■ Canonical Perturbation Theory 12.1 Introduction 526 12.2 Time-dependent Perturbation Theory 527 12.3 Illustrations of Time-dependent Perturbation Theory 533 12.4 Time-independent Perturbation Theory 541 12.5 Adiabatic Invariants 549

526

13 ■ Introduction to the Lagrangian and Hamiltonian Formulations for Continuous Systems and Fields 13.1 The Transition from a Discrete to a Continuous System 558 13.2 The Lagrangian Formulation for Continuous Systems 561 13.3 The Stress-energy Tensor and Conservation Theorems 566 13.4 Hamiltonian Formulation 572 13.5 Relativistic Field Theory 577 13.6 Examples of Relativistic Field Theories 583 13.7 Noether's Theorem 589

558

Appendix A ■ Euler Angles in Alternate Conventions and Cayley-Klein Parameters Appendix B ■ Groups and Algebras

601 605

Selected Bibliography Author Index

617 623

Subject index

625

Preface to the Third Edition

The first edition of this text appeared in 1950, and it was so well received that it went through a second printing the very next year. Throughout the next three decades it maintained its position as the acknowledged standard text for the intro­ ductory Classical Mechanics course in graduate level physics curricula through­ out the United States, and in many other countries around the world. Some major institutions also used it for senior level undergraduate Mechanics. Thirty years later, in 1980, a second edition appeared which was "a through-going revision of thefirstedition." The preface to the second edition contains the following state­ ment: "I have tried to retain, as much as possible, the advantages of thefirstedition while taking into account the developments of the subject itself, its position in the curriculum, and its applications to otherfields."This is the philosophy which has guided the preparation of this third edition twenty more years later. The second edition introduced one additional chapter on Perturbation Theory, and changed the ordering of the chapter on Small Oscillations. In addition it added a significant amount of new material which increased the number of pages by about 68%. This third edition adds still one more new chapter on Nonlinear Dy­ namics or Chaos, but counterbalances this by reducing the amount of material in several of the other chapters, by shortening the space allocated to appendices, by considerably reducing the bibliography, and by omitting the long lists of symbols. Thus the third edition is comparable in size to the second. In the chapter on relativity we have abandoned the complex Minkowski space in favor of the now standard real metric. Two of the authors prefer the complex metric because of its pedagogical advantages (HG) and because itfitsin well with Clifford Algebra formulations of Physics (CPP), but the desire to prepare students who can easily move forward into other areas of theory such as field theory and general relativity dominated over personal preferences. Some modern notation such as 1-forms, mapping and the wedge product is introduced in this chapter. The chapter on Chaos is a necessary addition because of the current interest in nonlinear dynamics which has begun to play a significant role in applications of classical dynamics. The majority of classical mechanics problems and appli­ cations in the real world include nonlinearities, and it is important for the student to have a grasp of the complexities involved, and of the new properties that can emerge. It is also important to realize the role of fractal dimensionality in chaos. New sections have been added and others combined or eliminated here and there throughout the book, with the omissions to a great extent motivated by the desire not to extend the overall length beyond that of the second edition. A section ix

Preface to the Third Edition

was added on the Euler and Lagrange exact solutions to the three body problem. In several places phase space plots and Lissajous figures were appended to illus­ trate solutions. The damped-driven pendulum was discussed as an example that explains the workings of Josephson junctions. The symplectic approach was clar­ ified by writing out some of the matrices. The harmonic oscillator was treated with anisotropy, and also in polar coordinates. The last chapter on continua and fields was formulated in the modern notation introduced in the relativity chap­ ter. The significances of the special unitary group in two dimensions SU(2) and the special orthogonal group in three dimensions SO(3) were presented in more up-to-date notation, and an appendix was added on groups and algebras. Special tables were introduced to clarify properties of ellipses, vectors, vector fields and 1-forms, canonical transformations, and the relationships between the spacetime and symplectic approaches. Several of the new features and approaches in this third edition had been men­ tioned as possibilities in the preface to the second edition, such as properties of group theory, tensors in non-Euclidean spaces, and "new mathematics" of theoret­ ical physics such as manifolds. The reference to "One area omitted that deserves special attention—nonlinear oscillation and associated stability questions" now constitutes the subject matter of our new Chapter 11 "Classical Chaos." We de­ bated whether to place this new chapter after Perturbation theory where it fits more logically, or before Perturbation theory where it is more likely to be covered in class, and we chose the latter. The referees who reviewed our manuscript were evenly divided on this question. The mathematical level of the present edition is about the same as that of the first two editions. Some of the mathematical physics, such as the discussions of hermitean and unitary matrices, was omitted because it pertains much more to quantum mechanics than it does to classical mechanics, and little used nota­ tions like dyadics were curtailed. Space devoted to power law potentials, CayleyKlein parameters, Routh's procedure, time independent perturbation theory, and the stress-energy tensor was reduced. In some cases reference was made to the second edition for more details. The problems at the end of the chapters were divided into "derivations" and "exercises," and some new ones were added. The authors are especially indebted to Michael A. Unseren and Forrest M. Hoffman of the Oak Ridge National laboratory for their 1993 compilation of errata in the second edition that they made available on the Internet. It is hoped that not too many new errors have slipped into this present revision. We wish to thank the students who used this text in courses with us, and made a number of useful suggestions that were incorporated into the manuscript. Professors Thomas Sayetta and the late Mike Schuette made helpful comments on the Chaos chapter, and Professors Joseph Johnson and James Knight helped to clarify our ideas on Lie Algebras. The following professors reviewed the manuscript and made many helpful suggestions for improvements: Yoram Alhassid, Yale University; Dave Ellis, University of Toledo; John Gruber, San Jose State; Thomas Handler, University of Tennessee; Daniel Hong, Lehigh University; Kara Keeter, Idaho State University; Carolyn Lee; Yannick Meurice, University of Iowa; Daniel

xi

Preface to the Third Edition

Marlow, Princeton University; Julian Noble, University of Virginia; Muhammad Numan, Indiana University of Pennsylvania; Steve Ruden, University of Califor­ nia, Irvine; Jack Semura, Portland State University; Tammy Ann Smecker-Hane, University of California, Irvine; Daniel Stump, Michigan State University; Robert Wald, University of Chicago; Doug Wells, Idaho State University. We thank E. Barreto, P. M. Brown, C. Chien, C. Chou, F. Du, R. F. Gans, I. R. Gatland, C. G. Gray, E. J. Guala, Jr, S. Gutti, D. H. Hartmann, M. Horbatsch, J. Howard, K. Jagannathan, R. Kissmann, L. Kramer, O. Lehtonen, N. A. Lemos, J. Palacios, R. E. Reynolds, D. V. Sathe, G. T. Seidler, J. Suzuki, A. TenneSens, J. Williams, and T. Yu for providing us with corrections in previous print­ ings. We also thank Martin Tiersten for pointing out the errors in Figures 3.7 and 3.13 that occurred in the earlier editions and the first few printings of this edition. A list of corrections for all printings are on the Web at (http://astro.physics.sc. edu/goldstein/goldstein.html). Additions to this listing may be emailed to the ad­ dress given on that page. It has indeed been an honor for two of us (CPP and JLS) to collaborate as co-authors of this third edition of such a classic book fifty years after its first ap­ pearance. We have admired this text since we first studied Classical Mechanics from the first edition in our graduate student days (CPP in 1953 and JLS in 1960), and each of us used the first and second editions in our teaching throughout the years. Professor Goldstein is to be commended for having written and later en­ hanced such an outstanding contribution to the classic Physics literature. Above all we register our appreciation and acknolwedgement in the words of Psalm 19,1: Oi ovpavoi SLrjyovvrat 8oi-av ®€ov Flushing, New York Columbia, South Carolina Columbia, South Carolina July, 2002

HERBERT GOLDSTEIN CHARLES P. POOLE, JR. JOHN L. SAFKO

Preface to the Second Edition

The prospect of a second edition of Classical Mechanics, almost thirty years af­ ter initial publication, has given rise to two nearly contradictory sets of reactions. On the one hand it is claimed that the adjective "classical" implies the field is complete, closed, far outside the mainstream of physics research. Further, the first edition has been paid the compliment of continuous use as a text since it first ap­ peared. Why then the need for a second edition? The contrary reaction has been that a second edition is long overdue. More important than changes in the sub­ ject matter (which have been considerable) has been the revolution in the attitude towards classical mechanics in relation to other areas of science and technology. When it appeared, the first edition was part of a movement breaking with older ways of teaching physics. But what were bold new ventures in 1950 are the com­ monplaces of today, exhibiting to the present generation a slightly musty and oldfashioned air. Radical changes need to be made in the presentation of classical mechanics. In preparing this second edition, I have attempted to steer a course somewhere between these two attitudes. I have tried to retain, as much as possible, the ad­ vantages of the first edition (as I perceive them) while taking some account of the developments in the subject itself, its position in the curriculum, and its applica­ tions to other fields. What has emerged is a thorough-going revision of the first edition. Hardly a page of the text has been left untouched. The changes have been of various kinds: Errors (some egregious) that I have caught, or which have been pointed out to me, have of course been corrected. It is hoped that not too many new ones have been introduced in the revised material. The chapter on small oscillations has been moved from its former position as the penultimate chapter and placed immediately after Chapter 5 on rigid body motion. This location seems more appropriate to the usual way mechanics courses are now being given. Some material relating to the Hamiltonian formulation has therefore had to be removed and inserted later in (the present) Chapter 8. A new chapter on perturbation theory has been added (Chapter 11). The last chapter, on continuous systems and fields, has been greatly expanded, in keeping with the implicit promise made in the Preface to the first edition. New sections have been added throughout the book, ranging from one in Chap­ ter 3 on Bertrand's theorem for the central-force potentials giving rise to closed orbits, to the final section of Chapter 12 on Noether's theorem. For the most part these sections contain completely new material.

Preface to the Second Edition

xiii

In various sections arguments and proofs have been replaced by new ones that seem simpler and more understandable, e.g., the proof of Euler's theorem in Chap­ ter 4. Occasionally, a line of reasoning presented in the first edition has been supplemented by a different way of looking at the problem. The most important example is the introduction of the symplectic approach to canonical transforma­ tions, in parallel with the older technique of generating functions. Again, while the original convention for the Euler angles has been retained, alternate conventions, including the one common in quantum mechanics, are mentioned and detailed formulas are given in an appendix. As part of the fruits of long experience in teaching courses based on the book, the body of exercises at the end of each chapter has been expanded by more than a factor of two and a half. The bibliography has undergone similar expansion, reflecting the appearance of many valuable texts and monographs in the years since the first edition. In deference to—but not in agreement with—the present neglect of foreign languages in graduate education in the United States, references to foreign-language books have been kept down to a minimum. The choices of topics retained and of the new material added reflect to some degree my personal opinions and interests, and the reader might prefer a different selection. While it would require too much space (and be too boring) to discuss the motivating reasons relative to each topic, comment should be made on some general principles governing my decisions. The question of the choice of mathe­ matical techniques to be employed is a vexing one. Thefirstedition attempted to act as a vehicle for introducing mathematical tools of wide usefulness that might be unfamiliar to the student. In the present edition the attitude is more one of cau­ tion. It is much more likely now than it was 30 years ago that the student will come to mechanics with a thorough background in matrix manipulation. The sec­ tion on matrix properties in Chapter 4 has nonetheless been retained, and even expanded, so as to provide a convenient reference of needed formulas and tech­ niques. The cognoscenti can, if they wish, simply skip the section. On the other hand, very little in the way of newer mathematical tools has been introduced. El­ ementary properties of group theory are given scattered mention throughout the book. Brief attention is paid in Chapters 6 and 7 to the manipulation of tensors in non-Euclidean spaces. Otherwise, the mathematical level in this edition is pretty much the same as in thefirst.It is more than adequate for the physics content of the book, and alternate means exist in the curriculum for acquiring the mathematics needed in other branches of physics. In particular the "new mathematics" of theo­ retical physics has been deliberately excluded. No mention is made of manifolds or diffeomorphisms, of tangent fibre bundles or invariant tori. There are certain highly specialized areas of classical mechanics where the powerful tools of global analysis and differential topology are useful, probably essential. However, it is not clear to me that they contribute to the understanding of the physics of classi­ cal mechanics at the level sought in this edition. To introduce these mathematical concepts, and their applications, would swell the book beyond bursting, and serve, probably, only to obscure the physics. Theoretical physics, current trends to the contrary, is not merely mathematics.

Preface to the Second Edition

In line with this attitude, the complex Minkowski space has been retained for most of the discussion of special relativity in order to simplify the mathematics. The bases for this decision (which it is realized goes against the present fashion) are given in detail on pages 292-293. It is certainly true that classical mechanics today is far from being a closed subject. The last three decades have seen an efflorescence of new developments in classical mechanics, the tackling of new problems, and the application of the techniques of classical mechanics to far-flung reaches of physics and chemistry. It would clearly not be possible to include discussions of all of these developments here. The reasons are varied. Space limitations are obviously important. Also, popular fads of current research often prove ephemeral and have a short lifetime. And some applications require too extensive a background in other fields, such as solid-state physics or physical chemistry. The selection made here represents something of a personal compromise. Applications that allow simple descriptions and provide new insights are included in some detail. Others are only briefly men­ tioned, with enough references to enable the student to follow up his awakened curiosity. In some instances I have tried to describe the current state of research in a field almost entirely in words, without mathematics, to provide the student with an overall view to guide further exploration. One area omitted deserves spe­ cial mention—nonlinear oscillation and associated stability questions. The im­ portance of the field is unquestioned, but it was felt that an adequate treatment deserves a book to itself. With all the restrictions and careful selection, the book has grown to a size probably too large to be covered in a single course. A number of sections have been written so that they may be omitted without affecting later developments and have been so marked. It was felt however that there was little need to mark special "tracks" through the book. Individual instructors, familiar with their own special needs, are better equipped to pick and choose what they feel should be included in the courses they give. I am grateful to many individuals who have contributed to my education in classical mechanics over the past thirty years. To my colleagues Professors Frank L. DiMaggio, Richard W. Longman, and Dean Peter W. Likins I am indebted for many valuable comments and discussions. My thanks go to Sir Edward Bullard for correcting a serious error in the first edition, especially for the gentle and gra­ cious way he did so. Professor Boris Garfinkel of Yale University very kindly read and commented on several of the chapters and did his best to initiate me into the mysteries of celestial mechanics. Over the years I have been the grateful re­ cipient of valuable corrections and suggestions from many friends and strangers, among whom particular mention should be made of Drs. Eric Ericsen (of Oslo University), K. Kalikstein, J. Neuberger, A. Radkowsky, and Mr. W. S. Pajes. Their contributions have certainly enriched the book, but of course I alone am responsible for errors and misinterpretations. I should like to add a collective ac­ knowledgment and thanks to the authors of papers on classical mechanics that have appeared during the last three decades in the American Journal of Physics, whose pages I hope I have perused with profit.

Preface to the Second Edition

xv

The staff at Addison-Wesley have been uniformly helpful and encouraging. I want especially to thank Mrs. Laura R. Finney for her patience with what must have seemed a never-ending process, and Mrs. Marion Howe for her gentle but persistent cooperation in the fight to achieve an acceptable printed page. To my father, Harry Goldstein ?"?, I owe more than words can describe for his lifelong devotion and guidance. But I wish at least now to do what he would not permit in his lifetime—to acknowledge the assistance of his incisive criticism and careful editing in the preparation of the first edition. I can only hope that the present edition still reflects something of his insistence on lucid and concise writing. I wish to dedicate this edition to those I treasure above all else on this earth, and who have given meaning to my life—to my wife, Channa, and our children, Penina Perl, Aaron Meir, and Shoshanna. And above all I want to register the thanks and acknowledgment of my heart, in the words of Daniel (2:23):

ru» I-DWDI *niriD TirDN rtfK •$ ^ ram arniMi ariDDn H Kew Gardens Hills, New York January 1980

HERBERT GOLDSTEIN

Preface to the First Edition

An advanced course in classical mechanics has long been a time-honored part of the graduate physics curriculum. The present-day function of such a course, however, might well be questioned. It introduces no new physical concepts to the graduate student. It does not lead him directly into current physics research. Nor does it aid him, to any appreciable extent, in solving the practical mechanics problems he encounters in the laboratory. Despite this arraignment, classical mechanics remains an indispensable part of the physicist's education. It has a twofold role in preparing the student for the study of modern physics. First, classical mechanics, in one or another of its ad­ vanced formulations, serves as the springboard for the various branches of mod­ ern physics. Thus, the technique of action-angle variables is needed for the older quantum mechanics, the Hamilton-Jacobi equation and the principle of least ac­ tion provide the transition to wave mechanics, while Poisson brackets and canon­ ical transformations are invaluable in formulating the newer quantum mechan­ ics. Secondly, classical mechanics affords the student an opportunity to master many of the mathematical techniques necessary for quantum mechanics while still working in terms of the familiar concepts of classical physics. Of course, with these objectives in mind, the traditional treatment of the sub­ ject, which was in large measure fixed some fifty years ago, is no longer adequate. The present book is an attempt at an exposition of classical mechanics which does fulfill the new requirements. Those formulations which are of importance for modern physics have received emphasis, and mathematical techniques usually associated with quantum mechanics have been introduced wherever they result in increased elegance and compactness. For example, the discussion of central force motion has been broadened to include the kinematics of scattering and the classical solution of scattering problems. Considerable space has been devoted to canonical transformations, Poisson bracket formulations, Hamilton-Jacobi theory, and action-angle variables. An introduction has been provided to the variational principle formulation of continuous systems and fields. As an illustration of the application of new mathematical techniques, rigid body rotations are treated from the standpoint of matrix transformations. The familiar Euler's theorem on the mo­ tion of a rigid body can then be presented in terms of the eigenvalue problem for an orthogonal matrix. As a consequence, such diverse topics as the inertia tensor, Lorentz transformations in Minkowski space, and resonant frequencies of small oscillations become capable of a unified mathematical treatment. Also, by this

Preface to the First Edition

xvii

technique it becomes possible to include at an early stage the difficult concepts of reflection operations and pseudotensor quantities, so important in modern quan­ tum mechanics. A further advantage of matrix methods is that "spinors" can be introduced in connection with the properties of Cay ley-Klein parameters. Several additional departures have been unhesitatingly made. All too often, special relativity receives no connected development except as part of a highly specialized course which also covers general relativity. However, its vital impor­ tance in modern physics requires that the student be exposed to special relativ­ ity at an early stage in his education. Accordingly, Chapter 6 has been devoted to the subject. Another innovation has been the inclusion of velocity-dependent forces. Historically, classical mechanics developed with the emphasis on static forces dependent on position only, such as gravitational forces. On the other hand, the velocity-dependent electromagnetic force is constantly encountered in modern physics. To enable the student to handle such forces as early as possible, velocitydependent potentials have been included in the structure of mechanics from the outset, and have been consistently developed throughout the text. Still another new element has been the treatment of the mechanics of continu­ ous systems and fields in Chapter 11, and some comment on the choice of material is in order. Strictly interpreted, the subject could include all of elasticity, hydro­ dynamics, and acoustics, but these topics lie outside the prescribed scope of the book, and adequate treatises have been written for most of them. In contrast, no connected account is available on the classical foundations of the variational prin­ ciple formulation of continuous systems, despite its growing importance in the field theory of elementary particles. The theory of fields can be carried to consid­ erable length and complexity before it is necessary to introduce quantization. For example, it is perfectly feasible to discuss the stress-energy tensor, microscopic equations of continuity, momentum space representations, etc., entirely within the domain of classical physics. It was felt, however, that an adequate discussion of these subjects would require a sophistication beyond what could naturally be expected of the student. Hence it was decided, for this edition at least, to limit Chapter 11 to an elementary description of the Lagrangian and Hamiltonian for­ mulation of fields. The course for which this text is designed normally carries with it a prerequisite of an intermediate course in mechanics. For both the inadequately prepared grad­ uate student (an all too frequent occurrence) and the ambitious senior who desires to omit the intermediate step, an effort was made to keep the book self-contained. Much of Chapters 1 and 3 is therefore devoted to material usually covered in the preliminary courses. With few exceptions, no more mathematical background is required of the stu­ dent than the customary undergraduate courses in advanced calculus and vec­ tor analysis. Hence considerable space is given to developing the more compli­ cated mathematical tools as they are needed. An elementary acquaintance with Maxwell's equations and their simpler consequences is necessary for understand­ ing the sections on electromagnetic forces. Most entering graduate students have had at least one term's exposure to modern physics, and frequent advantage has

xviii

Preface to the First Edition

been taken of this circumstance to indicate briefly the relation between a classical development and its quantum continuation. A large store of exercises is available in the literature on mechanics, easily accessible to all, and there consequently seemed little point to reproducing an extensive collection of such problems. The exercises appended to each chapter therefore have been limited, in the main, to those which serve as extensions of the text, illustrating some particular point or proving variant theorems. Pedantic museum pieces have been studiously avoided. The question of notation is always a vexing one. It is impossible to achieve a completely consistent and unambiguous system of notation that is not at the same time impracticable and cumbersome. The customary convention has been followed by indicating vectors by bold face Roman letters. In addition, matrix quantities of whatever rank, and tensors other than vectors, are designated by bold face sans serif characters, thus: A. An index of symbols is appended at the end of the book, listing the initial appearance of each meaning of the important symbols. Minor characters, appearing only once, are not included. References have been listed at the end of each chapter, for elaboration of the material discussed or for treatment of points not touched on. The evaluations ac­ companying these references are purely personal, of course, but it was felt neces­ sary to provide the student with some guide to the bewildering maze of literature on mechanics. These references, along with many more, are also listed at the end of the book. The list is not intended to be in any way complete, many of the older books being deliberately omitted. By and large, the list contains the references used in writing this book, and must therefore serve also as an acknowledgement of my debt to these sources. The present text has evolved from a course of lectures on classical mechanics that I gave at Harvard University, and I am grateful to Professor J. H. Van Vleck, then Chairman of the Physics Department, for many personal and official encour­ agements. To Professor J. Schwinger, and other colleagues I am indebted for many valuable suggestions. I also wish to record my deep gratitude to the students in my courses, whose favorable reaction and active interest provided the continuing impetus for this work.

Cambridge, Mass. March 1950

HERBERT GOLDSTEIN

R Survey of the Elementary Principles

The motion of material bodies formed the subject of some of the earliest research pursued by the pioneers of physics. From their efforts there has evolved a vast field known as analytical mechanics or dynamics, or simply, mechanics. In the present century the term "classical mechanics" has come into wide use to denote this branch of physics in contradistinction to the newer physical theories, espe­ cially quantum mechanics. We shall follow this usage, interpreting the name to include the type of mechanics arising out of the special theory of relativity. It is the purpose of this book to develop the structure of classical mechanics and to outline some of its applications of present-day interest in pure physics. Basic to any presentation of mechanics are a number of fundamental physical concepts, such as space, time, simultaneity, mass, and force. For the most part, however, these concepts will not be analyzed critically here; rather, they will be assumed as undefined terms whose meanings are familiar to the reader.

1.1 ■ MECHANICS OF A PARTICLE Let r be the radius vector of a particle from some given origin and v its vector velocity:

.-£ The linear momentum p of the particle is defined as the product of the particle mass and its velocity: p = my.

(1.2)

In consequence of interactions with external objects and fields, the particle may experience forces of various types, e.g., gravitational or electrodynamic; the vec­ tor sum of these forces exerted on the particle is the total force F. The mechanics of the particle is contained in Newton's second law of motion, which states that there exist frames of reference in which the motion of the particle is described by the differential equation

F = f.p,

(1.3) 1

2

Chapter 1

Survey of the Elementary Principles

or F=^(mv). at

(1.4)

In most instances, the mass of the particle is constant and Eq. (1.4) reduces to d\ F = m — = ma, at

(1.5)

where a is the vector acceleration of the particle defined by

■-& The equation of motion is thus a differential equation of second order, assuming F does not depend on higher-order derivatives. A reference frame in which Eq. (1.3) is valid is called an inertial or Galilean system. Even within classical mechanics the notion of an inertial system is some­ thing of an idealization. In practice, however, it is usually feasible to set up a co­ ordinate system that comes as close to the desired properties as may be required. For many purposes, a reference frame fixed in Earth (the "laboratory system") is a sufficient approximation to an inertial system, while for some astronomical purposes it may be necessary to construct an inertial system (or inertial frame) by reference to distant galaxies. Many of the important conclusions of mechanics can be expressed in the form of conservation theorems, which indicate under what conditions various mechan­ ical quantities are constant in time. Equation (1.3) directly furnishes the first of these, the Conservation Theorem for the Linear Momentum of a Particle: If the total force, F, is zero, then p = 0 and the linear momentum, p, is conserved. The angular momentum of the particle about point 0 , denoted by L, is defined as L = r x p,

(1.7)

where r is the radius vector from O to the particle. Notice that the order of the factors is important. We now define the moment of force or torque about O as N = r x F.

(1.8)

The equation analogous to (1.3) for N is obtained by forming the cross product of r with Eq. (1.4): d r x F = N = r x — (mv). dt

(1.9)

1.1

3

Mechanics of a Particle

Equation (1.9) can be written in a different form by using the vector identity: d d — (r x rav) = v x mv + r x —{my), dt dt

(1.10)

where the first term on the right obviously vanishes. In consequence of this iden­ tity, Eq. (1.9) takes the form d dL N = — (r x mv) = — = L. dt dt

(1.11)

Note that both N and L depend on the point 0 , about which the moments are taken. As was the case for Eq. (1.3), the torque equation, (1.11), also yields an imme­ diate conservation theorem, this time the Conservation Theorem for the Angular Momentum of a Particle: If the total torque, N, is zero then L = 0, and the angular momentum L is conserved. Next consider the work done by the external force F upon the particle in going from point 1 to point 2. By definition, this work is

Wn= I F-ds.

(1.12)

For constant mass (as will be assumed from now on unless otherwise specified), the integral in Eq. (1.12) reduces to [ d\ / and therefore

m f d

F - ds = m / — -vdt = — /

J dt

o —(vz)dt,

2 J dr

Wl2 = ™(vj-v2l).

(1.13)

The scalar quantity mv2/2 is called the kinetic energy of the particle and is de­ noted by T, so that the work done is equal to the change in the kinetic energy: Wn = T2-T1.

(1.14)

If the force field is such that the work Wn is the same for any physically possible path between points 1 and 2, then the force (and the system) is said to be conservative. An alternative description of a conservative system is obtained by imagining the particle being taken from point 1 to point 2 by one possible path and then being returned to point 1 by another path. The independence of W\2 on the particular path implies that the work done around such a closed circuit is zero, i.e.:

i

F-ds = 0.

(1.15)

4

Chapter 1 Survey of the Elementary Principles

Physically it is clear that a system cannot be conservative if friction or other dis­ sipation forces are present, because F • ds due to friction is always positive and the integral cannot vanish. By a well-known theorem of vector analysis, a necessary and sufficient condi­ tion that the work, W\2, be independent of the physical path taken by the particle is that F be the gradient of some scalar function of position: F = -VV(r),

(1.16)

where V is called the potential, or potential energy. The existence of V can be inferred intuitively by a simple argument. If W\2 is independent of the path of integration between the end points 1 and 2, it should be possible to express W\2 as the change in a quantity that depends only upon the positions of the end points. This quantity may be designated by — V, so that for a differential path length we have the relation F • ds = -dV or

which is equivalent to Eq. (1.16). Note that in Eq. (1.16) we can add to V any quantity constant in space, without affecting the results. Hence the zero level ofV is arbitrary. For a conservative system, the work done by the forces is Wn = Vi-

V2.

(1.17)

Combining Eq. (1.17) with Eq. (1.14), we have the result 7i + Vi = r 2 + V2,

(1.18)

which states in symbols the Energy Conservation Theorem for a Particle: If the forces acting on a particle are conservative, then the total energy of the particle, T + V, is conserved. The force applied to a particle may in some circumstances be given by the gradient of a scalar function that depends explicitly on both the position of the particle and the time. However, the work done on the particle when it travels a distance ds, F« ds =

dV ds, ds

is then no longer the total change in —V during the displacement, since V also changes explicitly with time as the particle moves. Hence, the work done as the

1.2

5

Mechanics of a System of Particles

particle goes from point 1 to point 2 is no longer the difference in the function V between those points. While a total energy T + V may still be defined, it is not conserved during the course of the particle's motion.

1.2 ■ MECHANICS OF A SYSTEM OF PARTICLES

In generalizing the ideas of the previous section to systems of many particles, we must distinguish between the external forces acting on the particles due to sources outside the system, and internal forces on, say, some particle i due to all other particles in the system. Thus, the equation of motion (Newton's second law) for the ith particle is written as ][>,•,•+F

X^

dTi

AAdR

(1.23)

dt

is the total mass of the system times the velocity of the center of mass. Conse­ quently, the equation of motion for the center of mass, (1.23), can be restated as the Conservation Theorem for the Linear Momentum of a System of Particles: If the total external force is zero, the total linear momentum is conserved. We obtain the total angular momentum of the system by forming the cross product r,- x p,- and summing over /. If this operation is performed in Eq. (1.19), there results, with the aid of the identity, Eq. (1.10),

Yfit xM = J2Tt(Ti

xB)

= L = j> x *f+1> x¥ Ji- a- 24 ) 1.7

The last term on the right in (1.24) can be considered a sum of the pairs of the form r; x ¥ji + Tj x ¥tj = (r,- - 1 7 ) x F/,-,

(1.25)

1.2

Mechanics of a System of Particles

7

FIGURE 1.2 The vector Tjj between the ith and j\h particles.

using the equality of action and reaction. But r,- — r ; is identical with the vector Ttj from j to i (cf. Fig. 1.2), so that the right-hand side of Eq. (1.25) can be written as rtjxFji.

If the internal forces between two particles, in addition to being equal and oppo­ site, also lie along the line joining the particles—a condition known as the strong law of action and reaction—then all of these cross products vanish. The sum over pairs is zero under this assumption and Eq. (1.24) may be written in the form £=N*>

(1 .26)

The time derivative of the total angular momentum is thus equal to the moment of the external force about the given point. Corresponding to Eq. (1.26) is the Conservation Theorem for Total Angular Momentum: L is constant in time if the applied (external) torque is zero. (It is perhaps worthwhile to emphasize that this is a vector theorem; i.e., Lz will be conserved if Nz is zero, even if N-f and Ny are not zero.) Note that the conservation of linear momentum in the absence of applied forces assumes that the weak law of action and reaction is valid for the internal forces. The conservation of the total angular momentum of the system in the absence of applied torques requires the validity of the strong law of action and reaction—that the internal forces in addition be central. Many of the familiar physical forces, such as that of gravity, satisfy the strong form of the law. But it is possible to find forces for which action and reaction are equal even though the forces are not central (see below). In a system involving moving charges, the forces between the charges predicted by the Biot-Savart law may indeed violate both forms of

8

Chapter 1 Survey of the Elementary Principles

the action and reaction law.* Equations (1.23) and (1.26), and their corresponding conservation theorems, are not applicable in such cases, at least in the form given here. Usually it is then possible to find some generalization of P or L that is conserved. Thus, in an isolated system of moving charges it is the sum of the mechanical angular momentum and the electromagnetic "angular momentum" of the field that is conserved. Equation (1.23) states that the total linear momentum of the system is the same as if the entire mass were concentrated at the center of mass and moving with it. The analogous theorem for angular momentum is more complicated. With the origin O as reference point, the total angular momentum of the system is

L = J]r, xp,. Let R be the radius vector from O to the center of mass, and let r'( be the radius vector from the center of mass to the ith particle. Then we have (cf. Fig. 1.3)

r,- = r{ + R

(1.27)

and v,- = v • + v where V =

dR ~dt

Center of mass

FIGURE 1.3 tum.

The vectors involved in the shift of reference point for the angular momen­

*If two charges are moving uniformly with parallel velocity vectors that are not perpendicular to the line joining the charges, then the net mutual forces are equal and opposite but do not lie along the vector between the charges. Consider, further, two charges moving (instantaneously) so as to "cross the T," i.e., one charge moving directly at the other, which in turn is moving at right angles to the first. Then the second charge exerts a nonvanishing magnetic force on the first, without experiencing any magnetic reaction force at that instant.

1.2

Mechanics of a System of Particles

9

is the velocity of the center of mass relative to O, and

*-£ 1

dt is the velocity of the f th particle relative to the center of mass of the system. Using Eq. (1.27), the total angular momentum takes on the form L = ^ R x m,-v + J ^ r ; i i

x m

i^i + I y^Jmix\ I x v + R x — y ^ m / r - . dt \ i I i

The last two terms in this expression vanish, for both contain the factor ^ m,-r[., which, it will be recognized, defines the radius vector of the center of mass in the very coordinate system whose origin is the center of mass and is therefore a null vector. Rewriting the remaining terms, the total angular momentum about O is L = R x M v + ^ r J x pj.

(1.28)

i

In words, Eq. (1.28) says that the total angular momentum about a point O is the angular momentum of motion concentrated at the center of mass, plus the angular momentum of motion about the center of mass. The form of Eq. (1.28) emphasizes that in general L depends on the origin O, through the vector R. Only if the center of mass is at rest with respect to O will the angular momentum be independent of the point of reference. In this case, the first term in (1.28) vanishes, and L always reduces to the angular momentum taken about the center of mass. Finally, let us consider the energy equation. As in the case of a single particle, we calculate the work done by all forces in moving the system from an initial configuration 1, to a final configuration 2:

*f} • dm + J2 f FP ' d*i'

Wi2 = J2f ¥t • dsi =J2f

(L29>

Again, the equations of motion can be used to reduce the integrals to

Y^

¥

i' dsi = J 2

mi

^ ' Yidt = 12 /

d

( 2m,'Ul?) '

Hence, the work done can still be written as the difference of the final and initial kinetic energies: Wi2 =

T2-Tl,

where T, the total kinetic energy of the system, is r = ^mI-U?. i

(1.30)

10

Chapter 1

Survey of the Elementary Principles

Making use of the transformations to center-of-mass coordinates, given in Eq. (1.27), we may also write T as T = -Y^mi{y + Vi)-(y + vfi) i

= lJ2miv2+\l2miV?+y-

jt tem'r/j -

and by the reasoning already employed in calculating the angular momentum, the last term vanishes, leaving r =

iMl)2+l^m.l)/2

(131)

i

The kinetic energy, like the angular momentum, thus also consists of two parts: the kinetic energy obtained if all the mass were concentrated at the center of mass, plus the kinetic energy of motion about the center of mass. Consider now the right-hand side of Eq. (1.29). In the special case that the external forces are derivable in terms of the gradient of a potential, the first term can be written as

where the subscript i on the del operator indicates that the derivatives are with respect to the components of r;. If the internal forces are also conservative, then the mutual forces between the ith and jth particles, F/7 and F ; / , can be obtained from a potential function Vtj. To satisfy the strong law of action and reaction, Vtj can be a function only of the distance between the particles: Vij = Vij(\Ti-rJ\).

(1.32)

The two forces are then automatically equal and opposite, Vji = -ViVtj = +X?jVij = -Ftj,

(1.33)

and lie along the line joining the two particles, Wi7-(|rl--ri/|) = ( r / - r ; ) / ,

(1.34)

where / is some scalar function. If Vtj were also a function of the difference of some other pair of vectors associated with the particles, such as their velocities or (to step into the domain of modern physics) their intrinsic "spin" angular mo­ menta, then the forces would still be equal and opposite, but would not necessarily lie along the direction between the particles.

1.2

11

Mechanics of a System of Particles

When the forces are all conservative, the second term in Eq. (1.29) can be rewritten as a sum over pairs of particles, the terms for each pair being of the form 2

/

WVij-dSi + VjVij-dSj).

If the difference vector r,- — r ; is denoted by r^-, and if V;; stands for the gradient with respect to r,y, then

and dsi — dsj = dvi — dYj = dr^, so that the term for the ij pair has the form - / * « « ,

drtj.

The total work arising from internal forces then reduces to |2

(1.35) l

The factor \ appears in Eq. (1.35) because in summing over both i and j each member of a given pair is included twice, first in the i summation and then in the j summation. From these considerations, it is clear that if the external and internal forces are both derivable from potentials it is possible to define a total potential energy, V, of the system,

I

ij

such that the total energy T + V is conserved, the analog of the conservation theorem (1.18) for a single particle. The second term on the right in Eq. (1.36) will be called the internal potential energy of the system. In general, it need not be zero and, more important, it may vary as the system changes with time. Only for the particular class of systems known as rigid bodies will the internal potential always be constant. Formally, a rigid body can be defined as a system of particles in which the distances r^ are fixed and cannot vary with time. In such case, the vectors drtj can only be perpendicular to the corresponding r,-7-, and therefore to the F^-. Therefore, in a rigid body the internal forces do no work, and the internal potential must remain

Chapter 1 Survey of the Elementary Principles

constant. Since the total potential is in any case uncertain to within an additive constant, an unvarying internal potential can be completely disregarded in dis­ cussing the motion of the system.

1.3 ■ CONSTRAINTS From the previous sections one might obtain the impression that all problems in mechanics have been reduced to solving the set of differential equations (1.19):

j

One merely substitutes the various forces acting upon the particles of the system, turns the mathematical crank, and grinds out the answers! Even from a purely physical standpoint, however, this view is oversimplified. For example, it may be necessary to take into account the constraints that limit the motion of the system. We have already met one type of system involving constraints, namely rigid bod­ ies, where the constraints on the motions of the particles keep the distances 77/ unchanged. Other examples of constrained systems can easily be furnished. The beads of an abacus are constrained to one-dimensional motion by the supporting wires. Gas molecules within a container are constrained by the walls of the ves­ sel to move only inside the container. A particle placed on the surface of a solid sphere is subject to the constraint that it can move only on the surface or in the region exterior to the sphere. Constraints may be classified in various ways, and we shall use the following system. If the conditions of constraint can be expressed as equations connecting the coordinates of the particles (and possibly the time) having the form /(ri,r2,r3,...,0=0,

(1.37)

then the constraints are said to be holonomic. Perhaps the simplest example of holonomic constraints is the rigid body, where the constraints are expressed by equations of the form (r,--r;)2-c?.=0. A particle constrained to move along any curve or on a given surface is another obvious example of a holonomic constraint, with the equations defining the curve or surface acting as the equations of a constraint. Constraints not expressible in this fashion are called nonholonomic. The walls of a gas container constitute a nonholonomic constraint. The constraint involved in the example of a particle placed on the surface of a sphere is also nonholo­ nomic, for it can be expressed as an inequality r2 - a2 > 0

1.3

13

Constraints

(where a is the radius of the sphere), which is not in the form of (1.37). Thus, in a gravitational field a particle placed on the top of the sphere will slide down the surface part of the way but will eventually fall off. Constraints are further classified according to whether the equations of con­ straint contain the time as an explicit variable (rheonomous) or are not explicitly dependent on time (scleronomous). A bead sliding on a rigid curved wire fixed in space is obviously subject to a scleronomous constraint; if the wire is moving in some prescribed fashion, the constraint is rheonomous. Note that if the wire moves, say, as a reaction to the bead's motion, then the time dependence of the constraint enters in the equation of the constraint only through the coordinates of the curved wire (which are now part of the system coordinates). The overall constraint is then scleronomous. Constraints introduce two types of difficulties in the solution of mechanical problems. First, the coordinates r,- are no longer all independent, since they are connected by the equations of constraint; hence the equations of motion (1.19) are not all independent. Second, the forces of constraint, e.g., the force that the wire exerts on the bead (or the wall on the gas particle), is not furnished a pri­ ori. They are among the unknowns of the problem and must be obtained from the solution we seek. Indeed, imposing constraints on the system is simply another method of stating that there are forces present in the problem that cannot be spec­ ified directly but are known rather in terms of their effect on the motion of the system. In the case of holonomic constraints, the first difficulty is solved by the intro­ duction of generalized coordinates. So far we have been thinking implicitly in terms of Cartesian coordinates. A system of N particles, free from constraints, has 3N independent coordinates or degrees offreedom. If there exist holonomic constraints, expressed in k equations in the form (1.37), then we may use these equations to eliminate k of the 3N coordinates, and we are left with 3N — k inde­ pendent coordinates, and the system is said to have 3N — k degrees of freedom. This elimination of the dependent coordinates can be expressed in another way, by the introduction of new, 3N — k9 independent variables #i, #2, • • •» q3N-k in terms of which the old coordinates ri, r 2 , . . . , r # are expressed by equations of the form 1*1 = r\(q\,q2,

• • •, q3N-k, t)

! *N = TN(q\,q2,

(1.38) .. •, q3N-k, 0

containing the constraints in them implicitly. These are transformation equations from the set of (r/) variables to the (qi) set, or alternatively Eqs. (1.38) can be con­ sidered as parametric representations of the (17) variables. It is always assumed that we can also transform back from the (qi) to the (17) set, i.e., that Eqs. (1.38) combined with the k equations of constraint can be inverted to obtain any qt as a function of the (17) variable and time.

Chapter 1 Survey of the Elementary Principles

Usually the generalized coordinates, qu unlike the Cartesian coordinates, will not divide into convenient groups of three that can be associated together to form vectors. Thus, in the case of a particle constrained to move on the surface of a sphere, the two angles expressing position on the sphere, say latitude and longi­ tude, are obvious possible generalized coordinates. Or, in the example of a double pendulum moving in a plane (two particles connected by an inextensible light rod and suspended by a similar rod fastened to one of the particles), satisfactory gen­ eralized coordinates are the two angles 6\, #2- (Cf. Fig. 1.4.) Generalized coordi­ nates, in the sense of coordinates other than Cartesian, are often useful in systems without constraints. Thus, in the problem of a particle moving in an external cen­ tral force field (V = V(r)), there is no constraint involved, but it is clearly more convenient to use spherical polar coordinates than Cartesian coordinates. Do not, however, think of generalized coordinates in terms of conventional orthogonal po­ sition coordinates. All sorts of quantities may be invoked to serve as generalized coordinates. Thus, the amplitudes in a Fourier expansion of r ; may be used as generalized coordinates, or we may find it convenient to employ quantities with the dimensions of energy or angular momentum. If the constraint is nonholonomic, the equations expressing the constraint can­ not be used to eliminate the dependent coordinates. An oft-quoted example of a nonholonomic constraint is that of an object rolling on a rough surface with­ out slipping. The coordinates used to describe the system will generally involve angular coordinates to specify the orientation of the body, plus a set of coordi­ nates describing the location of the point of contact on the surface. The constraint of "rolling" connects these two sets of coordinates; they are not independent. A change in the position of the point of contact inevitably means a change in its orientation. Yet we cannot reduce the number of coordinates, for the "rolling" condition is not expressible as a equation between the coordinates, in the manner of (1.37). Rather, it is a condition on the velocities (i.e., the point of contact is stationary), a differential condition that can be given in an integrated form only after the problem is solved.

FIGURE 1.4 Double pendulum.

1.3

15

Constraints

FIGURE 1.5

Vertical disk rolling on a horizontal plane.

A simple case will illustrate the point. Consider a disk rolling on the horizontal xy plane constrained to move so that the plane of the disk is always vertical. The coordinates used to describe the motion might be the x, y coordinates of the center of the disk, an angle of rotation 0 about the axis of the disk, and an angle 0 between the axis of the disk and say, the x axis (cf. Fig 1.5). As a result of the constraint the velocity of the center of the disk, v, has a magnitude proportional to0, v = a,qi,q2,..-,t)

= constant,

(2.43)

which are first-order differential equations. These first integrals are of interest because they tell us something physically about the system. They include, in fact, the conservation laws obtained in Chapter 1. Let us consider as an example a system of mass points under the influence of forces derived from potentials dependent on position only. Then dL _ dT _ dV_ _ dT _ = miXi =

d ^

1

/.2

.2

.2\

pix,

which is the x component of the linear momentum associated with the ith particle. This result suggests an obvious extension to the concept of momentum. The generalized momentum associated with the coordinate qj shall be defined as Pj = ^ .

(2.44)

dqj

The terms canonical momentum and conjugate momentum are often also used for Pj. Notice that if qj is not a Cartesian coordinate, pj does not necessarily have the dimensions of a linear momentum. Further, if there is a velocity-dependent potential, then even with a Cartesian coordinate qj the associated generalized *In this and succeeding sections it will be assumed, unless otherwise specified, the system is such that its motion is completely described by a Hamilton's principle of the form (2.2).

56

Chapter 2 Variational Principles and Lagrange's Equations

momentum will not be identical with the usual mechanical momentum. Thus, in the case of a group of particles in an electromagnetic field, the Lagrangian is (cf. 1.63)

i

i

i

(qi here denotes charge) and the generalized momentum conjugate to xt is dL Pix = 7 T = mtXi + qtAx,

(2.45)

OX(

i.e., mechanical momentum plus an additional term. If the Lagrangian of a system does not contain a given coordinate qj (although it may contain the corresponding velocity qj), then the coordinate is said to be cyclic or ignorable. This definition is not universal, but it is the customary one and will be used here. The Lagrange equation of motion, d dL dt dqj

dL _ dqj

reduces, for a cyclic coordinate, to

dt dqj or dt which mean that Pj = constant.

(2.46)

Hence, we can state as a general conservation theorem that the generalized momentum conjugate to a cyclic coordinate is conserved. Note that the derivation of Eq. (2.46) assumes that qj is a generalized coordi­ nate; one that is linearly independent of all the other coordinates. When equations of constraint exist, all the coordinates are not linearly independent. For exam­ ple, the angular coordinate 6 is not present in the Lagrangian of a hoop rolling without slipping in a horizontal plane that was previously discussed, but the angle appears in the constraint equations rdO = dx. As a result, the angular momentum, PQ = mr26, is not a constant of the motion. Equation (2.46) constitutes a first integral of the form (2.43) for the equations of motion. It can be used formally to eliminate the cyclic coordinate from the problem, which can then be solved entirely in terms of the remaining general-

2.6

Conservation Theorems and Symmetry Properties

57

ized coordinates. Briefly, the procedure, originated by Routh, consists in modify­ ing the Lagrangian so that it is no longer a function of the generalized velocity corresponding to the cyclic coordinate, but instead involves only its conjugate momentum. The advantage in so doing is that pj can then be considered one of the constants of integration, and the remaining integrations involve only the noncyclic coordinates. We shall defer a detailed discussion of Routh's method until the Hamiltonian formulation (to which it is closely related) is treated. Note that the conditions for the conservation of generalized momenta are more general than the two momentum conservation theorems previously derived. For example, they furnish a conservation theorem for a case in which the law of ac­ tion and reaction is violated, namely, when electromagnetic forces are present. Suppose we have a single particle in a field in which neither (/> nor A depends on x. Then x nowhere appears in L and is therefore cyclic. The corresponding canon­ ical momentum px must therefore be conserved. From (1.63) this momentum now has the form px = mx + qAx = constant.

(2.47)

In this case, it is not the mechanical linear momentum mx that is conserved but rather its sum with q Ax .* Nevertheless, it should still be true that the conservation theorems of Chapter 1 are contained within the general rule for cyclic coordinates; with proper restrictions (2.46) should reduce to the theorems of Section 1.2. We first consider a generalized coordinate qj, for which a change dqj repre­ sents a translation of the system as a whole in some given direction. An example would be one of the Cartesian coordinates of the center of mass of the system. Then clearly qj cannot appear in T, for velocities are not affected by a shift in the origin, and therefore the partial derivative of T with respect to qj must be zero. Further, we will assume conservative systems for which V is not a function of the velocities, so as to eliminate such complications as electromagnetic forces. The Lagrange equation of motion for a coordinate so defined then reduces to d dT

dV = Pi =

dtdqj

FJ

dqj

= Qi^J

(2.48) V

}

We will now show that (2.48) is the equation of motion for the total linear momentum, i.e., that Qj represents the component of the total force along the di­ rection of translation of qj, and pj is the component of the total linear momentum along this direction. In general, the generalized force Qj is given by Eq. (1.49):

Since dqj corresponds to a translation of the system along some axis, the vectors Tiiqj) and r,-(#y + dqj) are related as shown in Fig. 2.7. By the definition of a *It can be shown from classical electrodynamics that under these conditions, i.e., neither A nor depending on JC, that qAx is exactly the *-component of the electromagnetic linear momentum of the field associated with the charge q.

Chapter 2

Variational Principles and Lagrange's Equations

FIGURE 2.7 Change in a position vector under translation of the system. derivative, we have 9£L = dqj

l i m r,fa+ n = n.F, which (as was stated) is the component of the total force in the direction of n. To prove the other half of the statement, note that with the kinetic energy in the form

T = ^mtf' the conjugate momentum is dT dqj

v ^ • d*i d( Y iJ

= Y^rm\i

dqj'

using Eq. (1.51). Then from Eq. (2.49) Pj = n . ^ m / v / , i

which again, as predicted, is the component of the total system linear momentum along n. Suppose now that the translation coordinate qj that we have been discussing is cyclic. Then qj cannot appear in V and therefore

2.6

Conservation Theorems and Symmetry Properties

59

But this is simply the familiar conservation theorem for linear momentum—that if a given component of the total applied force vanishes, the corresponding com­ ponent of the linear momentum is conserved. In a similar fashion, it can be shown that if a cyclic coordinate qj is such that dqj corresponds to a rotation of the system of particles around some axis, then the conservation of its conjugate momentum corresponds to conservation of an angular momentum. By the same argument used above, T cannot contain qj, for a rotation of the coordinate system cannot affect the magnitude of the velocities. Hence, the partial derivative of T with respect to qj must again be zero, and since V is independent of qj, we once more get Eq. (2.48). But now we wish to show that with qj a rotation coordinate the generalized force is the component of the total applied torque about the axis of rotation, and pj is the component of the total angular momentum along the same axis. The generalized force Qj is again given by

G;=£F*

dqj

only the derivative now has a different meaning. Here the change in qj must cor­ respond to an infinitesimal rotation of the vector r z , keeping the magnitude of the vector constant. From Fig. 2.8, the magnitude of the derivative can easily be obtained: \dr( | = r,- sin 0 dqj

and

OIL

= rt sin#,

dqj

FIGURE 2.8 Change of a position vector under rotation of the system.

Chapter 2

Variational Principles and Lagrange's Equations

and its direction is perpendicular to both r* and n. Clearly, the derivative can be written in vector form as -^=nx

r /

.

(2.50)

dqj

With this result, the generalized force becomes

e,- = I>- nxr ' i

= ]Tn.r, xFl5 i

reducing to Qj = n . ^ N / = n . N , i

which proves the first part. A similar manipulation of pj with the aid of Eq. (2.50) provides proof of the second part of the statement: Pj

= — ^J

= ^mtVi i

• — L = ^ n • r, x m,-v,- = n . ^ L , - = n • L. "J

i

i

Summarizing these results, we see that if the rotation coordinate qj is cyclic, then Qj, which is the component of the applied torque along n, vanishes, and the component of L along n is constant. Here we have recovered the angular momentum conservation theorem out of the general conservation theorem relating to cyclic coordinates. The significance of cyclic translation or rotation coordinates in relation to the properties of the system deserves some comment at this point. If a generalized co­ ordinate corresponding to a displacement is cyclic, it means that a translation of the system, as if rigid, has no effect on the problem. In other words, if the system is invariant under translation along a given direction, the corresponding linear momentum is conserved. Similarly, the fact that a generalized rotation coordinate is cyclic (and therefore the conjugate angular momentum conserved) indicates that the system is invariant under rotation about the given axis. Thus, the momen­ tum conservation theorems are closely connected with the symmetry properties of the system. If the system is spherically symmetric, we can say without further ado that all components of angular momentum are conserved. Or, if the system is symmetric only about the z axis, then only Lz will be conserved, and so on for the other axes. These symmetry considerations can often be used with relatively complicated problems to determine by inspection whether certain constants of the motion exist, (cf. Noether's theorem—Sec. 13.7.) Suppose, for example, the system consists of a set of mass points moving in a potential field generated by fixed sources uniformly distributed on an infinite plane, say, the z = 0 plane. (The sources might be a mass distribution if the forces

2.7 Energy Function and the Conservation of Energy

61

were gravitational, or a charge distribution for electrostatic forces.) Then the sym­ metry of the problem is such that the Lagrangian is invariant under a translation of the system of particles in the x- or j-directions (but not in the z-direction) and also under a rotation about the z axis. It immediately follows that the x- and ycomponents of the total linear momentum, Px and Py, are constants of the motion along with Lz, the z-component of the total angular momentum. However, if the sources were restricted only to the half plane, x > 0, then the symmetry for trans­ lation along the x axis and for rotation about the z axis would be destroyed. In that case, Px and Lz could not be conserved, but Py would remain a constant of the motion. We will encounter the connections between the constants of motion and the symmetry properties of the system several times in the following chapters.

2.7 ■ ENERGY FUNCTION AND THE CONSERVATION OF ENERGY Another conservation theorem we should expect to obtain in the Lagrangian for­ mulation is the conservation of total energy for systems where the forces are derivable from potentials dependent only upon position. Indeed, it is possible to demonstrate a conservation theorem for which conservation of total energy repre­ sents only a special case. Consider a general Lagrangian, which will be a function of the coordinates qj and the velocities

(2-53)

dq

J

and Eq. (2.52) can be looked on as giving the total time derivative of h:

If the Lagrangian is not an explicit function of time, i.e., if t does not appear in L explicitly but only implicitly through the time variation of q and q, then Eq. (2.54) says that h is conserved. It is one of the first integrals of the motion and is sometimes referred to as Jacobi's integral.^ Under certain circumstances, the function h is the total energy of the system. To determine what these circumstances are, we recall that the total kinetic energy of a system can always be written as r = 7b + ri + r 2 ,

(1.73)

where TQ is a function of the generalized coordinates only, 7i(^r, ^) is linear in the generalized velocities, and T2(q, q) is a quadratic function of the g's. For a very wide range of systems and sets of generalized coordinates, the Lagrangian can be similarly decomposed as regards its functional behavior in the q variables: L(q, q; t) = L0(q, t) + Lx(q, q, t) + L2(q, q, t).

(2.55)

Here L2 is a homogeneous function of the second degree (not merely quadratic) in q, while L\ is homogeneous of the first degree in q. There is no reason intrinsic to mechanics that requires the Lagrangian to conform to Eq. (2.55), but in fact it does for most problems of interest. The Lagrangian clearly has this form when the forces are derivable from a potential not involving the velocities. Even with the velocity-dependent potentials, we note that the Lagrangian for a charged particle in an electromagnetic field, Eq. (1.63), satisfies Eq. (2.55). Now, recall that Euler's theorem states that if / is a homogeneous function of degree n in the variables JC,- , then

Yxt^=nf. t-r1 1

(2.56)

axi l

*The energy function h is identical in value with the Hamiltonian H (See Chapter 8). It is given a different name and symbol here to emphasize that h is considered a function of n independent variables qj and their time derivatives qj (along with the time), whereas the Hamiltonian will be treated as a function of 2n independent variables, qj, pj (and possibly the time). *i"This designation is most often confined to a first integral in the restricted three-body problem. How­ ever, the integral there is merely a special case of the energy function h, and there is some historical precedent to apply the name Jacobi integral to the more general situation.

2.7

63

Energy Function and the Conservation of Energy

Applied to the function h, Eq. (2.53), for the Lagrangians of the form (2.55), this theorem implies that h = 2L2 + Li-L

= L2-

L0.

(2.57)

If the transformation equations defining the generalized coordinates, Eqs. (1.38), do not involve the time explicitly, then by Eqs. (1.73) T = T2. If, further, the potential does not depend on the generalized velocities, then L2 = T and LQ = -V, so that h = T + V = E,

(2.58)

and the energy function is indeed the total energy. Under these circumstances, if V does not involve the time explicitly, neither will L. Thus, by Eq. (2.54), h (which is here the total energy), will be conserved. Note that the conditions for conservation of h are in principle quite distinct from those that identify h as the total energy. We can have a set of generalized coordinates such that in a particular problem h is conserved but is not the total energy. On the other hand, h can be the total energy, in the form T + V, but not be conserved. Also note that whereas the Lagrangian is uniquely fixed for each system by the prescription L = T — U independent of the choice of generalized coordinates, the energy function h depends in magnitude and functional form on the specific set of generalized coordinates. For one and the same system, various energy functions h of different physical content can be generated depending on how the generalized coordinates are chosen. The most common case that occurs in classical mechanics is one in which the kinetic energy terms are all of the form mqf/2 or pf/2m and the potential energy depends only upon the coordinates. For these conditions, the energy function is both conserved and is also the total energy. Finally, note that where the system is not conservative, but there are factional forces derivable from a dissipation function J7, it can be easily shown that T is re­ lated to the decay rate of h. When the equations of motion are given by Eq. (1.70), including dissipation, then Eq. (2.52) has the form dh

dL

^

^F •

By the definition of J7, Eq. (1.67), it is a homogeneous function of the g's of degree 2. Hence, applying Euler's theorem again, we have dh dL — = -2T-—. dt dt

(2.59)

Chapter 2

Variational Principles and Lagrange's Equations

If L is not an explicit function of time, and the system is such that h is the same as the energy, then Eq. (2.59) says that 2 T is the rate of energy dissipation, dE — = -2F, dt

(2.60)

a statement proved above (cf. Sec. 1.5) in less general circumstances.

DERIVATIONS 1. Complete the solution of the brachistochrone problem begun in Section 2.2 and show that the desired curve is a cycloid with a cusp at the initial point at which the particle is released. Show also that if the particle is projected with an initial kinetic energy \mvQ that the brachistochrone is still a cycloid passing through the two points with a cusp at a height z above the initial point given by v^ = 2gz. 2. Show that if the potential in the Lagrangian contains velocity-dependent terms, the canonical momentum corresponding to a coordinate of rotation 0 of the entire system is no longer the mechanical angular momentum LQ but is given by

P0 =Le-J2n'ri

x VV|.tf,

i

where Vv is the gradient operator in which the derivatives are with respect to the velocity components and n is a unit vector in the direction of rotation. If the forces are electromagnetic in character, the canonical momentum is therefore

i

3. Prove that the shortest distance between two points in space is a straight line. 4. Show that the geodesies of a spherical surface are great circles, i.e., circles whose centers lie at the center of the sphere.

EXERCISES 5. A particle is subjected to the potential V(x) = —Fx, where F is a constant. The particle travels from * = 0 t o * = a i n a time interval tQ. Assume the motion of the particle can be expressed in the form x(t) = A + Bt + C t2. Find the values of A, B, and C such that the action is a minimum. 6. Find the Euler-Lagrange equation describing the brachistochrone curve for a particle moving inside a spherical Earth of uniform mass density. Obtain a first integral for this differential equation by analogy to the Jacobi integral h. With the help of this integral, show that the desired curve is a hypocycloid (the curve described by a point on a circle rolling on the inside of a larger circle). Obtain an expression for the time of travel along the brachistochrone between two points on Earth's surface. How long

65

Exercises

would it take to go from New York to Los Angeles (assumed to be 4800 km apart on the surface) along a brachistochrone tunnel (assuming no friction) and how far below the surface would the deepest point of the tunnel be? 7. In Example 2 of Section 2.1 we considered the problem of the minimum surface of revolution. Examine the symmetric case x\ = *2, J2 = —y\ > 0, and express the condition for the parameter a as a transcendental equation in terms of the dimensionless quantities k = X2/a, and a = y2/*2- Show that for a greater than a certain value «0 two values of k are possible, for a = Q?O only one value of k is possible, while if a < ao no real value of k (or a) can be found, so that no catenary solution exists in this region. Find the value of OLQ, numerically if necessary. 8. The broken-segment solution described in the text (cf. p. 42), in which the area of revolution is only that of the end circles of radius y\ and V2, respectively, is known as the Goldschmidt solution. For the symmetric situation discussed in Exercise 7, obtain an expression for the ratio of the area generated by the catenary solutions to that given by the Goldschmidt solution. Your result should be a function only of the parameters k and a. Show that for sufficiently large values of a at least one of the catenaries gives an area below that of the Goldschmidt solution. On the other hand, show that if a = otQ, the Goldschmidt solution gives a lower area than the catenary. 9. A chain or rope of indefinite length passes freely over pulleys at heights y\ and J2 above the plane surface of Earth, with a horizontal distance X2 — x\ between them. If the chain or rope has a uniform linear mass density, show that the problem of finding the curve assumed between the pulleys is identical with that of the problem of mini­ mum surface of revolution. (The transition to the Goldschmidt solution as the heights y\ and y2 are changed makes for a striking lecture demonstration. See Exercise 8.) 10. Suppose it is known experimentally that a particle fell a given distance VQ in a time fy = V^yoJS' The times of fall for distances other than yo are not known. Suppose further that the Lagrangian for the problem is known, but that instead of solving the equation of motion for y as a function of t, it is guessed that the functional form is y = at + bt2. If the constants a and b are adjusted always so that the time to fall yo *s correctly given by to, show directly that the integral pto / Ldt JO

is an extremum for real values of the coefficients only when a = 0 and b = g/2. 11. When two bilHard balls collide, the instantaneous forces between them are very large but act only in an infinitesimal time At, in such a manner that the quantity

f Fdt JAt

remains finite. Such forces are described as impulsive forces, and the integral over At is known as the impulse of the force. Show that if impulsive forces are present Lagrange's equations may be transformed into

Chapter 2

Variational Principles and Lagrange's Equations

\9qj)f

Sj

\9qjJi

'

where the subscripts / and / refer to the state of the system before and after the impulse, Sj is the impulse of the generalized impulsive force corresponding to qj, and L is the Lagrangian including all the nonimpulsive forces. 12. The term generalized mechanics has come to designate a variety of classical mechan­ ics in which the Lagrangian contains time derivatives of qi higher than the first. Prob­ lems for which x = f(x,x,x,t) have been referred to as "jerky" mechanics. Such equations of motion have interesting applications in chaos theory (cf. Chapter 11). By applying the methods of the calculus of variations, show that if there is a Lagrangian of the form L{qi, qi, 'q\, t), and Hamilton's principle holds with the zero variation of both qi and q\ at the end points, then the corresponding Euler-Lagrange equations are d2 {dL\ dt2\dqiJ

d (dL\ dt\dqi)

8L dqt

„

Apply this result to the Lagrangian L

m .. k 2 = ~2qq ~ 2q ■

Do you recognize the equations of motion? 13. A heavy particle is placed at the top of a vertical hoop. Calculate the reaction of the hoop on the particle by means of the Lagrange's undetermined multipliers and Lagrange's equations. Find the height at which the particle falls off. 14. A uniform hoop of mass m and radius r rolls without slipping on a fixed cylinder of radius R as shown in the figure. The only external force is that of gravity. If the smaller cylinder starts rolling from rest on top of the bigger cylinder, use the method of Lagrange mulipliers to find the point at which the hoop falls off the cylinder.

LA 15. A form of the Wheatstone impedance bridge has, in addition to the usual four resis­ tances, an inductance in one arm and a capacitance in the opposite arm. Set up L and T for the unbalanced bridge, with the charges in the elements as coordinates. Using the Kirchhoff junction conditions as constraints on the currents, obtain the Lagrange equations of motion, and show that eliminating the X's reduces these to the usual net­ work equations.

67

Exercises

16. In certain situations, particularly one-dimensional systems, it is possible to incorpo­ rate frictional effects without introducing the dissipation function. As an example, find the equations of motion for the Lagrangian

i"" (=£-*)• How would you describe the system? Are there any constants of motion? Suppose a point transformation is made of the form s=

ey"2q.

What is the effective Lagrangian in terms of si Find the equation of motion for s. What do these results say about the conserved quantities for the system? 17. It sometimes occurs that the generahzed coordinates appear separately in the kinetic energy and the potential energy in such a manner that T and V may be written in the form

T = Y,Mqi)q?

and

V = £ V*(ft).

i

i

Show that Lagrange's equations then separate, and that the problem can always be reduced to quadratures. 18. A point mass is constrained to move on a massless hoop of radius a fixed in a vertical plane that rotates about its vertical symmetry axis with constant angular speed co. Obtain the Lagrange equations of motion assuming the only external forces arise from gravity. What are the constants of motion? Show that if co is greater than a critical value a>o, there can be a solution in which the particle remains stationary on the hoop at a point other than at the bottom, but that if co < COQ, the only stationary point for the particle is at the bottom of the hoop. What is the value of COQ! 19. A particle moves without friction in a conservative field of force produced by various mass distributions. In each instance, the force generated by a volume element of the distribution is derived from a potential that is proportional to the mass of the volume element and is a function only of the scalar distance from the volume element. For the following fixed, homogeneous mass distributions, state the conserved quantities in the motion of the particle: (a) The mass is uniformly distributed in the plane z = 0. (b) The mass is uniformly distributed in the half-plane z = 0, y > 0. (c) The mass is uniformly distributed in a circular cylinder of infinite length, with axis along the z axis. (d) The mass is uniformly distributed in a circular cylinder of finite length, with axis along the z axis. (e) The mass is uniformly distributed in a right cylinder of elliptical cross section and infinite length, with axis along the z axis. (f) The mass is uniformly distributed in a dumbbell whose axis is oriented along the z axis.

Chapter 2

Variational Principles and Lagrange's Equations

(g) The mass is in the form of a uniform wire wound in the geometry of an infinite helical solenoid, with axis along the z axis. 20. A particle of mass m slides without friction on a wedge of angle a and mass M that can move without friction on a smooth horizontal surface, as shown in the figure. Treating the constraint of the particle on the wedge by the method of Lagrange multipliers, find the equations of motion for the particle and wedge. Also obtain an expression for the forces of constraint. Calculate the work done in time t by the forces of constraint acting on the particle and on the wedge. What are the constants of motion for the system? Contrast the results you have found with the situation when the wedge is fixed. [Suggestion: For the particle you may either use a Cartesian coordinate system with y vertical, or one with y normal to the wedge or, even more instructively, do it in both systems.]

21. A carriage runs along rails on a rigid beam, as shown in the figure below. The carriage is attached to one end of a spring of equilibrium length r$ and force constant k, whose other end is fixed on the beam. On the carriage, another set of rails is perpendicular to the first along which a particle of mass m moves, held by a spring fixed on the beam, of force constant k and zero equilibrium length. Beam, rails, springs, and carriage are assumed to have zero mass. The whole system is forced to move in a plane about the point of attachment of the first spring, with a constant angular speed co. The length of the second spring is at all times considered small compared to TQ. (a) What is the energy of the system? Is it conserved? (b) Using generalized coordinates in the laboratory system, what is the Jacobi integral for the system? Is it conserved? (c) In terms of the generalized coordinates relative to a system rotating with the angu­ lar speed &>, what is the Lagrangian? What is the Jacobi integral? Is it conserved? Discuss the relationship between the two Jacobi integrals.

Exercises

69

22. Suppose a particle moves in space subject to a conservative potential V(r) but is constrained to always move on a surface whose equation is a (r, t) = 0. (The explicit dependence on t indicates that the surface may be moving.) The instantaneous force of constraint is taken as always perpendicular to the surface. Show analytically that the energy of the particle is not conserved if the surface moves in time. What physically is the reason for nonconservation of the energy under this circumstance? 23. Consider two particles of masses m\ and rri2. Let m\ be confined to move on a circle of radius a in the z = 0 plane, centered at x = y = 0. Let mi be confined to move on a circle of radius b in the z — c plane, centered at x = y = 0. A light (massless) spring of spring constant k is attached between the two particles. (a) Find the Lagrangian for the system. (b) Solve the problem using Lagrange multipliers and give a physical interpretation for each multiplier. 24. The one-dimensional harmonic oscillator has the Lagrangian L = mx /2 — kx /2. Suppose you did not know the solution to the motion, but realized that the motion must be periodic and therefore could be described by a Fourier series of the form x(t) = Y^ cij cos jcot, (taking t = 0 at a turning point) where co is the (unknown) angular frequency of the motion. This representation for x{t) defines a many-parameter path for the system point in configuration space. Consider the action integral / for two points, t\ and ^ separated by the period T = 2jt/co. Show that with this form for the system path, / is an extremum for nonvanishing x only if aj = 0, for j ^ 1, and only if co2 = k/m. 25. A disk of radius R rolls without slipping inside the stationary parabola y = ax2. Find the equations of constraint. What condition allows the disk to roll so that it touches the parabola at one and only one point independent of its position? 26. A particle of mass m is suspended by a massless spring of length L. It hangs, without initial motion, in a gravitational field of strength g. It is struck by an impulsive hor­ izontal blow, which introduces an angular velocity co. If co is sufficiently small, it is obvious that the mass moves as a simple pendulum. If co is sufficiently large, the mass will rotate about the support. Use a Lagrange multiplier to determine the conditions under which the string becomes slack at some point in the motion. 27. (a) Show that the constraint Eq. (2.29) is truly nonholonomic by showing that it can­ not be integrated to a homonomic form. (b) Show that the corresponding constraint forces do no virtual work. (c) Find one or more solutions to (2.29)-(2.30) for V = 0 and show that they con­ serve energy.

CHAPTER The Central Force Problem

In this chapter we shall discuss the problem of two bodies moving under the in­ fluence of a mutual central force as an application of the Lagrangian formulation. Not all the problems of central force motion are integrable in terms of well-known functions. However, we shall attempt to explore the problem as thoroughly as is possible with the tools already developed. In the last section of this chapter we consider some of the complications that follow by the presence of a third body.

3.1 ■ REDUCTION TO THE EQUIVALENT ONE-BODY PROBLEM Consider a monogenic system of two mass points, m\ and mi (cf. Fig. 3.1), where the only forces are those due to an interaction potential U. We will assume at first that U is any function of the vector between the two particles, T2 — ri, or of their relative velocity, r2 — i*i, or of any higher derivatives of r2 — r i . Such a system has six degrees of freedom and hence six independent generalized coordinates. We choose these to be the three components of the radius vector to the center of mass, R, plus the three components of the difference vector r = T2 — ri. The Lagrangian will then have the form L = T(R, r) - U(r, r , . . . ) .

FIGURE 3.1 Coordinates for the two-body problem. 70

(3.1)

3.1

Reduction to the Equivalent One-Body Problem

71

The kinetic energy T can be written as the sum of the kinetic energy of the motion of the center of mass, plus the kinetic energy of motion about the center of mass, T\ r = i(mi+m2)R2 + r/ with

Tf = \mxx't + \m2Y%. Here r^ and Tr2 are the radii vectors of the two particles relative to the center of mass and are related to r by r

l =

r

T

>

m\ +m2 *2 = } r m\ +ni2

(3.2)

Expressed in terms of r by means of Eq. (3.2), T takes on the form Tt

1 =

m m

l 2

^2

2m\ +m,2 and the total Lagrangian (3.1) is L=

2

\

R2 + -

2

' r 2 - t/(r, r , . . . ) . 2m\+m2

(3.3)

It is seen that the three coordinates R are cyclic, so that the center of mass is either at rest or moving uniformly. None of the equations of motion for r will contain terms involving R or R. Consequently, the process of integration is par­ ticularly simple here. We merely drop the first term from the Lagrangian in all subsequent discussion. The rest of the Lagrangian is exactly what would be expected if we had a fixed center of force with a single particle at a distance r from it, having a mass H=

\

,

(3.4)

where ii is known as the reduced mass. Frequently, Eq. (3.4) is written in the form 1 1 1 - = — + —. \x m\ rri2

(3.5)

Thus, the central force motion of two bodies about their center of mass can always be reduced to an equivalent one-body problem.

72

Chapter 3 The Central Force Problem 3.2 ■ THE EQUATIONS OF M O T I O N A N D FIRST INTEGRALS

We now restrict ourselves to conservative central forces, where the potential is V (r), a function of r only, so that the force is always along r. By the results of the preceding section, we need only consider the problem of a single particle of reduced mass m moving about a fixed center of force, which will be taken as the origin of the coordinate system. Since potential energy involves only the radial distance, the problem has spherical symmetry; i.e., any rotation, about any fixed axis, can have no effect on the solution. Hence, an angle coordinate representing rotation about a fixed axis must be cyclic. These symmetry properties result in a considerable simplification in the problem. Since the problem is spherically symmetric, the total angular momentum vec­ tor, L = r x p, is conserved. It therefore follows that r is always perpendicular to the fixed direc­ tion of L in space. This can be true only if r always lies in a plane whose normal is parallel to L. While this reasoning breaks down if L is zero, the motion in that case must be along a straight line going through the center of force, for L = 0 requires r to be parallel to r, which can be satisfied only in straight-line motion.* Thus, central force motion is always motion in a plane. Now, the motion of a single particle in space is described by three coordinates; in spherical polar coordinates these are the azimuth angle 0, the zenith angle (or colatitude) i//, and the radial distance r. By choosing the polar axis to be in the direction of L, the motion is always in the plane perpendicular to the polar axis. The coordinate ty then has only the constant value 7r/2 and can be dropped from the subsequent discussion. The conservation of the angular momentum vector fur­ nishes three independent constants of motion (corresponding to the three Carte­ sian components). In effect, two of these, expressing the constant direction of the angular momentum, have been used to reduce the problem from three to two de­ grees of freedom. The third of these constants, corresponding to the conservation of the magnitude of L, remains still at our disposal in completing the solution. Expressed now in plane polar coordinates, the Lagrangian is L = T

-V

= \m(r2 + r202) - V(r).

(3.6)

As was forseen, 0 is a cyclic coordinate, whose corresponding canonical momen­ tum is the angular momentum of the system: Pe = —- = mr 9. d6 *Formally: r = rnr + rOriQ, hence r x f = 0 requires 0 = 0 .

3.2

The Equations of Motion and First Integrals

73

One of the two equations of motion is then simply

Pe =

jt(mrH)=0.

(3.7)

with the immediate integral mr26 = /,

(3.8)

where / is the constant magnitude of the angular momentum. From (3.7) is also follows that

s(H"

0.

(3.9)

The factor ^ is inserted because \r20 is just the areal velocity—the area swept out by the radius vector per unit time. This interpretation follows from Fig. 3.2, the differential area swept out in time dt being

dA=

\r(rd0),

and hence dA _ 1 dt ~ 2

2d0

dt'

The conservation of angular momentum is thus equivalent to saying the areal velocity is constant. Here we have the proof of the well-known Kepler's second law of planetary motion: The radius vector sweeps out equal areas in equal times. It should be emphasized however that the conservation of the areal velocity is a general property of central force motion and is not restricted to an inverse-square law of force.

FIGURE 3.2 The area swept out by the radius vector in a time dt.

74

Chapter 3 The Central Force Problem

The remaining Lagrange equation, for the coordinate r, is d

.9

dV

— (mr) - mrO2 + — = 0. at dr

(3.10)

Designating the value of the force along r, —dV/dr, by f(r) the equation can be rewritten as mr-mrO2

= f(r).

(3.11)

By making use of the first integral, Eq. (3.8), 0 can be eliminated from the equa­ tion of motion, yielding a second-order differential equation involving r only: mr

I2

3 =/(r).

(3.12)

There is another first integral of motion available, namely the total energy, since the forces are conservative. On the basis of the general energy conservation theorem, we can immediately state that a constant of the motion is E = \m(r2 + r202) + V(r),

(3.13)

where E is the energy of the system. Alternatively, this first integral could be derived again directly from the equations of motion (3.7) and (3.12). The latter can be written as 1 I2 d ( v + \v -—=■z • 1 + o—2 = -;r— 2mr dr \ dr \ 2mrz I If both sides of Eq. (3.14) are multiplied by f the left side becomes m?

(3 14)

-

... d (I .2\ mrr = — ( -mr ) dt \2 ) The right side similarly can be written as a total time derivative, for if g(r) is any function of r, then the total time derivative of g has the form

— ( \ — ^i_^L dt

dr dt

Hence, Eq. (3.14) is equivalent to d (\

.2\

d /

1 I2 \

or _ -mr2 + - — 2 + V =0, dt \ 2 2 mrL /

3.2

The Equations of Motion and First Integrals

75

and therefore 1 1 Z2 -mr2 + -—=■ + V = constant. 2 2mr

(3.15)

Equation (3.15) is the statement of the conservation of total energy, for by us­ ing (3.8) for /, the middle term can be written

-— = J - m w = — ,

2mr2 2mr2 2 and (3.15) reduces to (3.13). These first two integrals give us in effect two of the quadratures necessary to complete the problem. As there are two variables, r and 0, a total of four inte­ grations are needed to solve the equations of motion. The first two integrations have left the Lagrange equations as two first-order equations (3.8) and (3.15); the two remaining integrations can be accomplished (formally) in a variety of ways. Perhaps the simplest procedure starts from Eq. (3.15). Solving for r, we have

or dr dt =

.

(3.17)

^ (

E

-

v

- ^ )

At time t = 0, let r have the initial value r$. Then the integral of both sides of the equation from the initial state to the state at time t takes the form

-I

dT

(3.18)

As it stands, Eq. (3.18) gives t as a function of r and the constants of integration E, /, and ro- However, it may be inverted, at least formally, to give r a s a function of t and the constants. Once the solution for r is found, the solution 0 follows immediately from Eq. (3.8), which can be written as Idt d6 = —^.z rar

(3.19)

If the initial value of 0 is 0Q, then the integral of (3.19) is simply dt 0 = 1 f - ^2 7 + 0 mr (t)

O

.

(3.20)

Chapter 3 The Central Force Problem

Equations (3.18) and (3.20) are the two remaining integrations, and formally the problem has been reduced to quadratures (evaluating integrals), with four con­ stants of integration E, /, ro, #o- These constants are not the only ones that can be considered. We might equally as well have taken ro, Oo, ro, Oo, but of course E and / can always be determined in terms of this set. For many applications, how­ ever, the set containing the energy and angular momentum is the natural one. In quantum mechanics, such constants as the initial values of r and 0, or of r and 6, become meaningless, but we can still talk in terms of the system energy or of the system angular momentum. Indeed, two salient differences between classical and quantum mechanics appear in the properties of E and / in the two theories. To dis­ cuss the transition to quantum theories it is important that the classical description of the system be in terms of its energy and angular momentum.

3.3 ■ THE EQUIVALENT ONE-DIMENSIONAL PROBLEM, AND CLASSIFICATION OF ORBITS Although we have solved the one-dimensional problem formally, practically speaking the integrals (3.18) and (3.20) are usually quite unmanageable, and in any specific case it is often more convenient to perform the integration in some other fashion. But before obtaining the solution for any specific force laws, let us see what can be learned about the motion in the general case, using only the equations of motion and the conservation theorems, without requiring explicit solutions. For example, with a system of known energy and angular momentum, the mag­ nitude and direction of the velocity of the particle can be immediately determined in terms of the distance r. The magnitude v follows at once from the conservation of energy in the form E = \mv2 + V(r) or

v = J- (E - V(r)).

(3.21)

V m The radial velocity—the component of r along the radius vector—has been given in Eq. (3.16). Combined with the magnitude v, this is sufficient information to furnish the direction of the velocity.* These results, and much more, can also be obtained from consideration of an equivalent one-dimensional problem. The equation of motion in r, with 6 expressed in terms of/, Eq. (3.12), involves only r and its derivatives. It is the same equation as would be obtained for a * Alternatively, the conservation of angular momentum furnishes 0, the angular velocity, and this to­ gether with r gives both the magnitude and direction of r.

3.3

77

The Equivalent One-Dimensional Problem

fictitious one-dimensional problem in which a particle of mass m is subject to a force I2 /' = / + —3.

(3.22)

The significance of the additional term is clear if it is written as mrO2 = mvg/r, which is the familiar centrifugal force. An equivalent statement can be obtained from the conservation theorem for energy. By Eq. (3.15) the motion of the particle in r is that of a one-dimensional problem with a fictitious potential energy: 1 I2

V = V+ --

T

.

(3.220

As a check, note that

i2

av f' = —7r- = f(r) +

which agrees with Eq. (3.22). The energy conservation theorem (3.15) can thus also be written as E = V' + \mr2.

(3.150

As an illustration of this method of examining the motion, consider a plot of V against r for the specific case of an attractive inverse-square law of force:

/~4rz

(For positive k, the minus sign ensures that the force is toward the center of force.) The potential energy for this force is

v—t, r and the corresponding fictitious potential is k I2 r 2mr2 Such a plot is shown in Fig. 3.3; the two dashed lines represent the separate com­ ponents k r and the solid line is the sum Vf.

, and

I2 - — z^ 2mr

78

Chapter 3 The Central Force Problem

FIGURE 3.3 The equivalent one-dimensional potential for attractive inverse-square law of force.

Let us consider now the motion of a particle having the energy E\, as shown in Figs. 3.3 and 3.4. Clearly this particle can never come closer than r\ (cf. Fig. 3.4). Otherwise with r < r\, Vf exceeds E\ and by Eq. (3.150 the kinetic energy would have to be negative, corresponding to an imaginary velocity! On the other hand, there is no upper limit to the possible value of r, so the orbit is not bounded. A particle will come in from infinity, strike the "repulsive centrifugal barrier," be repelled, and travel back out to infinity (cf. Fig. 3.5). The distance between E and V is \mir2, i.e., proportional to the square of the radial velocity, and becomes zero, naturally, at the turning point r\. At the same time, the distance between E and V on the plot is the kinetic energy \mv2 at the given value of r. Hence, the distance between the V and V curves is ^mr202. These curves therefore supply the magnitude of the particle velocity and its components for any distance r, at the given energy and angular momentum. This information is sufficient to produce an approximate picture of the form of the orbit. For the energy £"2 = 0 (cf. Fig. 3.3), a roughly similar picture of the orbit behavior is obtained. But for any lower energy, such as £3 indicated in Fig. 3.6, we have a different story. In addition to a lower bound r\, there is also a maximum value V2 that cannot be exceeded by r with positive kinetic energy. The motion is then "bounded," and there are two turning points, r\ and ri, also known as apsidal distances. This does not necessarily mean that the orbits are closed. All that can be said is that they are bounded, contained between two circles of radius r\ and r2 with turning points always lying on the circles (cf. Fig. 3.7).

3.3

The Equivalent One-Dimensional Problem

79

A V

FIGURE 3.4

Unbounded motion at positive energies for inverse-square law of force.

FIGURE 3.5 The orbit for E\ corresponding to unbounded motion.

80

Chapter 3 The Central Force Problem

FIGURE 3.6 The equivalent one-dimensional potential for inverse-square law of force, illustrating bounded motion at negative energies.

If the energy is £4 at the minimum of the fictitious potential as shown in Fig. 3.8, then the two bounds coincide. In such case, motion is possible at only one radius; r = 0, and the orbit is a circle. Remembering that the effective "force" is the negative of the slope of the V curve, the requirement for circular orbits is simply that / ' be zero, or /2

f(r) =

=■ =

-mrO2.

We have here the familiar elementary condition for a circular orbit, that the ap­ plied force be equal and opposite to the "reversed effective force" of centripetal

FIGURE 3.7 The nature of the orbits for bounded motion. (/3 = 3 from Section 3.6.)

3.3

The Equivalent One-Dimensional Problem

i

81

FIGURE 3.8 The equivalent one-dimensional potential of inverse-square law of force, illustrating the condition for circular orbits. acceleration.* The properties of circular orbits and the conditions for them will be studied in greater detail in Section 3.6. Note that all of this discussion of the orbits for various energies has been at one value of the angular momentum. Changing / changes the quantitative details of the V7 curve, but it does not affect the general classification of the types of orbits. For the attractive inverse-square law of force discussed above, we shall see that the orbit for E\ is a hyperbola, for E2 a parabola, and for £3 an ellipse. With other forces the orbits may not have such simple forms. However, the same general qualitative division into open, bounded, and circular orbits will be true for any attractive potential that (1) falls off slower than 1/r2 as r -> 00, and (2) becomes infinite slower than 1/r2 as r -> 0. The first condition ensures that the potential predominates over the centrifugal term for large r, while the second condition is such that for small r it is the centrifugal term that is important. The qualitative nature of the motion will be altered if the potential does not sat­ isfy these requirements, but we may still use the method of the equivalent poten­ tial to examine features of the orbits. As an example, let us consider the attractive potential a 3a V(r) = — j , with / = — T . r5 r4 The energy diagram is then as shown in Fig. 3.9. For an energy E, there are two possible types of motion, depending upon the initial value of r. If ro is less than r\ the motion will be bounded, r will always remain less than r\, and the particle will pass through the center of force. If r is initially greater than 7*2, then it will *The case E < £4 does not correspond to physically possible motion, for then r 2 would have to be negative, or r imaginary.

82

Chapter 3 The Central Force Problem

FIGURE 3.9 The equivalent one-dimensional potential for an attractive inverse-fourth law of force. always remain so; the motion is unbounded, and the particle can never get inside the "potential" hole. The initial condition r\ < ro < n is again not physically possible. Another interesting example of the method occurs for a linear restoring force (isotropic harmonic oscillator): / =

-kr,

V =

\kr2.

For zero angular momentum, corresponding to motion along a straight line, V' = V and the situation is as shown in Fig. 3.10. For any positive energy the motion is bounded and, as we know, simple harmonic. If / ^ 0, we have the state of affairs shown in Fig. 3.11. The motion then is always bounded for all physically possible

FIGURE 3.10 Effective potential for zero angular momentum.

3.4

83

The Virial Theorem

t V

r

\

r

r2

FIGURE 3.11 The equivalent one-dimensional potential for a linear restoring force.

energies and does not pass through the center of force. In this particular case, it is easily seen that the orbit is elliptic, for if f = — Ax, the x- and y-components of the force are fx = -kx,

fy =

-ky.

The total motion is thus the resultant of two simple harmonic oscillations at right angles, and of the same frequency, which in general leads to an elliptic orbit. A well-known example is the spherical pendulum for small amplitudes. The familiar Lissajous figures are obtained as the composition of two sinusoidal os­ cillations at right angles where the ratio of the frequencies is a rational number. For two oscillations at the same frequency, the figure is a straight line when the oscillations are in phase, a circle when they are 90° out of phase, and an elliptic shape otherwise. Thus, central force motion under a linear restoring force there­ fore provides the simplest of the Lissajous figures.

3.4 ■ THE VIRIAL THEOREM Another property of central force motion can be derived as a special case of a general theorem valid for a large variety of systems—the virial theorem. It differs in character from the theorems previously discussed in being statistical in nature; i.e., it is concerned with the time averages of various mechanical quantities. Consider a general system of mass points with position vectors r,- and applied forces F| (including any forces of constraint). The fundamental equations of mo­ tion are then P/=F;. We are interested in the quantity

(1.3)

84

Chapter 3 The Central Force Problem

i

where the summation is over all particles in the system. The total time derivative of this quantity is AC*

-^ = J2 *«• • p< + 1 2 p« •r< • i

(3 23

- >

i

The first term can be transformed to

Y^h • Pi = Ylmiii ' ^ = YlmiV? i

i

= 2r

'

i

while the second term by (1.3) is

i

i

Equation (3.23) therefore reduces to

j t J2& • r; = 2T + ^ F / • r,, i

(3.24)

i

The time average of Eq. (3.24) over a time interval r is obtained by integrating both sides with respect to t from 0 to r, and dividing by r : 1 Cx dG , 1G — ^ R - / —dt = —- = IT + y ¥t . r,r J0 dt dt ^ or

2f + Jy^Vi

= - [G(r) - G(0)].

(3.25)

i

If the motion is periodic, i.e., all coordinates repeat after a certain time, and if r is chosen to be the period, then the right-hand side of (3.25) vanishes. A similar conclusion can be reached even if the motion is not periodic, provided that the coordinates and velocities for all particles remain finite so that there is an upper bound to G. By choosing r sufficiently long, the right-hand side of Eq. (3.25) can be made as small as desired. In both cases, it then follows that T=—^F/.r,-.

(3.26)

i

Equation (3.26) is known as the virial theorem, and the right-hand side is called the virial of Clausius. In this form the theorem is imporant in the kinetic theory

3.4

The Virial Theorem

85

of gases since it can be used to derive ideal gas law for perfect gases by means of the following brief argument. We consider a gas consisting of N atoms confined within a container of vol­ ume V. The gas is further assumed to be at a Kelvin temperature T (not to be confused with the symbol for kinetic energy). Then by the equipartition theorem of kinetic theory, the average kinetic energy of each atom is given by \kBT, kB being the Boltzmann constant, a relation that in effect is the definition of temper­ ature. The left-hand side of Eq. (3.26) is therefore \NkBT. On the right-hand side of Eq. (3.26), the forces F* include both the forces of interaction between atoms and the forces of constraint on the system. A perfect gas is defined as one for which the forces of interaction contribute negligibly to the virial. This occurs, e.g., if the gas is so tenuous that collisions between atoms occur rarely, compared to collisions with the walls of the container. It is these walls that constitute the constraint on the system, and the forces of constraint, F c , are localized at the wall and come into existence whenever a gas atom collides with the wall. The sum on the right-hand side of Eq. (3.26) can therefore be re­ placed in the average by an integral over the surface of the container. The force of constraint represents the reaction of the wall to the collision forces exerted by the atoms on the wall, i.e., to the pressure P. With the usual outward convention for the unit vector n in the direction of the normal to the surface, we can therefore write d¥( =

-PndA,

or i

p p

- ^ F / . r / = - - y n-rdA. But, by Gauss's theorem, I n-rdA=

fv-rdV

= 3V.

The virial theorem, Eq. (3.26), for the system representing a perfect gas can there­ fore be written lNkBT

=

\PV,

which, cancelling the common factor of \ on both sides, is the familiar ideal gas law. Where the interparticle forces contribute to the virial, the perfect gas law of course no longer holds. The virial theorem is then the principal tool, in classical kinetic theory, for calculating the equation of state corresponding to such imperfect gases.

Chapter 3 The Central Force Problem We can further show that if the forces F* are the sum of nonfrictional forces F^ and frictional forces f,- proportional to the velocity, then the virial depends only on the F^; there is no contribution from the f;. Of course, the motion of the system must not be allowed to die down as a result of the frictional forces. Energy must constantly be pumped into the system to maintain the motion; otherwise all time averages would vanish as r increases indefinitely (cf. Derivation 1.) If the forces are derivable from a potential, then the theorem becomes -

1^

T = -J2w-ri,

(3.27)

i

and for a single particle moving under a central force it reduces to 1 3V T = -—r. 2 or

(3.28)

If V is a power-law function of r, V

=arn+l,

where the exponent is chosen so that the force law goes as rn, then dV -r = (/i + l)V, or and Eq. (3.28) becomes — n + l— T = —^—V.

(3.29)

By an application of Euler's theorem for homogeneous functions (cf. p. 62), it is clear that Eq. (3.29) also holds whenever V is a homogeneous function in r of degree n + l. For the further special case of inverse-square law forces, n is —2, and the virial theorem takes on a well-known form: T = -\~V.

(3.30)

3.5 ■ THE DIFFERENTIAL EQUATION FOR THE ORBIT, AND INTEGRABLE POWER-LAW POTENTIALS In treating specific details of actual central force problems, a change in the orien­ tation of our discussion is desirable. Hitherto solving a problem has meant finding r and 0 as functions of time with E, /, etc., as constants of integration. But most often what we really seek is the equation of the orbit, i.e., the dependence of r upon 0, eliminating the parameter t. For central force problems, the elimination is particularly simple, since t occurs in the equations of motion only as a variable of differentiation. Indeed, one equation of motion, (3.8), simply provides a definite

3.5

The Differential Equation for the Orbit

87

relation between a differential change dt and the corresponding change dO: ldt = mr2d0.

(3.31)

The corresponding relation between derivatives with respect to t and 0 is d dt

I d mr2 dO

(3.32)

These relations may be used to convert the equation of motion (3.12) or (3.16) to a differential equation for the orbit. A substitution into Eq. (3.12) gives a secondorder differential equation, while a substitution into Eq. (3.17) gives a simpler first-order differential equation. The substitution into Eq. (3.12) yields I2 — = fir), mr

/ d / I dr\ r^a^\m~r^a7o)

(3.33)

which upon substituting u = 1/r and expressing the results in terms of the poten­ tial gives

l^+U

= -fTuV\u)-

(334)

The preceding equation is such that the resulting orbit is symmetric about two adjacent turning points. To prove this statement, note that if the orbit is symmet­ rical it should be possible to reflect it about the direction of the turning angle without producing any change. If the coordinates are chosen so that the turning point occurs for 0 = 0, then the reflection can be effected mathematically by substituting — 0 for 0. The differential equation for the orbit, (3.34), is obviously invariant under such a substitution. Further the initial conditions, here u = M(0),

I'du\ an \ ( — ) =0,

for 6 = 0,

will likewise be unaffected. Hence, the orbit equation must be the same whether expressed in terms of 0 or —0, which is the desired conclusion. The orbit is there­ fore invariant under reflection about the apsidal vectors. In effect, this means that the complete orbit can be traced if the portion of the orbit between any two turning points is known. Reflection of the given portion about one of the apsidal vectors produces a neighboring stretch of the orbit, and this process can be repeated in­ definitely until the rest of the orbit is completed, as illustrated in Fig. 3.12. For any particular force law, the actual equation of the orbit can be obtained by eliminating t from the solution (3.17) by means of (3.31), resulting in dO =

Ur

'-M w - £0

(3.35)

88

Chapter 3

The Central Force Problem

FIGURE 3.12 Extension of the orbit by reflection of a portion about the apsidal vectors.

With slight rearrangements, the integral of (3.35) is [r

dr

•%,

(3.36)

Jrn r 2 /TmJL _ 2mV _ J_ r

2

I

V

1

1

I

r

or, if the variable of integration is changed to u = 1/r, du

r

n »

Ju0 y2m£ _ 2mV _ u2

(3.37)

As in the case of the equation of motion, Eq. (3.37), while solving the problem formally, is not always a practicable solution, because the integral often cannot be expressed in terms of well-known functions. In fact, only certain types of force laws have been investigated. The most important are the power-law functions of r, V = arn+l

(3.38)

so that the force varies at the nth power of r.* With this potential, (3.37) becomes r 0=00-

du

UQ

s> ,

j2mE_2mau-n-l

(3.39)

This again is integrable in terms of simple functions only in certain cases. The particular power-law exponents for which the results can be expressed in terms of trigonometric functions are TI = 1 , - 2 , - 3 . *The case n = —1 is to be excluded from the discussion. In the potential (3.38), it corresponds to a constant potential, i.e., no force at all. It is an equally anomalous case if the exponent is used in the force law directly, since a force varying as r - 1 corresponds to a logarithmic potential, which is not a power law at all. A logarithmic potential is unusual for motion about a point; it is more characteristic of a line source. Further details of these cases are given in the second edition of this text.

3.6

Conditions for Closed Orbits (Bertrand's Theorem)

89

The results of the integral for n = 5, 3,0, - 4 , - 5 , - 7 can be expressed in terms of elliptic functions. These are all the possibilities for an integer exponent where the formal integrations are expressed in terms of simple well-known functions. Some fractional exponents can be shown to lead to elliptic functions, and many other exponents can be expressed in terms of the hypergeometric function. The trigonometric and elliptical functions are special cases of generalized hypergeometric function integrals. Equation (3.39) can of course be numerically integrated for any nonpathological potential, but this is beyond the scope of the text.

3.6 ■ CONDITIONS FOR CLOSED ORBITS (BERTRAND'S THEOREM)

We have not yet extracted all the information that can be obtained from the equiv­ alent one-dimensional problem or from the orbit equation without explicitly solv­ ing for the motion. In particular, it is possible to derive a powerful and thoughtprovoking theorem on the types of attractive central forces that lead to closed orbits, i.e., orbits in which the particle eventually retraces its own footsteps. Conditions have already been described for one kind of closed orbit, namely a circle about the center of force. For any given /, this will occur if the equivalent potential V'(r) has a minimum or maximum at some distance ro and if the energy E is just equal to Vf(ro). The requirement that V have an extremum is equiva­ lent to the vanishing of / ' at ro, leading to the condition derived previously (cf. Section 3.3). /(ro) =

I2

3,

(3.40)

which says the force must be attractive for circular orbits to be possible. In addi­ tion, the energy of the particle must be given by I2 E = V(r0) + — j , 2mrQ

(3.41)

which, by Eq. (3.15), corresponds to the requirement that for a circular orbit r is zero. Equations (3.40) and (3.41) are both elementary and familiar. Between them they imply that for any attractive central force it is possible to have a circular orbit at some arbitrary radius ro, provided the angular momentum / is given by Eq. (3.40) and the particle energy by Eq. (3.41). The character of the circular orbit depends on whether the extremum of V' is a minimum, as in Fig. 3.8, or a maximum, as if Fig. 3.9. If the energy is slightly above that required for a circular orbit at the given value of /, then for a minimum in V the motion, though no longer circular, will still be bounded. However, if

90

Chapter 3

The Central Force Problem

Vf exhibits a maximum, then the slightest raising of E above the circular value, Eq. (3.34), results in motion that is unbounded, with the particle moving both through the center of force and out to infinity for the potential shown in Fig. 3.9. Borrowing the terminology from the case of static equilibrium, the circular orbit arising in Fig. 3.8 is said to be stable', that in Fig. 3.9 is unstable. The stability of the circular orbit is thus determined by the sign of the second derivative of Vf at the radius of the circle, being stable for positive second derivative (V concave up) and unstable for Vf concave down. A stable orbit therefore occurs if d2V dr2

3/2

' dr

r=r0

+ ■

>0.

(3.42)

mrZ

r=r0

Using Eq. (3.40), this condition can be written

Wo) dr

ro

r=r0

(3.43)

or

dlnf dlnr

> -3

(3.430

r=r0

where /(ro)/ro is assumed to be negative and given by dividing Eq. (3.40) by ro. If the force behaves like a power law of r in the vicinity of the circular radius ro,

/ = ~krn, then the stability condition, Eq. (3.43), becomes -knrn~x

< 3krn~l

or n > —3,

(3.44)

where k is assumed to be positive. A power-law attractive potential varying more slowly than 1/r2 is thus capable of stable circular orbits for all values of ro. If the circular orbit is stable, then a small increase in the particle energy above the value for a circular orbit results in only a slight variation of r about ro. It can be easily shown from (3.34) that for such small deviations from the circularity conditions, the particle executes a simple harmonic motion in u(= 1/r) about UQ: u=

UQ

+ a cos j36.

(3.45)

Here a is an amplitude that depends upon the deviation of the energy from the value for circular orbits, and f$ is a quantity arising from a Taylor series expansion

3.6

Conditions for Closed Orbits (Bertrand's Theorem)

91

of the force law f(r) about the circular orbit radius ro. Direct substitution into the force law gives

P2 = 3 +

r_d£ fdr

(3.46) r=r0

As the radius vector of the particle sweeps completely around the plane, u goes through p cycles of its oscillation (cf. Fig. 3.13). If ^ is a rational number, the ratio of two integers, p/q, then after q revolutions of the radius vector the orbit would begin to retrace itself so that the orbit is closed. At each ro such that the inequality in Eq. (3.43) is satisfied, it is possible to establish a stable circular orbit by giving the particle an initial energy and angular momentum prescribed by Eqs. (3.40) and (3.41). The question naturally arises as to what form the force law must take in order that the slightly perturbed orbit about any of these circular orbits should be closed. It is clear that under these conditions f$ must not only be a rational number, it must also be the same rational number at all distances that a circular orbit is possible. Otherwise, since f$ can take on only discrete values, the number of oscillatory periods would change discontinuously with ro, and indeed the orbits could not be closed at the discontinuity. With fi2 everywhere constant, the defining equation for fi2, Eq. (3.46), becomes in effect a differential equation for the force law / in terms of the independent variable roWe can indeed consider Eq. (3.46) to be written in terms of r if we keep in mind that the equation is valid only over the ranges in r for which stable circular orbits are possible. A slight rearrangement of Eq. (3.46) leads to the equation dln

f dlnr

P

=62_3

(3.47)

FIGURE 3.13 Orbit for motion in a central force deviating slightly from a circular orbit for 0 = 5.

Chapter 3 The Central Force Problem

which can be immediately integrated to give a force law: f(r) = ~

.

0.48)

All force laws of this form, with f3 a rational number, lead to closed stable orbits for initial conditions that differ only slightly from conditions defining a circular orbit. Included within the possibilities allowed by Eq. (3.48) are some familiar forces such as the inverse-square law (/? = 1), but of course many other behaviors, such as / = —kr~~2/9(P = | ) , are also permitted. Suppose the initial conditions deviate more than slightly from the requirements for circular orbits; will these same force laws still give circular orbits? The ques­ tion can be answered directly by keeping an additional term in the Taylor series expansion of the force law and solving the resultant orbit equation. J. Bertrand solved this problem in 1873 and found that for more than first-order deviations from circularity, the orbits are closed only for /32 = 1 and /32 = 4. The first of these values of /?2, by Eq. (3.48), leads to the familiar attractive inversesquare law; the second is an attractive force proportional to the radial distance— Hooke's law! These force laws, and only these, could possibly produce closed orbits for any arbitrary combination of / and E(E < 0), and in fact we know from direct solution of the orbit equation that they do. Hence, we have Bertrand's theorem: The only central forces that result in closed orbits for all bound particles are the inverse-square law and Hooke 's law. This is a remarkable result, well worth the tedious algebra required. It is a com­ monplace astronomical observation that bound celestial objects move in orbits that are in first approximation closed. For the most part, the small deviations from a closed orbit are traceable to perturbations such as the presence of other bodies. The prevalence of closed orbits holds true whether we consider only the solar sys­ tem, or look to the many examples of true binary stars that have been observed. Now, Hooke's law is a most unrealistic force law to hold at all distances, for it implies a force increasing indefinitely to infinity. Thus, the existence of closed orbits for a wide range of initial conditions by itself leads to the conclusion that the gravitational force varies as the inverse-square of the distance. We can phrase this conclusion in a slightly different manner, one that is of somewhat more significance in modern physics. The orbital motion in a plane can be looked on as compounded of two oscillatory motions, one in r and one in 0 with the same period. The character of orbits in a gravitational field fixes the form of the force law. Later on we shall encounter other formulations of the relation between degeneracy and the nature of the potential.

3.7 ■ THE KEPLER PROBLEM: INVERSE-SQUARE LAW OF FORCE

The inverse-square law is the most important of all the central force laws, and it deserves detailed treatment. For this case, the force and potential can be written

3.7

The Kepler Problem: Inverse-Square Law of Force

93

as v

/ = -4

= -~-

(3-49)

r There are several ways to integrate the equation for the orbit, the simplest being to substitute (3.49) in the differential equation for the orbit (3.33). Another approach is to start with Eq. (3.39) with n set equal to —2 for the gravitational force du

0 = 0' - I -7==^===, llmE

V

| 2mku

2

"*"

I

~2'

(3.50)

2

I

where the integral is now taken as indefinite. The quantity 0f appearing in (3.50) is a constant of integration determined by the initial conditions and will not nec­ essarily be the same as the initial angle #o at time t = 0. The indefinite integral is of the standard form, dx /

1 2

y/a + fix + yx

arc c o s -

V-Y

6 + lyx _; , Vv

(3.51)

where q = ft2 - 4ay. To apply this to (3.50), we must set 2mE

2mk

n

« = — ,

(* = -£-

Y = ~h

(3.52)

and the discriminant q is therefore / .1 +.

■ ) • 2( '

_ /2mfc\ l, e = l, e < 1,

E > 0 E =0 E o sin 0/(1 — s2) vanishes at the two apsidal distances, while VQ attains its maximum value at perihelion and its minimum at aphelion. Table 3.1 lists angular velocity values at the ap­ sidal distances for several eccentricities. Figure 3.16 presents plots of the ra­ dial velocity component vr versus the radius vector r for the half cycle when vT points outward, i.e., it is positive. During the remaining half cycle vr is nega-

FIGURE 3.15 Dependence of normalized apsidal distances r\ (lower line) and r^ (upper line) on the eccentricity s.

3.7

97

The Kepler Problem: Inverse-Square Law of Force

TABLE 3.1 Normalized angular speeds 6 and VQ = rO at perihelion (r\) and aphelion (>2), respectively, in Keplerian orbits of various eccentricities (s). The normalized radial distances at perihelion and aphelion are listed in columns 2 and 3, respectively. The normalization is with respect to motion in a circle with the radius a and the angular momentum / = mavo = ma 6Q. Eccentricity

0 0.1 0.3 0.5 0.7 0.9

Angular speed

Perihelion

Aphelion

n/a

rila

61/60

1-e

l +e

1 (1 - s)2

1 1.1 1.3 1.5 1.7 1.9

1 1.234 2.041 4.000 11.111 100.000

1 0.9 0.7 0.5 0.3 0.1

62/60 1 (l+s)2 1 0.826 0.592 0.444 0.346 0.277

Linear angular speed U01/UO

1 1-6

1 1.111 1.429 2.000 3.333 10.000

U02/^O

1 l+e 1 0.909 0.769 0.667 0.588 0.526

tive, and the plot of Fig. 3.16 repeats itself for the negative range below vr = 0 (not shown). Figure 3.17 shows analogous plots of the angular velocity com­ ponent v