Geometry, Mechanics, and Control in Action for the Falling Cat [1 ed.] 9789811606878, 9789811606885

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Lecture Notes in Mathematics  2289

Toshihiro Iwai

Geometry, Mechanics, and Control in Action for the Falling Cat

Lecture Notes in Mathematics Volume 2289

Editors-in-Chief Jean-Michel Morel, CMLA, ENS, Cachan, France Bernard Teissier, IMJ-PRG, Paris, France Series Editors Karin Baur, University of Leeds, Leeds, UK Michel Brion, UGA, Grenoble, France Camillo De Lellis, IAS, Princeton, NJ, USA Alessio Figalli, ETH Zurich, Zurich, Switzerland Annette Huber, Albert Ludwig University, Freiburg, Germany Davar Khoshnevisan, The University of Utah, Salt Lake City, UT, USA Ioannis Kontoyiannis, University of Cambridge, Cambridge, UK Angela Kunoth, University of Cologne, Cologne, Germany Ariane Mézard, IMJ-PRG, Paris, France Mark Podolskij, University of Luxembourg, Esch-sur-Alzette, Luxembourg Sylvia Serfaty, NYU Courant, New York, NY, USA Gabriele Vezzosi, UniFI, Florence, Italy Anna Wienhard, Ruprecht Karl University, Heidelberg, Germany

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Toshihiro Iwai

Geometry, Mechanics, and Control in Action for the Falling Cat

Toshihiro Iwai Kyoto University Kyoto, Japan

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-981-16-0687-8 ISBN 978-981-16-0688-5 (eBook) https://doi.org/10.1007/978-981-16-0688-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The falling cat and the falling apple of Newton share the same mechanical property, i.e., vanishing angular momentum. We begin by giving a brief review of the falling apple of Newton. An apple fell down at his feet, when Newton was sitting in an orchard on a summer afternoon. This fact is supposed to have inspired Newton to find the universal law of gravitation on the basis of Kepler’s laws of planetary motion. For us, people of today, the falling of an apple is a two-body (an apple and the Earth) problem, which reduces to a one-body problem with respect to a mass-weighted difference vector between two bodies. As the initial velocity of the present difference vector vanishes, the angular momentum vanishes as well. Then, the conservation law of the angular momentum provides the equation r×

dr = 0, dt

(1)

where r denotes the mass-weighted difference vector. This implies that r and dr/dt are in parallel, so that there exists a scalar function λ such that dr/dt = λr. This means that the motion of the apple takes place along the line segment joining the respective centers of mass of the Earth and the apple. In contrast to this, the moon does not fall to Earth. This is because the angular momentum of the difference vector between the moon and the Earth does not vanish. In order to find the trajectories of the apple and the moon in respective cases, we need to solve Newton’s equations of motion with gravitational force. One of Kepler’s laws refers to the non-vanishing of the angular momentum of planets. The first law says that the orbit of every planet is an ellipse with the Sun at one of the two foci, and the second law says that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. These laws imply that the angular momentum is conserved and the force is central. The planet in question (or the moon) has a non-vanishing angular momentum. Further, the central force is shown to be inversely proportional to the square of the distance. In order to show

v

vi

Preface

that the proportional constant is a universal constant, the third law is needed, which says that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit (see Appendix 5.1 for details). In contrast to the falling apple of Newton, when interest attaches to the falling cat, the gravitational force may be put aside. We then reconsider the condition of the vanishing angular momentum in view of the fact that the angular momentum of the cat surely vanishes in the air. The angular momentum of an N-particle system with N ≥ 3 can bring about a complicated motion of each particle. For an N-particle system, there are N − 1 mass-weighted vectors (Jacobi vectors to be introduced in the text), in terms of which Eq. (1) is extended to N−1  i=1

ri ×

dr i = 0. dt

(2)

An initial question we are interested in concerns a property exhibited by a motion satisfying (2). Physically speaking, any motion satisfying this equation is viewed as vibrational motion, as the total angular momentum of the particle system vanishes. In contrast to solutions to (1), solutions to (2) exhibit interesting motions with remarkable geometric properties and are used to explain a reason why vibrations can give rise to effective rotations. In addition, a further question arises as to whether the following partial differential equations may have solutions or not: N−1 

r i × dr i = 0.

(3)

i=0

In other words, the question is stated as follows: Are there (3N − 6)-dimensional surfaces satisfying the above equations in R3N−3 ? This equation has been of central interest in the study on the possibility of the separation of vibration from rotation. However, Eq. (3) is not integrable. The quantity associated with the non-integrability has a geometric meaning. Since the falling cat makes vibrational motions only in the air, a solution to the falling cat problem must involve questions regarding Eqs. (2) and (3) without reference to gravitational force. As is well known, cats always can land on their feet when launched in the air. If cats are not given a non-vanishing angular moment at an initial instant, they cannot rotate during their motion, but the motion they can make in the air is vibration only. However, cats accomplish a turn after a vibrational motion, when landing on their feet. As is alluded to above, in order to solve this apparent mystery, one needs to gain a strict understanding of rotations and vibrations. The connection theory in differential geometry can provide rigorous definitions of rotation and vibration for many particle systems [22] and shows that vibrational motions can result in rotations, without performing rotational motions. The deformable bodies of cats

Preface

vii

are not easy to describe or to analyze in terms of mechanics. A feasible way to approach the question about the falling cat is to start with many particle systems and then proceed to systems of rigid bodies, and further to jointed rigid bodies, which can approximate the body of a cat. Since the present model of the falling cat is a mechanical object, mechanics of many-body systems and of jointed rigid bodies need to be set up. In order to take into account the fact that cats can contort their bodies, torque inputs should be applied as control inputs, which are to be suitably designed. In this book, the port-controlled Hamiltonian method will be adopted for the jointed rigid bodies to perform a turn and to halt the motion at the instance of landing. A brief review of control systems will be given through simple examples to explain the role of control inputs. This book consists of four chapters, the headings of which refer, in accordance with the title of this book, to Geometry, Mechanics, Control, and the Falling Cat, respectively. In the first chapter, geometries of many-body systems are set up, starting with planar many-body systems. As planar many-body systems have rather simple geometric structures, they will serve as an informative guide to the geometry of spatial many-body systems. After a review of the rotation group SO(3), a geometric setting is provided for spatial many-body systems. In this chapter, the main interest centers on the non-separability of rotation and vibration. The second chapter deals with mechanics of many-body systems on the basis of the geometric setting. Rigid bodies are treated as special cases of many-body systems, and rigid body mechanics is reformulated on the variational principle in the Lagrangian and Hamiltonian formalisms. Subsequently, Lagrangian and Hamiltonian mechanics for spatial many-body systems are set up on the variational principle as well. The third chapter contains examples of mechanical control systems with interest in the understanding of the design of control inputs from mechanical point of view. The last chapter is devoted to the falling cat problem with two jointed cylinders as a model system. Geometry, mechanics, and control set up in the preceding chapters are employed in the analysis of the falling cat. Advanced material concerning manybody systems and related topics together with Newton’s law of gravitation are discussed and reviewed in the appendices. A short remark on notations used in this book is to be made in advance of the text. The space of real n × m matrices is denoted by Rn×m , and the transpose of a matrix A by AT . Before beginning with the text, the author would like to stress that the present book gives a theory for the falling cat and does not recommend readers to make experiments on cats that test the bonds of trust between cats and human beings. The author would like to thank his old students and colleagues who were interested in the falling cat problem for discussions with, questions from, and suggestions by them. This book is a result of those activities. In particular, thanks go to Mr. Matsunaka. The graphs in Figs. 4.3, 4.4, 4.5 and in Figs. 4.6, 4.7, 4.8 and the snapshots in Fig. 4.9 given in Chap. 4 are newly produced for the present book by Mr. Matsunaka, who was a coauthor of an old joint paper on the falling cat [39].

viii

Preface

Fig. 1 A sketch of the falling cat and a model cat using jointed cylinders. Small rods attached to the cylinders are indicators to the attitude

The author is indebted to his wife and daughter for Fig. 1. Figure 1 is a sketch of the idea for the falling cat and Fig. 4.9 shows a realization of the idea developed in this book. Kyoto, Japan

Toshihiro Iwai

Contents

1

Geometry of Many-Body Systems. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Planar Many-Body Systems . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Rotation and Vibration of Planar Many-Body Systems .. . . . . . . . . . . . . 1.3 Vibrations Induce Rotations in Two Dimensions . . . . . . . . . . . . . . . . . . . . 1.4 Planar Three-Body Systems . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 The Rotation Group SO(3) . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6 Spatial Many-Body Systems . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.7 Rotation and Vibration for Spatial Many-Body Systems . . . . . . . . . . . . 1.8 Local Description of Spatial Many-Body Systems . . . . . . . . . . . . . . . . . . 1.8.1 Local Product Structure.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.8.2 Local Description in the Space Frame . .. . . . . . . . . . . . . . . . . . . . 1.8.3 Local Description in the Rotated Frame . . . . . . . . . . . . . . . . . . . . 1.9 Spatial Three-Body Systems . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.10 Non-separability of Vibration from Rotation . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 10 12 16 23 33 42 51 51 53 57 60 65

2 Mechanics of Many-Body Systems . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Equations of Motion for a Free Rigid Body . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Variational Principle for a Free Rigid Body .. . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Lagrangian Mechanics of Many-Body Systems .. . . . . . . . . . . . . . . . . . . . 2.4 Hamel’s Approach .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Hamiltonian Mechanics of Many-Body Systems . . . . . . . . . . . . . . . . . . . .

69 69 74 78 86 89

3 Mechanical Control Systems.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 95 3.1 Electron Motion in an Electromagnetic Field . . . .. . . . . . . . . . . . . . . . . . . . 95 3.2 The Inverted Pendulum on a Cart . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 100 3.3 Port-Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 105 3.4 Remarks on Optimal Hamiltonians . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 108 4 The Falling Cat .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 111 4.1 Modeling of the Falling Cat . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 111 4.2 Geometric Setting for Rigid Body Systems . . . . . .. . . . . . . . . . . . . . . . . . . . 112

ix

x

Contents

4.3

Geometric Setting for Two Jointed Cylinders .. . .. . . . . . . . . . . . . . . . . . . . 4.3.1 The Configuration Space . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.2 Geometric Quantities . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.3 Summary and a Remark on the Geometric Setting .. . . . . . . . A Lagrangian Model of the Falling Cat . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A Port-Controlled Hamiltonian System . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Execution of Somersaults . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Remarks on Control Problems . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

114 114 117 122 124 126 129 133

5 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Newton’s Law of Gravitation, Revisited .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Principal Fiber Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Spatial N-Body Systems with N ≥ 4 . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 The Orthogonal Group O(n) . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5 Many-Body Systems in n Dimensions . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.6 Holonomy for Many-Body Systems . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.7 Rigid Bodies in n Dimensions . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.8 Kaluza–Klein Formalism .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.9 Symplectic Approach to Hamilton’s Equations . .. . . . . . . . . . . . . . . . . . . . 5.10 Remarks on Related Topics . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.10.1 Quantum Many-Body Systems . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.10.2 Geometric Phases and Further Reading .. . . . . . . . . . . . . . . . . . . . 5.10.3 Open Dynamical Systems and Developments . . . . . . . . . . . . . .

137 137 142 143 152 155 162 164 169 172 174 174 175 176

4.4 4.5 4.6 4.7

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 177 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 181

Chapter 1

Geometry of Many-Body Systems

This chapter deals with the geometric setting for planar and spatial many-body systems on the basis of connection theory. Rather, the contents of this chapter may be of practical help in understanding the connection theory.

1.1 Planar Many-Body Systems Suppose we are given a system of N point particles in the plane R2 . Let x α and mα > 0, α = 1, 2, . . . , N, denote the position and the mass of each particle. Then, the configuration of these particles is described as (x 1 , x 2 , · · · , x N ) (Fig. 1.1). The totality of the configurations is called the configuration space, which we denote by X = {x; x = (x 1 , x 2 , · · · , x N ), x α ∈ R2 }.

(1.1)

The X is a linear space of dimension 2n, in which addition and scalar multiplication are performed componentwise: (x 1 + y 1 , x 2 + y 2 , · · · , x N + y N ) = (x 1 , x 2 , · · · , x N ) + (y 1 , y 2 , · · · , y N ), λ(x 1 , x 2 , · · · , x N ) = (λx 1 , λx 2 , · · · , λx N ). (1.2) We can view this system as a vector space of 2×N matrices consisting of N column vectors x α ∈ R2 . Let us denote the canonical basis vectors of R2 by   1 e1 = , 0

  0 e2 = . 1

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 T. Iwai, Geometry, Mechanics, and Control in Action for the Falling Cat, Lecture Notes in Mathematics 2289, https://doi.org/10.1007/978-981-16-0688-5_1

1

2

1 Geometry of Many-Body Systems

x2

Fig. 1.1 Position vectors

x1

xN

xi Then, a basis of X is given, for example, by (e1 , 0, · · · , 0),

(e2 , 0, · · · , 0),

··· ,

(0, · · · , e2 ).

(1.3)

The X is endowed with the mass-weighted inner product (or inner product, for short) defined to be K(x, y) =

N 

mα x α · y α ,

x, y ∈ X,

(1.4)

α=1

where x · y denotes the standard inner product on R2 . As is easily verified, the K(x, y) has the properties K(x, y) = K(y, x),

(1.5a)

K(λx + μy, z) = λK(x, z) + μK(y, z), K(x, x) ≥ 0,

λ, μ ∈ R,

K(x, x) = 0 ⇐⇒ x = 0.

(1.5b) (1.5c)

The center-of-mass system is the linear subspace of X which is defined to be Q = {x ∈ X;

N 

mα x α = 0}.

(1.6)

α=1

In order to characterize the Q in X, it is of help to introduce the center-of-mass vector for an arbitrary configuration y = (y 1 , · · · , y N ) ∈ X through ⎛ b=⎝

N 

β=1

⎞−1 mβ ⎠

N 

mα y α .

(1.7)

α=1

Then, by setting x α = y α − b, one has the decomposition of y, (y 1 , y 2 , · · · , y N ) = (x 1 , x 2 , · · · , x N ) + (b, b, · · · , b).

(1.8)

1.1 Planar Many-Body Systems

3

We can easily verify that x = (x 1 , x 2 , · · · , x N ) ∈ Q, and further that (x 1 , x 2 , · · · , x N ) ⊥ (b, b, · · · , b), where the orthogonality is defined with respect to the inner product K. On setting b = (b, · · · , b), Eq. (1.8) is written as y = x + b, which is an orthogonal decomposition of y ∈ X. Let us denote the orthogonal complement of Q by Q⊥ := {y ∈ X; K(y, x) = 0, x ∈ Q}.

(1.9)

Then, we have the orthogonal decomposition of X, X = Q ⊕ Q⊥ ,

(1.10)

to which y = x + b is subject. It is easily shown that Q⊥ = {x ∈ X| x = (c, c, · · · , c), c ∈ R2 }.

(1.11)

The basis (1.3) is not adequate to the decomposition (1.10). In fact, the basis belongs to neither Q nor Q⊥ . We wish to find a basis adapted to (1.10). An orthonormal basis of Q⊥ is easy to find on account of (1.11). In order to find a basis of Q, we can start, for example, with (−m2 e1 , m1 e1 , 0, · · · , 0), (−m2 e2 , m1 e2 , 0, · · · , 0), which clearly belong to Q, and apply the Schmidt method. Performing a similar procedure for other vectors in Q, we can find an orthonormal basis. Thus, we obtain the orthonormal system as follows: Proposition 1.1.1 The configuration space X for the planar N-body system admits the following orthonormal system of basis vectors: c1 = N0 (e1 , e1 , · · · , e1 ), c2 = N0 (e2 , e2 , · · · , e2 ), f2j −1

j terms j

  = Nj (−mj +1 e1 , · · · , −mj +1 e1 , ( mα )e1 , 0, · · · , 0),

(1.12)

α=1

terms j

  = Nj (−mj +1 e2 , · · · , −mj +1 e2 , ( mα )e2 , 0, · · · , 0), j

f2j

α=1

where j = 1, 2, . . . , N − 1, and where

N0 =

n  α=1

⎛

−1/2 mα

,

⎛

Nj = ⎝mj +1 ⎝

j 

α=1

⎞⎛ mα ⎠ ⎝

j +1  α=1

⎞⎞−1/2 mα ⎠ ⎠

.

4

1 Geometry of Many-Body Systems

It is straightforward to verify that K(ca , cb ) = δab ,

K(ca , fk ) = 0,

K(fk , f ) = δk ,

(1.13)

a, b = 1, 2, k,  = 1, 2, . . . , 2(N − 1). This means that Q⊥ ∼ = R2 ,

Q∼ = R2(N−1) ,

(1.14)

where Q⊥ and Q have basis vectors ca , a = 1, 2, and fk , k = 1, 2, . . . , 2(N − 1), respectively. For any configuration y = x + b ∈ Q ⊕ Q⊥ , the components of x ∈ Q and b ∈ Q⊥ with respect to these basis vectors are determined by pa = K(y, ca ) = K(b, ca ),

a = 1, 2,

qk = K(y, fk ) = K(x, fk ),

k = 1, 2, . . . , 2(N − 1),

(1.15)

and (pa , qk ) serve as the Cartesian coordinates of X = Q⊥ ⊕ Q. In what follows, we take up Q endowed with the Cartesian coordinates (qk ). 2(N−1) Any x ∈ Q is expressed as x = k=1 qk fk . We now show that the coordinates (qk ) determine the Jacobi vectors on the plane R2 . In fact, the vectors r j defined by r j := q2j −1 e1 + q2j e2 , j = 1, 2, . . . , N − 1, are written out as r j := q2j −1 e1 + q2j e2 = K(x, f2j −1 )e1 + K(x, f2j )e2 j +1 j j j 1/2   −1/2  −1      x j +1 − = mj +1 mα mα mα mα x α . α=1

α=1

α=1

α=1

(1.16) The present expression of r j shows that r j is the vector with the initial point at the center-of-mass of the set of the particles at x α , α = 1, · · · , j , to the end point at the position of the particle x j +1 , within a constant multiple. The vectors r j are exactly the Jacobi vectors (Fig. 1.2). Conversely, any of x 1 , · · · , x N−1 can be expressed as a linear combination of r 1 , · · · , r N−1 , by using (1.16), and hence any of x 1 , · · · , x N−1 , x N can beexpressed as a linear combination of (r 1 , · · · , r N−1 ) on account of x N = − m1N N−1 k=1 mk x k . It then turns out that the center-of-mass system Q is viewed as the totality of the Jacobi vectors (r 1 , · · · , r N−1 ).

1.1 Planar Many-Body Systems

5

x4

Fig. 1.2 Planar Jacobi vectors

x3

x5 x2 x1 In particular, for N = 3, the Jacobi vectors are described as  r 1 = q1 e 1 + q2 e 2 =  r 2 = q3 e 1 + q4 e 2 =

m1 m2 (x 2 − x 1 ), m1 + m2 m1 x 1 + m2 x 2  m3 (m1 + m2 )  x3 − . m1 + m2 + m3 m1 + m2

(1.17)

Conversely, the position vectors are expressed as   m  m3 −1 2 x1 = √ r1 + r2 , m1 m1 + m2 + m3 m1 + m2     m1 m3 1 x2 = √ r1 − r2 , m2 m1 + m2 + m3 m1 + m2 1 x 3 = − (m1 x 1 + m2 x 2 ). m3

(1.18)

In the center-of-mass system, any configuration coming from x ∈ Q by a rotation has the same shape as the initial one x. We do not have to distinguish these configurations. This is because if we consider the configuration as a molecule with each particle being an atom, then the molecule property is independent of its position in R2 , but depends on its shape only. In Euclidean geometry, congruence of triangles is discussed in a similar manner. Two triangles which are translated to one another by a translation and a rotation or a reflection are called congruent, which have the same shape. We start with the definition of the SO(2) action on the center-of-mass system Q. A reason why we do not treat the O(2) action is as follows: If we view the N-body system as a molecule, the reflected N-body system would have a different chemical property from the initial one. Now we define the SO(2) action by x = (x 1 , x 2 , · · · , x N ) −→ gx = (gx 1 , gx 2 , · · · , gx N ),

x ∈ Q, g ∈ SO(2), (1.19)

6

1 Geometry of Many-Body Systems

x3

Fig. 1.3 SO(2) action on a configuration

gx2 gx3 x2 x1

gx1

where   cos t − sin t g = g(t) = . sin t cos t  The SO(2) action is well-defined on Q (Fig. 1.3). In fact, if mα x α = 0, then  one has mα gx α = 0. The present SO(2) action is a linear transformation of Q, the representation of which is determined by gfk =

2(N−1) 

aj k fj ,

k = 1, 2, . . . , 2(N − 1),

j =1

where fk , k = 1, 2, . . . , 2(N −1), are the basis vectors already introduced in (1.12). Since aj k = K(fj , gfk ), we can evaluate aj k by using the definitions of the inner product K and of the basis vectors fk . After a calculation, we find that g(t)f2j −1 = f2j −1 cos t + f2j sin t, g(t)f2j = −f2j −1 sin t + f2j cos t.

(1.20)

In each linear subspace spanned by f2j −1 , f2j , the SO(2) action is described, in terms of the Cartesian coordinates (q2j −1 , q2j ), as      q2j −1 cos t − sin t q2j −1

−→ . q2j q2j sin t cos t Consequently, the matrix representation of the SO(2) is given by ⎛ ⎜ ⎜ (aj k ) = ⎜ ⎝

⎞

g(t)

⎟ ⎟ ⎟, ⎠

g(t) ..

. g(t)

which is a block diagonal matrix of size 2(N − 1) × 2(N − 1).

(1.21)

1.1 Planar Many-Body Systems

7

The present SO(2) action determines an equivalence relation on Q. For x, y ∈ Q, if there exists g ∈ SO(2) such that y = gx, then x and y are called congruent and denoted by x ∼ y. Then, it is easy to verify that (i) x ∼ x,

(ii) x ∼ y ⇒ y ∼ x,

(iii) x ∼ y, y ∼ z ⇒ x ∼ z.

This equivalence relation determines the factor space consisting of equivalence classes. Geometrically speaking, the present factor space is the space of shapes of molecules. We are interested in non-trivial shapes. Put anther way, if all the particles collect at the center-of-mass, its configuration is x = 0, which is considered as ˙ denote Q without x = 0, shapeless and we eliminate it from Q. Let Q ˙ = {x ∈ Q; x = 0}. Q

(1.22)

Then, the SO(2) action on Q˙ is free, i.e., if gx = x for x ∈ Q˙ then g = e, the ˙ identity of SO(2), as is easily verified. Thus, the factor space Q/SO(2) proves to be a manifold, which we call a shape space. We denote the shape space by M˙ and the natural projection by π: ˙ π : Q˙ −→ M˙ := Q/SO(2).

(1.23)

In other words, if we denote the equivalence class of x ∈ Q˙ by [x], the projection is ˙ defined to be π(x) = [x] ∈ Q/SO(2). By definition, [x] = [x  ] if and only if there  exists g ∈ SO(2) such that x = gx. The center-of-mass system Q has a rather simple structure with respect to the SO(2) action. In order to give a compact description, we introduce the complex vector space structure into Q ∼ = R2(N−1) by setting zj = q2j −1 + iq2j ,

j = 1, 2, . . . , N − 1,

i=

√ −1.

Then, we have the isomorphism Q ∼ = CN−1 . Accordingly, the SO(2) action on Q is represented as a U (1) action, z = (z1 , z2 , · · · , zN−1 ) −→ eit z = (eit z1 , eit z2 , · · · , eit zN−1 ).

(1.24)

This is because the transformation (1.21) is compactly represented as eit zj = (cos t + i sin t)(q2j −1 + iq2j ). ˙ N−1 . The U (1) action Now it is clear that Q˙ ∼ = CN−1 − {0}, which we denote by C N−1 it ∼ ˙ N−1 , then eit = 1. ˙ ˙ on Q = C is free, of course. In fact, if e z = z for z ∈ C

8

1 Geometry of Many-Body Systems

˙ N−1 → R+ × CN−1 to be We define a map C  zN−1  z1 (z1 , · · · , zN−1 ) −→ |z|, , · · · , , |z| |z|

|z|2 =

N−1 

|zj |2 ,

(1.25)

j =1

where R+ denotes the set of positive real numbers. Then, we verify that ˙ N−1 ∼ Q˙ ∼ = R+ × S 2N−3 , =C

(1.26)

where S

2N−3

= {z ∈ C

N−1

;

N−1 

|zj | =

j =1

2

2(N−1) 

(qk )2 = 1}.

k=1

The U (1) action leaves the space R+ invariant, and is free on S 2N−3 , so that the factor space (or the shape space) turns out to be isomorphic with ˙ M˙ = Q/U (1) ∼ = R+ × CP N−2 , = R+ × S 2N−3 /U (1) ∼

(1.27)

where CP N−2 = S 2N−3 /U (1) denotes the complex projective space of complex dimension N − 2, which is defined as the factor space by the equivalence relation defined on S 2N−3 through z ∼ w ⇐⇒ z = eiθ w. Initially, CP N−2 is defined to be the factor space by the ˙ N−1 through z ∼ w ⇐⇒ z = λw. equivalence relation on C Proposition 1.1.2 Except for a singular configuration in which all the particles collide at the center-of-mass, the shape space for the center-of-mass system Q˙ is diffeomorphic to R+ × CP N−2 . What CP N−2 means in the planar many-body system is described as follows:  2(N−1) 2 Since |z|2 = N−1 (qk )2 = K(x, x), the set S 2n−3 is viewed as j =1 |zj | = k=1 a normalization of molecular configurations, and thereby CP N−2 as the space of normalized shapes of the molecule. In what follows, we add more explanation of the complex projective space CP N−2 , which can be described as the space of (N −1)×(N −1) complex matrices of the form Z = (zj zk ) = zz∗ ,

z ∈ S 2N−3

j, k = 1, . . . , N − 1.

(1.28)

1.1 Planar Many-Body Systems

9

We first note that Z is invariant under the U (1) action (eit zj eit zk = zj zk ). Clearly, the map [z] → Z is surjective. Conversely, if Z = (zj zk ) = (wj w k ), then a manipulation so that [z] = [w], where  provides z = λw, |λ| = 1 (λ = (w|z)), ˙ N−1 . This implies that the map (w|z) = N−1 j =1 w j zj denotes the inner product on C [z] → Z is injective. Lemma 1.1.1 The matrix Z = (zj zk ) with z ∈ S 2N−3 ⊂ CN−1 is a projection of rank one, 1. Z ∗ = Z,

2. Z 2 = Z,

3. trZ = 1.

(1.29)

Conversely, for an (N − 1) × (N − 1) matrix Z satisfying the above properties, there exists a z ∈ S 2N−3 such that Z = (zj zk ). The complex projective space CP N−2 is now redefined to be the set of matrices satisfying the above properties. Proof It is trivial thatZ satisfies the property 1. For the proof of the property 2, one hasonly to use |zj |2 = 1. For the proof of the property 3, we note that 2 trZ = |zj | = 1. Conversely, suppose that Z has the properties 1, 2, 3. Then, the property 2 implies that the eigenvalues of Z are 1 and 0. On account of the properties 1 and 3, there exists a unitary matrix U such that U −1 ZU = diag(1, 0, . . . , 0). Let (z1 , . . . , zN−1 )T denote the first column vector of U . Then, the above equation is written as Z = (zj zk ) with |zk |2 = 1. This ends the proof. ˙ Let Uk be open In concluding this section, we introduce local coordinates of Q. ˙ N−1 defined to be subsets of C ˙ N−1 ; zk = 0}, Uk = {z = (z1 , z2 , · · · , zN−1 ) ∈ C

k = 1, 2, . . . , N − 1.

In the following, taking k = N − 1, we deal with UN−1 . We introduce local coordinates (θ, r, wa ), a = 1, 2, . . . , N − 2, in UN−1 ⊂ Q˙ through za = reiθ ρwa ,

a = 1, 2, . . . , N − 2,

zN−1 = reiθ ρ,

(1.30)

where r2 =

N−1  k=1

|zk |2 ,

wa =

za zN−1

,

ρ −2 = 1 +

N−2  a=1

|wa |2 .

(1.31)

10

1 Geometry of Many-Body Systems

The U (1) action is described in these coordinates as (θ, r, wa ) −→ (θ + t, r, wa ),

(1.32)

which means that (r, wa ) are invariant under the U (1) action, so that they serve as local coordinates of π(UN−1 ), which describe the shape of a molecule, and the variable θ describes the attitude of the molecule. In terms of (r, wa ), the matrix elements of Z are expressed, independently of θ , as za zb =

r 2 wa w b 1+

N−2 

,

r 2 wa

za zN−1 =

|wa |2

1+

a=1

N−2 

,

etc.

|wa |2

a=1

1.2 Rotation and Vibration of Planar Many-Body Systems We give rigorous definitions of rotational and vibrational vectors for planar manybody systems and then proceed to prove that vibrational motions give rise to rotations. Needless to say, we have to distinguish rotations (resp. vibrations) from rotational (resp. vibrational) vectors. We start with the definition of the tangent space. The tangent space to Q˙ at x ∈ Q˙ is defined to be ˙ = {(u1 , . . . , uN ); uα ∈ R2 , Tx (Q)

N 

mα uα = 0},

˙ x = (x 1 , · · · , x N ) ∈ Q,

α=1

(1.33) where each vector uα is viewed as a vector at x α (Fig. 1.4). Fig. 1.4 A tangent vector u = (u1 , u2 , u3 ) at x = (x 1 , x 2 , x 3 ) ∈ Q˙ in the case of a three-body system

u3

x3 x2 u2

u1 x1

1.2 Rotation and Vibration of Planar Many-Body Systems

11

The tangent space is endowed with a natural inner product. For tangent vectors ˙ the inner product of them is defined to be u, v ∈ Tx (Q), Kx (u, v) =

N 

mα uα · v α ,

u = (u1 , . . . , uN ), v = (v 1 , . . . , v N ),

(1.34)

α=1

˙ where the subscript x of Kx indicates that Kx is defined on the tangent space Tx (Q). We call Kx the mass-weighted metric (or metric, for short). Let tangent vectors u   and v be expressed as u = Uk fk and v = Vk fk , respectively. Then, the inner product of them is expressed, by the use of (1.13), as Kx (u, v) =

2(N−1) 

(1.35)

Uk V k .

k=1

We define a rotational vector to be an infinitesimal transformation of the SO(2) action. Since the SO(2) action on the frame fk is given in (1.20), the derivatives of g(t)fk with respect to t at t = 0 are evaluated as   d g(t)f2j −1  = f2j , dt t =0

  d g(t)f2j  = −f2j −1 , dt t =0

so that the rotational vector Fx at x ∈ Q˙ is found to be  N−1   d g(t)x  = (−q2j f2j −1 + q2j −1 f2j ), Fx := dt t =0

(1.36)

j =1

2(N−1) where the frame fk in the above equation is viewed as a frame at x = k=1 qk fk . According to the usual notation in differential geometry, the rotational vector field x → Fx is expressed as F =

N−1 

− q2j

j =1

∂ ∂q2j −1

+ q2j −1

∂  . ∂q2j

(1.37)

˙ 2(N−1) , F is defined through To be precise, for any smooth function f (x) on Q˙ ∼ =R  N−1   dq2j −1  dq2j ∂f ∂f  d f (g(t)x) (0) (0) = + dt dt ∂q2j −1 dt ∂q2j t =0 =

j =1 N−1  j =1

− q2j

∂f ∂f  + q2j −1 = (Ff )(x). ∂q2j −1 ∂q2j

Any rotational vector is a scalar multiple of Fx .

12

1 Geometry of Many-Body Systems

˙ is called a vibrational vector, if it is orthogonal to all A tangent vector u ∈ Tx (Q) ˙ Let u ∈ Tx (Q) ˙ have the components (Uk ) with respect rotational vectors in Tx (Q). to the frame {fj }. Then, on account of (1.36) and (1.35), the u is a vibrational vector, if and only if Kx (u, Fx ) =

N−1 

(−U2j −1 q2j + U2j q2j −1 ) = 0.

(1.38)

j =1

Let us introduce complex variables by ζj = U2j −1 + iU2j ,

i=

√ −1.

(1.39)

Then, Eq. (1.38) is rewritten as N−1 

(−U2j −1 q2j + U2j q2j −1 ) =

j =1

N−1 1  (ζj zj − ζ j zj ) = 0, 2i

(1.40)

j =1

˙ is decomposed which will be used in the next section. Thus, the tangent space Tx (Q) into the direct sum of the rotational and the vibrational subspaces: ˙ = Vx,rot ⊕ Vx,vib. Tx (Q)

(1.41)

Now that vibrational vectors are defined, we are in a position to deal with vibrational motions or vibrational curves. A curve x(t) in Q˙ is called a vibrational curve, if its tangent vector x(t) ˙ is always a vibrational vector at x(t). From (1.38), 2(N−1) it follows that in the Cartesian coordinates, a curve x(t) = k=1 qk (t)fk is a vibrational curve, if and only if N−1  j =1

− q2j

q2j −1 dq2j −1  + q2j = 0, dt dt

(1.42)

which means that the angular momentum of the planar N-body system vanishes.

1.3 Vibrations Induce Rotations in Two Dimensions ˙ Suppose we are given a closed curve C : ξ(t), 0 ≤ t ≤ L, in the shape space M. ˙ there exists a point x0 ∈ Q˙ such that π(x0 ) = ξ(0). Then, there For ξ(0) ∈ M, exists a vibrational curve C ∗ : x(t), 0 ≤ t ≤ L, starting at x0 = x(0) and covering C, π(C ∗ ) = C. In fact, the fundamental theory of ordinary differential equations ensures the existence of C ∗ as a solution to (1.42) or to (1.45), as will soon be seen. Though C is a closed curve by definition (ξ(0) = ξ(L)), the vibrational curve C ∗

1.3 Vibrations Induce Rotations in Two Dimensions

13

x(t)

x(L) x(0)

............................................................................................. .............. .......... .......... ....... ......... ...... ........ . .... . . . . . ... ..... . . . . . . ... .. . ... ...... ............ . . . . . . . . .............. ......... . . . . .................. . . . . . . . . . . ................................... . .......................................... ...............

Q˙

C∗ π

ξ(t)

ξ(0) = ξ(L)

.............................................................................................................. ............ ................... ....... ........ .... .... .. ..... . ....... ... .............. ........ ............................... .................................................................................................

M˙

C

Fig. 1.5 A vibrational motion gives rise to a rotation

is not necessarily closed, x(0) = x(L). By the definition of C ∗ , one has π(x(0)) = ξ(0) = ξ(L) = π(x(L)), which implies that x(0) and x(L) have the same shape as systems of many bodies. However, the conceivable fact that x(0) = x(L) means that x(0) and x(L) take different attitudes in the center-of-mass system. Since x(L) and x(0) are equivalent, [x(L)] = [x(0)], there exists a g ∈ SO(2) such that x(L) = gx(0),

g ∈ SO(2).

(1.43)

Put another way, the vibrational motion x(t) gives rise to the rotation g ∈ SO(2) after a cycle of the shape deformation ξ(t), 0 ≤ t ≤ L (Fig. 1.5). In what follows, we will evaluate rotation angles by using complex local coordinates. Let C be a closed curve in the shape space M˙ and C ∗ : z(t) ∈ ˙ N−1 its horizontal lift (i.e., a vibrational curve with π(C ∗ ) = C). Then, Q˙ ∼ = C from (1.40) with z˙ j = ζj , it follows that z(t) should be subject to N−1  j =1

dzj dzj  zj = 0. zj − dt dt

(1.44)

In order to evaluate the rotation angle, we adopt the local coordinates introduced in (1.30) with the assumption that C ⊂ π(UN−1 ). Let C ∗ be expressed as r = r(t), θ = θ (t), wa = wa (t), a = 1, . . . , N − 2, in terms of the local coordinates (r, θ, wa ). Then, on setting za (t) = r(t)eiθ(t ) ρ(t)wa (t), zN−1 (t) = r(t)eiθ(t )ρ(t), Eq. (1.44) is rewritten in terms of (θ, r, wa ) as N−2 

wa

i dθ + dt 2

a=1

dw a dwa  − wa dt dt

1+

N−2  a=1

|wa |

2

= 0.

(1.45)

14

1 Geometry of Many-Body Systems

We note here that the above equation does not include the term dr/dt, which means that r(t) does not contribute a change in the rotation angle. Hence, we may restrict r(t) to a constant value, r(t) = r0 , in the closed curve C. Equation (1.45) is easy to integrate. The gain of the rotation angle along the vibrational curve C ∗ is given by N−2 



L dθ

θ (L) − θ (0) = 0

dt

dt = −

i 2



L

wa

a=1

0

dwa dwa  − wa dt dt

1+

N−2 

dt. |wa |

(1.46)

2

a=1

The right-hand side of the above equation is a contour integral along the closed curve C. If we take a variety of closed curves, we can obtain any value of rotation angles. For example, if we take C:

wa (t) = ca eit , ca = const.,

0 ≤ t ≤ 2π,

then we have the rotation angle 2π θ (2π) − θ (0) = − 1+

N−2  a=1 N−2 

|ca |2 .

(1.47)

|ca |2

a=1

If we reverse the orientation of the curve, the rotation angle given above is reversed in sign. Then, if we choose an orientation of C and the numbers ca properly, we can obtain any rotation angles from −π to π. Proposition 1.3.1 The planar many-body system can realize any rotation angle by a suitable vibrational motion. Put another way, for any g ∈ SO(2), there exists a vibrational curve x(t) satisfying (1.43). In the rest of this section, we discuss connection and curvature (see Appendix 5.2 for definition, but the following is comprehensible without referring to it). The ˙ is decomposed into the direct sum of the rotational and the tangent space Tx (Q) vibrational subspaces (see Eq. (1.41)). This decomposition can be described in terms of differential forms. There exists a unique one-form ω satisfying ω(F ) = 1,

ω(v) = 0

for ∀v ∈ Vx,vib,

(1.48)

1.3 Vibrations Induce Rotations in Two Dimensions

15

where F is the rotational vector given in (1.37). The one-form ω is called a connection form. In terms of (q1 , · · · , q2(N−1) ) or (z1 , · · · , zN−1 ), the ω is expressed as ⎛ ω=⎝

⎞−1

2(N−1) 

⎛

(q )2 ⎠

=1

= i ⎝2

N−1 

N−1 

⎞−1 |zj |2 ⎠

(q2j −1 dq2j − q2j dq2j −1 )

j =1 N−1 

j =1

(1.49) (zj dzj − zj dzj ).

j =1

Further, in terms of (θ, r, wa ) given in (1.30), the ω is written as N−2 

i ω = dθ + 2

(wa dwa − wa dwa )

a=1

1+

N−2 

(1.50)

. |wa |2

a=1

For the connection form ω, the curvature form  is defined to be  = dω. A straightforward calculation with (1.49) provides ⎛

N−1 

=i⎝

⎞−2 ⎛

N−1 

|zj |2 ⎠

⎝

j =1

|zj |2

j =1

N−1 

dzk ∧ dzk −

k=1

N−1 

⎞ zk zj dzk ∧ dzj ⎠ .

j,k=1

(1.51) From (1.50), the  proves to take the form

1+ =i

N−2  b=1

|wb |

2

 N−2 

dwa ∧ dwa −

a=1

1+

N−2 

N−2  a,b=1 2

w a wb dwa ∧ dwb .

(1.52)

|wa |2

a=1

We note that the expression of  shows that  is independent of θ . Then,  is ˙ In conclusion, we remark that viewed as a two-form defined on π(UN−1 ) ⊂ M. by virtue of the Stokes theorem, the rotation angle produced by a vibrational curve  covering a closed curve C in π(UN−1 ) ⊂ M˙ is exactly equal to − S , where S is ˙ The rotation angle obtained in (1.47) a surface bounded by the closed curve C in M. is also interpreted in this manner.

16

1 Geometry of Many-Body Systems

1.4 Planar Three-Body Systems In order to gain a better understanding of rotation, vibration, connection, and curvature, we work with the planar three-body system. In view of Proposition 1.1.2, we start with a review of CP 1 . Lemma 1.1.1 with N = 3 provides a realization of CP 1 in the matrix form,  CP ∼ = 1

Z=

  |z1 |2 z1 z2  2 2  |z1 | + |z2 | = 1 . z2 z1 |z2 |2 

(1.53)

We introduce here the real variables (ξ1 , ξ2 , ξ3 ) through ξ1 + iξ2 = 2z1 z2 ,

ξ3 = |z1 |2 − |z2 |2 .

(1.54)

Then, a straightforward calculation provides ξ12 + ξ22 + ξ32 = (|z1 |2 + |z2 |2 )2 = 1,

(1.55)

which implies that Eq. (1.54) defines a mapping, Z → (ξ1 , ξ2 , ξ3 ), from CP 1 to the unit sphere S 2 = {(ξ1 , ξ2 , ξ3 ) ∈ R3 | ξ12 + ξ22 + ξ32 = 1}. 

|z1 |2 z1 z2 z2 z1 |z2 |2



This mapping is injective. In fact, suppose that both Z = and W =   |w1 |2 w1 w2 are mapped to (ξ1 , ξ2 , ξ3 ). Then, the equations z1 z2 = w1 w2 and w2 w 1 |w2 |2 |z1 |2 − |z2 |2 = |w1 |2 − |w2 |2 imply that there exists λ ∈ C such that z1 = λw1 , z2 = λw2 and |λ| = 1, which results in Z = W . Conversely, for any (ξ2 , ξ2 , ξ3 ) ∈ S 2 , one defines Z to be 1 Z= 2



1 + ξ3 ξ1 − iξ2 ξ1 + iξ2 1 − ξ3

 ,

ξ12 + ξ22 + ξ32 = 1.

Then, it is easy to verify that Z ∗ = Z, Z 2 = Z, trZ = 1 by a simple calculation, which means that Z ∈ CP 1 , and thereby the mapping Z → (ξ1 , ξ2 , ξ3 ) proves to be surjective. Thus, we verify that CP 1 ∼ = S 2 , and therefore, Proposition 1.1.2 with N = 3 is rewritten as: Proposition 1.4.1 Except for the configuration where three bodies collide at the center-of-mass, the shape space for the planar three-body system is diffeomorphic with R3 − {0}, M˙ ∼ = R+ × CP 1 ∼ = R+ × S 2 ∼ = R3 − {0}.

(1.56)

1.4 Planar Three-Body Systems

17

It then turns out that for the three-body system, the projection π defined in (1.23) is put in the form ˙ 4 := R4 − {0} −→ R ˙ 3 := R3 − {0}. π: R

(1.57)

We study the map π in more detail. Let (z1 , z2 ) denote the complex coordinates of C2 and qk , k = 1, . . . , 4, the Cartesian coordinates of R4 ∼ = C2 as before, i.e., z1 = q1 + iq2, z2 = q3 + iq4. Let (ξ1 , ξ2 , ξ3 ) defined in (1.54) be viewed as the Cartesian coordinates of R3 . Under the condition |z1 |2 + |z2 |2 = 1, the map (z1 , z2 ) → Z defines a map S 3 → CP 1 , and further, the map Z → (ξ1 , ξ2 , ξ3 ) also defines the diffeomorphism CP 1 → S 2 . Accordingly, under the condition |z1 |2 + |z2 |2 = 1, the composition (z1 , z2 ) → (ξ1 , ξ2 , ξ3 ) determines the map S 3 → S 2 , which is well ˙4 ∼ ˙3 ∼ known as the Hopf map [12]. Since R = R+ × S 3 and since R = R+ × S 2 , the map (z1 , z2 ) → (ξ1 , ξ2 , ξ3 ) without reference to the condition |z1 |2 + |z2 |2 = 1 is ˙ 4 −→ R ˙ 3 . Thus, we obtain the explicit expression of π viewed as giving a map R given in (1.57) as

˙ 3; ˙ 4 −→ R π: R

⎧ ξ1 = 2Re(z1 z2 ) = 2(q1q3 + q2 q4 ), ⎪ ⎪ ⎨ ξ2 = 2Im(z1 z2 ) = 2(q1q4 − q2 q3 ), ⎪ ⎪ ⎩ ξ3 = |z1 |2 − |z2 |2 = q12 + q22 − q32 − q42 .

(1.58)

We proceed to rotational and vibrational vectors for the planar three-body system. ˙ has been decomposed into the direct sum of Vx,rot and The tangent space Tx (Q) Vx,vib (see Eq. (1.41)). For the three-body system, those subspaces are explicitly described. By using the orthonormal frame {fj }, we define the tangent vectors, ˙ 4 to be F, vk , k = 1, 2, 3, at each point of Q˙ ∼ =R F = −q2 f1 + q1 f2 − q4 f3 + q3 f4 , v1 = q3 f1 + q4 f2 + q1 f3 + q2 f4 , v2 = q4 f1 − q3 f2 − q2 f3 + q1 f4 ,

(1.59)

v3 = q1 f1 + q2 f2 − q3 f3 − q4 f4 , respectively, where F is obtained from (1.36) with N = 3, and where we have ˙ A straightforward calculation shows that these vector omitted the subscript x ∈ Q. fields are orthogonal to one another, which implies that Vx,rot = span{F },

Vx,vib = span{v1 , v2 , v3 }.

(1.60)

18

1 Geometry of Many-Body Systems

It is to be noted that the existence of mutually orthogonal vector fields on the whole ˙ is special to the planar three-body system. In general or in the case of N > 3, of Q mutually orthogonal vector fields exist only locally. So far we have defined rotational and vibrational vectors in a rather abstract way. We now show that the tangent vectors given in (1.59) indeed give rotational and vibrational vectors in a visual manner. To this end, we first recall that the orthonormal system {fj } on Q˙ for the three-body system is given, from (1.12), by f1 = N1 (−m2 e1 , m1 e1 , 0), f2 = N1 (−m2 e2 , m1 e2 , 0), f3 = N2 (−m3 e1 , −m3 e1 , (m1 + m2 )e1 ), f4 = N2 (−m3 e2 , −m3 e2 , (m1 + m2 )e2 ). Now we restrict x = (r 1 , r 2 ) ∈ Q˙ to r 1 = ae1 , r 2 = be2 for notational simplicity. Then, by the definition (1.17) of r 1 , r 2 , the Cartesian coordinates take the values q1 = a, q2 = 0, q3 = 0, q4 = b, so that Eq. (1.59) reduces to F = af2 − bf3 ,

v1 = bf2 + af3 ,

v2 = bf1 + af4 ,

v3 = af1 − bf4 .

These vectors are illustrated in Fig. 1.6, which can visually explain rotational and vibrational modes of the planar three bodies at the position (r 1 , r 2 ) = (ae1 , be2 ). We proceed to the connection form. For the planar three-body system, the connection form defined in (1.49) takes the form ω = r −2 (−q2 dq1 + q1 dq2 − q4 dq3 + q3 dq4 ),

r2 =

4 

(qk )2 .

(1.61)

k=1

v3

v3 v2

v1 F

v3

v2

v1 v1

v1

v3

v2 v2

v2

v1 v3

Fig. 1.6 Rotational and vibrational vectors at the configuration (r 1 , r 2 ) = (ae1 , be 2 )

1.4 Planar Three-Body Systems

19

˙ 4 and calculate Here we view the variables (ξk ) given in (1.58) as functions on R their differentials to obtain dξ1 = 2(q3 dq1 + q4 dq2 + q1 dq3 + q2 dq4 ), dξ2 = 2(q4 dq1 − q3 dq2 − q2 dq3 + q1 dq4 ),

(1.62)

dξ3 = 2(q1 dq1 + q2 dq2 − q3 dq3 − q4 dq4 ). We note that the one-forms ω, dξk , k = 1, 2, 3, form a basis of the space of one˙ 3 , we need to use ˙ If we consider (ξk ) as the Cartesian coordinates of R forms on Q. ∗ the notation π dξ , etc., for the left-hand sides of the above equations, where π ∗ ˙4 → R ˙ 3 . However, we have omitted the is the symbol for the pull-back of π : R ∗ symbol π for simplicity. In a dual manner to ω, dξk , the tangent vectors F, vk given in (1.59) provide a basis of the space of tangent vector fields, if they are, respectively, expressed as ∂ ∂ ∂ ∂ + q1 − q4 + q3 , ∂q1 ∂q2 ∂q3 ∂q4 ∂ ∂ ∂ ∂ + q4 + q1 + q2 , v1 = q3 ∂q1 ∂q2 ∂q3 ∂q4 ∂ ∂ ∂ ∂ v2 = q4 − q3 − q2 + q1 , ∂q1 ∂q2 ∂q3 ∂q4 ∂ ∂ ∂ ∂ + q2 − q3 − q4 . v3 = q1 ∂q1 ∂q2 ∂q3 ∂q4 F = −q2

(1.63)

It is easy to verify that ω and dξk /(2r 2 ) are one-forms dual to the vector fields given in (1.63): ω(F ) = 1,

ω(vk ) = 0,

dξj (F ) = 0,

1 dξj (vk ) = δj k , 2r 2

j, k = 1, 2, 3. (1.64)

˙ 4 , the canonical Since these one-forms form a basis of the space of one-forms on R basis dqk , k = 1, . . . , 4, is expressed in terms of ω, dξk as 1 (q3 dξ1 + q4 dξ2 + q1 dξ3 ), 2r 2 1 dq2 = q1 ω + 2 (q4 dξ1 − q3 dξ2 + q2 dξ3 ), 2r 1 dq3 = −q4 ω + 2 (q1 dξ1 − q2 dξ2 − q3 dξ3 ), 2r 1 dq4 = q3 ω + 2 (q2 dξ1 + q1 dξ2 − q4 dξ3 ). 2r dq1 = −q2 ω +

(1.65)

20

1 Geometry of Many-Body Systems

We proceed to the curvature form  = dω. By differentiation of the connection ω given in (1.61), the  is written out as  = 2r −4 [((q3 )2 + (q4 )2 )dq1 ∧ dq2 + ((q1 )2 + (q2 )2 )dq3 ∧ dq4 − (q2 q3 − q1 q4 )(dq1 ∧ dq3 + dq2 ∧ dq4)

(1.66)

− (q1 q3 + q2 q4 )(dq1 ∧ dq4 − dq2 ∧ dq3)]. By the use of (1.65), the  is rewritten as =

1 ξ1 dξ2 ∧ dξ3 + ξ2 dξ3 ∧ dξ1 + ξ3 dξ1 ∧ dξ2 . 2 (ξ12 + ξ22 + ξ32 )3/2

(1.67)

Though we have chosen an elementary way to obtain the curvature form shown above, we can also obtain it in a smart manner. In view of (1.63), we can verify that π∗ vk = 2r 2

∂ , ∂ξk

k = 1, 2, 3.

(1.68)

˙ 3 , the defining equation of π∗ vk is In fact, for an arbitrary smooth function f on R expressed and arranged as (π∗ vk )f = vk (f ◦ π) =

4  3 

j

vk

j =1 k=1

∂ξk ∂f ∂f = 2r 2 , ∂qj ∂ξk ∂ξk

 j j vk ∂/∂qj . In other words, Eq. (1.68) means where vk are components of vk = that 2r12 vk are the horizontal lifts of ∂/∂ξk . We denote this fact by  ∂ ∗ 1 = 2 vk , ∂ξk 2r

k = 1, 2, 3.

(1.69)

Further, a straightforward calculation of the brackets among (∂/∂ξk )∗ results in ! ∂ ∗  ∂ ∗ "  ξk =− , εij k 6 F, ∂ξi ∂ξj 2r

i, j, k = 1, 2, 3.

(1.70)

k

Further, we apply the formula dω(X, Y ) = X(ω(Y )) − Y (ω(X)) − ω([X, Y ]) (see [75]) for X = (∂/∂ξi )∗ and Y = (∂/∂ξj )∗ to obtain dω

  1  ∂ ∗  ∂ ∗ = 6 , εij k ξk , ∂ξi ∂ξj 2r k

which gives the same result as (1.67).

1.4 Planar Three-Body Systems

21

As was mentioned above, the pull-back symbol π ∗ is missing in the description of dξk in (1.62). Accordingly, in the right-hand side of (1.67), the pull-back symbol π ∗ is also missing. However, Eq. (1.67) shows that the  can be viewed as a two˙ 3 , since  is invariant under the action of form defined on the shape space M˙ ∼ =R ˙ 3 . This SO(2). In what follows, we consider  as defined on the shape space M˙ ∼ =R curvature has an interesting feature from a physical point of view. Except for the factor 12 , the  corresponds to the magnetic flux of the Dirac monopole; B = ξ /|ξ |3 . In other words, the shape space of the planar three-body system is endowed with a monopole field due to rotation, while this does not mean the existence of the monopole as a particle [26]. We will express the connection and the curvature in other local coordinates. We ˙ 2 through introduce the coordinates (r, θ, φ, ψ) in C z1 = rei

−φ+ψ 2

θ cos , 2

z2 = rei

φ+ψ 2

θ sin . 2

(1.71)

The range of each variable will be discussed later, but for now it needs to be seen that (θ, φ, ψ) serve as coordinates on S 3 . The connection form (1.49) with N = 3 is expressed, in terms of (1.71), as ω=

1 (dψ − cos θ dφ), 2

(1.72)

which shows that the present ω is independent of r, so that it can be viewed as a ˙ 4 . We proceed to the curvature form to be expressed in one-form defined on S 3 ⊂ R the coordinates (θ, φ, ψ). Differentiation of ω results in  = dω =

1 sin θ dθ ∧ dφ. 2

(1.73)

This form can be viewed as 12 times the volume element of S 2 expressed in spherical coordinates. In fact, from (1.54) and (1.71), we obtain ξ1 = r 2 sin θ cos φ,

ξ2 = r 2 sin θ sin φ,

ξ3 = r 2 cos θ,

(1.74)

which implies that (r 2 , θ, φ) can be viewed as polar spherical coordinates of R3 . We now return to the question about the ranges of the angular variables given in (1.71). We define the ranges of r, θ, φ, ψ to be r > 0,

0 ≤ θ ≤ π,

0 ≤ φ ≤ 2π,

0 ≤ ψ ≤ 4π,

(1.75)

˙ 2 through (1.71). In this coordinate system, the action of eit ∈ U (1) which cover C (see (1.24) with N = 3) is expressed as (r, θ, φ, ψ) −→ (r, θ, φ, ψ + 2t).

22

1 Geometry of Many-Body Systems

This implies that the ψ is a coordinate describing the attitude of the planar three bodies, and the r, θ, φ are those for the shape, where the ψ ranges over R/4πZ. A further remark is in order. The coordinate system (1.75) fails to work when θ = 0, π. In fact, if θ = 0, one has z2 = 0, so that for any α ∈ R, the pairs (φ, ψ) and (φ +α, ψ −α) correspond to the same z1 . Further, if θ = π, then z1 = 0, and hence, for any α ∈ R, (φ, ψ) and (φ + α, ψ + α) correspond to the same z2 . In spite of this fact, the expression (1.72) of ω is valid, since it has the same value for (φ, ψ) and (φ + α, ψ ± α). We will discuss local expressions of ω in more detail. Let U1 and U2 denote ˙ 2 which are defined by z1 = 0 and z2 = 0, respectively. Then, open subsets of C (z1 , z2 ) ∈ U1 (resp. U2 ) if and only if θ = π (resp. θ = 0). Further, let D+ and D− denote the open subsets of S 2 defined to be D+ = S 2 − {S} and D− = S 2 − {N}, where S and N stand for the south and the north poles, respectively. The projection ˙4 → R ˙ 3 maps U1 and U2 to π: R π(U1 ) ∼ = R+ × D+ ,

π(U2 ) ∼ = R+ × D− ,

respectively, where R+ denotes the set of positive real numbers. Let σ1 : π(U1 ) → U1 and σ2 : π(U2 ) → U2 denote the local sections defined to be σ1 : (r 2 , θ, φ) → (r 2 , θ, φ, φ) and σ2 : (r 2 , θ, φ) → (r 2 , θ, φ, π − φ), respectively, or by ψ = φ and by ψ = π − φ, respectively. Then, Eq. (1.72) reduces to ω+ =

1 (1 − cos θ )dφ, 2

ω− =

1 (−1 − cos θ )dφ, 2

(1.76)

respectively, where ω+ = σ1∗ ω and ω− = σ2∗ ω denote the value of ω at σ1 (p) (p ∈ π(U1 )) and σ2 (p) (p ∈ π(U2 )), respectively. The connection forms ω± exactly correspond to the locally-defined vector potentials A± for the Dirac monopole B = ( 12 )ξ /|ξ |3 , respectively, A+ =

−ξ2 e1 + ξ1 e2 , 2ρ(ρ + ξ3 )

A− =

ξ2 e1 − ξ1 e2 , 2ρ(ρ − ξ3 )

(1.77)

where ρ = r 2 and where A± are defined under the conditions ρ(ρ ± ξ3 ) = 0, respectively. In fact, we can verify that A± ·dξ = ω± by straightforward calculation (see also (5.31) for a local description of the connection form). In the rest of this section, we discuss the metric defined on the shape space for the planar three-body system. Since (qk ) are the Cartesian coordinates with respect to the orthonormal system for the center-of-mass system Q, the canonical (massweighted) metric on Q is defined by (1.35) and arranged, by using (1.65), as ds 2 =

4  1 (dξ12 + dξ22 + dξ32 ), (dqj )2 = ρω2 + 4ρ j =1

ρ = |ξ | = r 2 ,

(1.78)

1.5 The Rotation Group SO(3)

23

which can be also obtained from Kx (F, F ) = r 2 , Kx (F, vi ) = 0, Kx (vi , vj ) = r 2 δij and from ω(F ) = 1, dξk (vj ) = 2r 2 δkj . The first term of the right-hand side of (1.78) takes non-zero values for vectors in Vx,rot, but vanishes for all vectors in Vx,vib. The second term vanishes for all the vectors in Vx,rot, but takes nonzero values for vectors in Vx,vib. Hence, the second term, which is SO(2)-invariant, ˙ 3 to endow R ˙ 3 with the metric naturally projects onto the shape space R dξ12 + dξ22 + dξ32 # . 4 ξ12 + ξ22 + ξ32

(1.79)

Mechanics of the planar three-body system will be discussed in Appendix 5.8.

1.5 The Rotation Group SO(3) In order to study rotation and vibration of many-body systems in three dimensions, we need to give a review of the rotation group SO(3) and its Lie algebra so(3). If we put X ∈ SO(3) in the form X = (x 1 , x 2 , x 3 ), where x j are column vectors in R3 , the conditions XT X = I with I the 3 × 3 identity and det X = 1 are rewritten as xj · xk = δj k and det(x 1 , x 2 , x 2 ) = 1, which imply that {x j } form a positivelyoriented orthonormal system in R3 . We are interested in how the set of all positively-oriented orthonormal systems on R3 is realized. Let x j , j = 1, 2, 3, be a positively-oriented orthonormal system. If we move the head of vector x 1 freely within the constraint |x 1 | = 1, it draws the unit sphere S 2 . We now parallel translate x 2 in R3 so that its tail may link to the head of x 1 . Then, the translated x 2 becomes a tangent vector to S 2 . If we fix x 1 arbitrarily and move freely the tangent vector x 2 within the restrictions x 1 · x 2 = 0, |x 2 | = 1, the head of x 2 draws the unit circle S 1 . If x 1 and x 2 are fixed arbitrarily, the vector x 3 is determined by x 3 = x 1 × x 2 , which means that x 3 is immobile. It then turns out that the set of all the positively-oriented orthonormal systems is realized as the set of all the unit tangent vectors to the unit sphere. This set is called the unit tangent bundle over S 2 and denoted by T1 (S 2 ) (Fig. 1.7): SO(3) ∼ = T1 (S 2 ) = {(x, y) ∈ R3 × R3 ; |x| = |y| = 1, x · y = 0}.

(1.80)

We turn to the Lie algebra so(3) of SO(3). Let A ∈ R3×3 . By definition, A is in so(3) if and only if et A ∈ SO(3) for t ∈ R. It then turns out that so(3) = {A ∈ R3×3 ; A + AT = 0}.

(1.81)

24

1 Geometry of Many-Body Systems

y

Fig. 1.7 A view of the unit tangent bundle over S 2

x

The so(3) is a real three-dimensional vector space, whose canonical basis is given by ⎛ ⎞ 00 0 L1 = ⎝0 0 −1⎠ , 01 0

⎛

0 0 L2 = ⎝ 0 0 −1 0

⎞ 1 0⎠ , 0

⎛

⎞ 0 −1 0 L3 = ⎝1 0 0⎠ . 0 0 0

(1.82)

 aj Lj , aj ∈ R. The isomorphism of  3 3 R with so(3), which is denoted by R : R → so(3), is given, for a = aj e j ∈ Then, any A ∈ so(3) is expressed as A =

R3 with {ej } being the standard basis of R3 , by ⎛

⎞ 0 −a3 a2 R(a) = aj R(ej ) = aj Lj = ⎝ a3 0 −a1 ⎠ . j =1 j =1 −a2 a1 0 3 

3 

(1.83)

Proposition 1.5.1 For a, b ∈ R3 and g ∈ SO(3), the following formulae hold: 1. R(a)x = a × x for x ∈ R3 , 2. Adg R(a) := gR(a)g −1 = R(ga), 3. R(a × b) = −abT + ba T . Proof The proof of the item 1 is clear from the definition of the vector product. To prove the item 2, we use the formula g(a × b) = ga × gb

for g ∈ SO(3),

(1.84)

which means that the SO(3) action preserves the vector product. This formula can be verified by linearly extending the easy-to-prove formula g(ei × ej ) = gek = gei × gej , where (i, j, k) is a cyclic permutation of (1, 2, 3). Then, we find that gR(a)g −1 x = g(a × g −1 x) = ga × x = R(ga)x for all x. The item 3 is a consequence of the formula a × (b × c) = (a · c)b − (a · b)c.

(1.85)

1.5 The Rotation Group SO(3)

25

In fact, by using this formula, we find that R(a × b)x = (−abT + ba T )x. This ends the proof. The so(3) is endowed with the inner product through A, B =

1 tr(AT B), 2

A, B ∈ so(3).

(1.86)

Proposition 1.5.2 With respect to the inner product (1.86), the Lj , j = 1, 2, 3, form an orthonormal system of so(3), and further so(3) and R3 are isometric: 1. Lj , Lk  = δj k , j, k = 1, 2, 3, 2. R(a), R(b) = a · b for a, b ∈ R3 . The proof is easily performed by using the definition of the inner product. Proposition 1.5.3 As Lie algebras, so(3) ∼ = R3 have the following properties: 1. [Lj , Lk ] =

3  =1

εj k L ,

ej × ek =

3 

εj k e ,

=1 R3 ,

2. [R(a), R(b)] = R(a × b), a, b ∈ 3. R(a), [R(b), R(c)] = a · (b × c), a, b, c ∈ R3 . Proof The item 1 is easy to prove by calculation. To prove the item 2, we use the formula (1.85). For any x ∈ R3 , we can show that [R(a), R(b)]x and R(a × b)x coincide by using the item 1 of Proposition 1.5.1 together with the formula (1.85). Note that since Li = R(ei ), the first equation of the item 1 is a consequence the item 2 and the second equation of the item 1. The item 3 is a consequence of the item 2 of the present proposition and the item 2 of Proposition 1.5.2. This ends the proof. ∞ t n n Applying the definition, exp(tA) = n=0 n! A , of matrix exponential, we obtain ⎛ ⎞ 1 0 0 exp(tR(e1 )) = ⎝0 cos t − sin t ⎠ , 0 sin t cos t ⎛ ⎞ cos t 0 sin t exp(tR(e2 )) = ⎝ 0 1 0 ⎠ , − sin t 0 cos t ⎛ ⎞ cos t − sin t 0 exp(tR(e3 )) = ⎝ sin t cos t 0⎠ . 0 0 1

(1.87)

26

1 Geometry of Many-Body Systems

We can obtain the same results by solving associated differential equations. For example, to find the matrix expression of exp(tR(e2 )), we have only to solve the linear differential equation dx/dt = R(e2 )x, x ∈ R3 . In fact, the coupled first-order differential equations reduce to the uncoupled second-order differential equations d 2 x3 /dt 2 = −x3 etc, which are easy to solve. We then find that x1 (t) = x3 (0) sin t + x1 (0) cos t, x2 (t) = x2 (0), x3 (t) = x3 (0) cos t − x1 (0) sin t. Putting these solutions in the form x(t) = M(t)x(0) and comparing it with the generic expression of the solution, x(t) = exp(tR(e2 ))x(0), we find that M(t) = exp(tR(e 2 )) on account of the uniqueness of solution. Clearly, exp(tR(e j )) describes a rotation about the ej -axis with j = 1, 2, 3. (Though the axis of the rotation is a line which is fixed under the rotation, we use the same word for a vector along the axis.) In fact, one has exp(tR(e j ))ej = ej , which means that the action of exp(tR(ej )) fixes the vector ej . In general, the following lemma is valid for rotation. Proposition 1.5.4 Let a be a unit vector (|a| = 1). Then, exp(tR(a)) is a rotation about the axis a. Conversely, any rotation about the axis a is expressed as exp(tR(a)). Proof It is easy to see that exp(tR(a)) ∈ O(3). Since det exp(tR(a)) = exp(ttr(R(a))) = 1, where the symbol exp in the right-hand side stands for the scalar exponential, we find that exp(tR(a)) ∈ SO(3). From R(a)a = a × a = 0, it follows that exp(tR(a))a = a. Conversely, to express a rotation about the axis a, we introduce mutual orthogonal unit vectors b and c which are also orthogonal to a in such a manner that the system b, c, a form a positively-oriented orthonormal system. Then, a rotation g about the axis a is put, with respect to the basis {b, c, a}, in the form gb = b cos t + c sin t, gc = −b sin t + c cos t, ga = a. These are also expressed as ⎛ cos t − sin t (gb, gc, ga) = (b, c, a) ⎝ sin t cos t 0 0

⎞ 0 0⎠ . 1

Here we denote by h the matrix whose column vectors are b, c, a. It is clear that h ∈ SO(3) and he3 = a. Then, the above equation takes the form gh = h exp(tR(e3 )), so that one has −1

g = het R(e3 ) h−1 = et hR(e3 )h

= et R(he3 ) = et R(a) ,

where use has been made of the formula 2 of Proposition 1.5.1. This ends the proof.

1.5 The Rotation Group SO(3)

27

We now describe the rotation et R(a) in an explicit form. We first note that = a, which means that a is on the axis of rotation. For any unit vector a, there exists a matrix g ∈ SO(3) such that ge3 = a. Then, et R(a) is arranged as et R(a) a

et R(a) = et R(ge3 ) = get R(e3 ) g −1 . We bring et R(e3 ) into the form ⎛

e

t R(e3 )

⎛ ⎛ ⎞⎞ 00 000 ⎝ ⎝ ⎝ ⎠ ⎠ = cos t I − 0 0 0 + sin t R(e3 ) + 0 0 00 001

⎞ 0 0⎠ . 1

Operating the both sides of the above equation with Adg and using the formula ⎛

⎞ 000 e3 eT3 = ⎝0 0 0⎠ , 001 we obtain   et R(a) = cos t I − ge3 (ge3 )T + sin tR(ge 3 ) + ge3 (ge3 )T = cos t (I − aa T ) + sin t R(a) + aa T ,

(1.88)

which is an explicit expression of the rotation about a unit vector a. We can prove further the following proposition on the rotation axis. Proposition 1.5.5 Any rotation g ∈ SO(3) has a rotation axis. In particular, if g = I , then the rotation axis for g is unique. Proof Since det(g − I ) = det(g T − I ) = det(g T ) det(I − g) = − det(g − I ), we see that det(g − I ) = 0. Then, there exists an eigenvector x = 0 associated with the eigenvalue 1; gx = x, which means that the line containing x is a rotation axis. Suppose that there are two axes for g, i.e., there exist mutually independent x and y such that gx = x, gy = y. Then, it follows that g(x × y) = gx × gy = x × y. This implies that g leaves invariant three mutually independent vectors x, y, x × y of R3 , so that it leaves all the vectors of R3 invariant, which means that g = I . Therefore, the rotation axis should be unique, if g = I . This ends the proof.

28

1 Geometry of Many-Body Systems

So far we have shown that any rotation has a rotation axis and that a rotation about the axis a is put in the form exp(tR(a)) with |a| = 1. Because of the periodicity of rotation, the totality of rotations forms the set B 3 = {x ∈ R3 ; |x| ≤ π}. In addition, on account of (1.88), every pair of antipodal points on the boundary of B 3 should be identified. The B 3 with the present identification, which we denote by Bˆ 3 , can be shown to be isomorphic with the real projective space RP 3 , where RP 3 is defined to be the factor space of S 3 by identification of pairs of antipodal points. Let S 3 (π) denote the three-sphere of radius π, which is realized in R4 by the condition |y| = π, y ∈ R4 . The real projective space RP 3 is realized through the identification of the antipodal points, y and −y, of S 3 (π). Put another way, RP 3 is the factor space, S 3 (π)/Z2 , by the action of the group Z2 = {±1}. We denote by [y] the equivalence class of y ∈ S 3 (π), so that [−y] = [y]. For [y] ∈ RP 3 with y4 = 0, either y or −y has a negative fourth coordinate. We assume that y has y4 < 0, without loss of generality. We draw a line segment joining the north pole (0, 0, 0, π) ∈ S 3 (π) and the y ∈ S 3 (π) in question. Let (x1 , x2 , x3 , 0) be the point at which the line segment crosses the plane y4 = 0 in R4 . Then, we have xk =

πyk , π − y4

k = 1, 2, 3.

Let us denote these points by x ∈ R3 ⊂ R4 with |x| < π, which correspond to the interior points of B 3 . If y4 = 0, then |y| = π defines the two-sphere S 2 (π) of radius π, corresponding to the boundary of B 3 . Since [y] ∈ RP 3 and since S 2 (π) in question is on the equator of S 3 (π), antipodal points of S 2 (π) should be identified. Thus, we have constructed the one-to-one correspondence between RP 3 and Bˆ 3 . We can show that any g ∈ SO(3) is composed of rotations about only two axes. Proposition 1.5.6 Let ej , j = 1, 2, 3, be the standard basis of R3 . (We may take any orthonormal system for ej .) Any g ∈ SO(3) can be generated by rotations about chosen two axes, say e2 , e3 . Proof The vector e3 is transformed to ge3 by the action of g ∈ SO(3). We carry ge3 to a vector in the e3 –e1 plane by a rotation h3 about the e3 axis and further move the vector in question to e3 by a rotation about the e2 axis. This procedure provides h2 h3 ge3 = e3 . This implies that h2 h3 g is a rotation about the e3 axis, which we −1 denote by k3 , so that we have h2 h3 g = k3 . Hence, we obtain g = h−1 3 h2 k3 , as is wanted. In the course of the proof, we can restrict the rotation angle of h3 to −2π ≤ −φ ≤ 0 so that h3 ge3 may be sitting on the half of the e3 –e1 plane with +e1 . Then, the rotation angle of h2 about the e2 axis may take the range −π ≤ −θ ≤ 0. The rotation angle of k3 about the e3 axis has the range 0 ≤ ψ ≤ 2π of course. Thus we have obtained the following theorem.

1.5 The Rotation Group SO(3)

29

Theorem 1.5.1 For any g ∈ SO(3), there are three angles φ, ψ, θ such that g = eφR(e3 ) eθR(e2 ) eψR(e3 ) , 0 ≤ φ ≤ 2π,

0 ≤ θ ≤ π,

0 ≤ ψ ≤ 2π.

(1.89)

These angles are called the Euler angles. We should be careful about the ranges of the Euler angles. If θ = 0, π, the Euler angles are uniquely determined, but if θ = 0, π, they are not. To see this, we denote by g(φ, θ, ψ) the rotation with the Euler angles (φ, θ, ψ). If g(φ, θ, ψ) has other Euler angles (φ  , θ  , ψ  ), the equation g(φ, θ, ψ)e 3 = g(φ  , θ  , ψ  )e3 provides φ = φ  , θ = θ  if θ = 0, π. Further, the equation g(φ, θ, ψ) = g(φ, θ, ψ  ) results in ψ = ψ  . In contrast to this, if θ = 0, π, the easy-to-prove equations eαR(e3 ) e−αR(e3 ) = I and eαR(e3 ) eπR(e2 ) eαR(e3 ) = eπR(e2 ) with α ∈ R imply that g(φ +α, 0, ψ −α) = g(φ, 0, ψ) and g(φ +α, π, ψ +α) = g(φ, π, ψ), respectively. Further remarks are in order. As is stated in Proposition 1.5.6, the choice of the rotation axes is not unique, so that the Euler angles can be defined in various ways. One may choose e1 , e3 as rotation axes. In this case, the procedure to obtain the Euler angles is as follows: We carry the vector ge3 to a vector in the e2 –e3 plane by a rotation about the e3 axis to get h3 ge3 , and further translate it to e3 by a rotation −1 about the e1 axis to obtain h1 h3 ge3 = e3 . Then, we have g = h−1 3 h1 k3 . In this φ$ e θ$ e ψ$ e 3 1 3 case, Eq. (1.89) takes the form g = e e e , where$ ei = R(ei ). If the two axes e1 , e2 are chosen, the same reasoning results in g = eφ$e2 eθ$e1 eψ$e2 . Furthermore, we may take three vectors ej as rotation axes. A procedure for the Euler angles is as follows: For any g ∈ SO(3), we first take ge1 . By h2 ∈ SO(3), we rotate ge1 about e2 to put it on the plane e1 –e2 with +e1 . We then rotate the vector h2 ge1 by a rotation h3 about e3 to place it on e1 . Then, we have h3 h2 ge1 = e1 . This means that h3 h2 g is a rotation about e1 , which we denote by k1 . Thus, the g proves to expressed −1 as g = h−1 2 h3 k1 . If we denote by χa , a = 1, 2, 3, three angles associated with −1 −1 h2 , h3 , k1 , respectively, we obtain g = eχ2 R(e2 ) eχ3 R(e3 ) eχ1 R(e1 ) , π π 0 ≤ χ1 ≤ 2π, 0 ≤ χ2 ≤ 2π, − ≤ χ3 ≤ , 2 2

(1.90)

which will be used in the falling cat problem. We now explain a geometric interpretation of the Euler angles (φ, θ, ψ) given in (1.89) from the viewpoint of the identification SO(3) = T1 (S 2 ) given in (1.80). The angles (φ, θ ) of the Euler angles give rise to the spherical coordinates of S 2 . In fact, from |ge3 | = 1, we obtain S 2 = {ge3 ; g ∈ SO(3)},

30

1 Geometry of Many-Body Systems

and the vector ge3 is expressed as ge3 = e1 sin θ cos φ + e2 sin θ sin φ + e3 cos θ,

0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π.

To see a geometric interpretation of the angular variable ψ, we put the vector ge1 in the form ge 1 = u1 cos ψ + u2 sin ψ,

(1.91)

where u1 = eφR(e3 ) eθR(e2 ) e1 = e1 cos θ cos φ + e2 cos θ sin φ − e3 sin θ, u2 = eφR(e3 ) eθR(e2 ) e2 = −e1 sin φ + e2 cos φ. As is easily seen, the vectors u1 and u2 are mutually orthogonal unit vectors, which are perpendicular to ge3 . We parallel translate u1 and u2 in R3 so that their tails link to the head of ge3 . Then, the translated vectors are viewed as an orthonormal system on the tangent plane to S 2 . From this point of view, Eq. (1.91) implies that the variable ψ is an angle variable of the unit circle drawn by ge1 on the tangent plane to S 2 at ge3 with (φ, θ ) fixed. Put another way, ψ (0 ≤ ψ ≤ 2π) is an angle variable for the unit circle over the sphere S 2 . We remark here that the vectors x, y in the definition (1.80) of T1 (S 2 ) can be taken as ge3 and ge1 , respectively. We do not have to talk about ge2 , since we have ge3 × ge1 = ge2 . In what follows, we describe the left-invariant one-forms and right-invariant oneforms and their dual vector fields, which are typically used in mechanics for rigid bodies. For g ∈ SO(3) viewed as a matrix variable, the one-forms g −1 dg,

dgg −1

(1.92)

are called the left- and the right-invariant (one-)forms, respectively. In fact, for a constant matrix h ∈ SO(3), we verify that (hg)−1 d(hg) = g −1 dg,

d(gh)(gh)−1 = dgg −1 ,

which means that the forms g −1 dg and dgg −1 are invariant under the left and the right translations, g → hg and g → gh, respectively. Differentiating g −1 g = I , we obtain dg −1 g+g −1 dg = (g −1 dg)T +g −1 dg = 0, which implies that g −1 dg is antisymmetric. In a similar manner, we see that dgg −1 is anti-symmetric as well. We denote by a and  a the components of the right- and the left-invariant one-forms, respectively: dgg −1 =

3  a=1

a R(ea ),

g −1 dg =

3  a=1

 a R(ea ).

(1.93)

1.5 The Rotation Group SO(3)

31

A straightforward calculation in terms of the Euler angles can provide ⎧ 1  = − sin φdθ + sin θ cos φdψ, ⎪ ⎪ ⎪ ⎨ 2 = cos φdθ + sin θ sin φdψ, ⎪ ⎪ ⎪ ⎩ 3 = dφ + cos θ dψ, ⎧ 1  = sin ψdθ − sin θ cos ψdφ, ⎪ ⎪ ⎨  2 = cos ψdθ + sin θ sin ψdφ, ⎪ ⎪ ⎩ 3  = dψ + cos θ dφ.

(1.94)

(1.95)

To show this, we first put the rotation matrix g in the form g = g3 (φ)g2 (θ )g3 (ψ), as is shown in (1.89), where g3 , g2 denote rotations about the e3 - and the e2 -axes, respectively. For the factor matrices, one verifies that dg3 (φ)g3−1 (φ) = R(e3 )dφ,

dg2 (θ )g2−1 (θ ) = R(e2 )dθ.

It then follows that dgg −1 is expressed as dgg −1 = R(e3 )dφ + R(g3 (φ)e2 )dθ + R(g3 (φ)g2 (θ )e3 )dψ. Further, by using g2 (θ )e3 = e3 cos θ + e1 sin θ, g3 (φ)e1 = e1 cos φ + e2 sin φ, g3 (φ)e2 = −e1 sin φ + e2 cos φ, we can obtain the explicit expression (1.94) of a , a = 1, 2, 3. A similar method can be applied to find the expression (1.95) of  a . From the definition (1.92) of the left- and right-invariant one-forms, we find, by differentiation, that d(g −1 dg) = −g −1 dg ∧ g −1 dg,

d(dgg −1 ) = dgg −1 ∧ dgg −1 ,

which are written out to give, respectively, d c = −

1 εabc  a ∧  b , 2

dc =

1 εabc a ∧ b . 2

(1.96)

32

1 Geometry of Many-Body Systems

In a dual manner to the left- and the right-invariant one-forms, we can define leftand right-invariant vector fields on SO(3). Let h(t) be a one-parameter subgroup of SO(3). Then, the vector fields defined to be     d d  gh(t) , Yg = h(t)g  , (1.97) Xg = dt dt t =0 t =0 are called a left- and a right-invariant vector field, respectively. If h(t) = et R(a) , one obtains Xg = gR(a),

Yg = R(a)g,

respectively. In what follows, we will find the left- and the right-invariant vector fields in terms of the Euler which are dual to  a and to a , respectively. angles, a For the vector  =  ea of the left-invariant one-forms, we have dg = gR(). In particular, for a curve g(t) passing g at t = 0, this equation reduces to ˙ where  ˙ = (d/dt), i.e.,  ˙ 1 = sin ψ θ˙ − sin θ cos ψ φ˙ etc. For dg/dt = gR(), t R(a) ˙ so that g(t) = ge , the present equation is evaluated to give gR(a) = gR() ˙ ˙ R(a) = R(). For a = e1 , the equation R(e1 ) = R() reads sin ψ θ˙ − sin θ cos ψ φ˙ = 1,

cos ψ θ˙ + sin θ sin ψ φ˙ = 0,

ψ˙ + cos θ φ˙ = 0,

˙ θ˙ , ψ˙ , and thereby the leftwhich are easily solved to give explicit expressions of φ, invariant vector field K1 is determined. The same procedure runs in parallel for a = e2 and a = e3 to determine K2 , K3 , respectively. It then turns out that the left-invariant vector fields Ka = gR(ea ) are put in the form ⎧ ∂ ∂ cos ψ ∂ ⎪ ⎪ K1 = − + sin ψ + cot θ cos ψ , ⎪ ⎪ sin θ ∂φ ∂θ ∂ψ ⎪ ⎨ ∂ ∂ sin ψ ∂ K2 = + cos ψ − cot θ sin ψ , ⎪ sin θ ∂φ ∂θ ∂ψ ⎪ ⎪ ⎪ ∂ ⎪ ⎩ K3 = . ∂ψ

(1.98)

As is easily verified, the Ka are dual to a :  a (Kb ) = δab ,

a, b = 1, 2, 3.

˙ for a = ea , a = In the same manner as above, by using the equation R(a) = R() 1, 2, 3, we find that the right-invariant vector fields Ja = R(ea )g are expressed as ⎧ ∂ cos φ ∂ ∂ ⎪ ⎪ J1 = −cos φ cot θ − sin φ + , ⎪ ⎪ ∂φ ∂θ sin θ ∂ψ ⎪ ⎨ ∂ sin φ ∂ ∂ J2 = −sin φ cot θ + cos φ + , (1.99) ⎪ ∂φ ∂θ sin θ ∂ψ ⎪ ⎪ ⎪ ∂ ⎪ ⎩ J3 = , ∂φ

1.6 Spatial Many-Body Systems

33

which are dual to a : a (Jb ) = δab ,

a, b = 1, 2, 3.

In a dual manner to (1.96), the commutation relations among Ka and among Ja are given by   εabc Kc , [Ja , Jb ] = − εabc Jc , (1.100) [Ka , Kb ] = c

c

respectively. These equations can be verified in a straightforward manner or by using the formula for the differential of a one-form κ, dκ(X, Y ) = X(κ(Y )) − Y (κ(X)) − κ([X, Y ]) (see [75]) together with (1.96). Since Ja = R(ea )g, Ka = gR(e a ), these vector fields are related by  Ja = R(ea )g = gR(g −1 ea ) = g gab R(eb )  = gab Kb , (1.101) where g = (gab ) and g −1 = (gab )T .

1.6 Spatial Many-Body Systems We proceed to systems of N particles in the space R3 . Though the geometric setting for spatial many-body systems is an extension of that for planar many-body systems, we need a modification accompanying the extension of the rotation group from SO(2) to SO(3). We denote by x α , α = 1, 2, . . . , N, the position of each particle with mass mα > 0, α = 1, 2, . . . , N. The configuration of the particles is described by x = (x 1 , x 2 , · · · , x N ). The totality of the configurations is denoted by X = {x; x = (x 1 , x 2 , · · · , x N ), x α ∈ R3 },

(1.102)

which is called a configuration space. The X is endowed with a linear space structure by introducing the addition and multiplication operations through (x 1 + y 1 , x 2 + y 2 , · · · , x n + y N ) = (x 1 , x 2 , · · · , x N ) + (y 1 , y 2 , · · · , y N ), λ(x 1 , x 2 , · · · , x N ) = (λx 1 , λx 2 , · · · , λx N ),

λ ∈ R.

We can think of the X as the space of 3 × N matrices, since x is viewed as a matrix consisting of N column vectors x α . Here we denote the standard basis vectors R3 by ⎛ ⎞ 1 e1 = ⎝ 0 ⎠ , 0

⎛ ⎞ 0 e2 = ⎝ 1 ⎠ , 0

⎛ ⎞ 0 e3 = ⎝ 0 ⎠ . 1

34

1 Geometry of Many-Body Systems

Then, a basis of X is formed, for example, by (ek , 0, · · · , 0),

(0, ek , 0, · · · , 0),

··· ,

(0, · · · , ek ),

k = 1, 2, 3. (1.103)

The X is endowed with the mass-weighted inner product (or inner product, for short) through K(x, y) =

N 

mα x α · y α ,

x, y ∈ X,

(1.104)

α=1

where the center dot · denotes the standard inner product on R3 . It is easily verified that the K satisfies the following: 1. K(x, y) = K(y, x), 2. K(λx + μy, z) = λK(x, z) + μK(y, z), λ, μ ∈ R, 3. K(x, x) ≥ 0, K(x, x) = 0 ⇐⇒ x = 0. For any configuration y = (y 1 , · · · , y N ) ∈ X, the center-of-mass vector for y is defined to be

b=

N 

−1 mα

α=1

N 

mβ y β .

β=1

The center-of-mass system is defined to be Q = {x ∈ X;

N 

mα x α = 0},

(1.105)

α=1

which is the set of all the configurations with the center-of-mass fixed at the origin of R3 . The Q is a linear subspace of X. By using the center-of-mass vector b, any configuration y is decomposed into (y 1 , y 2 , · · · , y N ) = (x 1 , x 2 , · · · , x N ) + (b, b, · · · , b), where x α = y α − b. Since Q and further that

N

α=1 mα x α

(1.106)

= 0, we find that x = (x 1 , x 2 , · · · , x N ) ∈

(x 1 , x 2 , · · · , x N ) ⊥ (b, b, · · · , b). Now we set b = (b, · · · , b). Then, the decomposition (1.106) of y is expressed as y = x + b, and proves to be an orthogonal decomposition. On introducing the orthogonal complement to Q by Q⊥ := {y ∈ X; K(y, x) = 0, x ∈ Q},

1.6 Spatial Many-Body Systems

35

the X is decomposed into X = Q ⊕ Q⊥ .

(1.107)

The decomposition y = x + b mentioned above is subject to this orthogonal decomposition of X. We note further that Q⊥ = {x ∈ X| x = (c, c, · · · , c), c ∈ R3 },

(1.108)

which is easy to prove. We now seek a basis which fits the decomposition (1.107). We note that the basis (1.103) of X does not fit the decomposition (1.107), since the basis vectors do not belong to the subspace Q or Q⊥ . A basis of Q⊥ is easily found from (1.108), which takes the form (ek , · · · , ek ). To find a basis of Q, we take from Q, for example, the vectors (−m2 e1 , m1 e1 , 0, · · · , 0), (−m2 e2 , m1 e2 , 0, · · · , 0), (−m2 e3 , m1 e3 , 0, · · · , 0). We can apply the Gram–Schmidt orthonormalization method to these vectors, and continue the procedure to obtain the following. Proposition 1.6.1 The configuration space X for a spatial N-body system has the orthonormal system, with respect to the inner product (1.104), given by c1 = N0 (e1 , · · · , e1 ), c2 = N0 (e2 , · · · , e2 ),

(1.109)

c3 = N0 (e3 , · · · , e3 ), f3j −2

j terms j

  = Nj (−mj +1 e1 , · · · , −mj +1 e1 , ( mα )e1 , 0, · · · , 0), α=1

f3j −1

j terms j

  = Nj (−mj +1 e2 , · · · , −mj +1 e2 , ( mα )e2 , 0, · · · , 0),

(1.110)

α=1

f3j

j terms j

  = Nj (−mj +1 e3 , · · · , −mj +1 e3 , ( mα )e3 , 0, · · · , 0), α=1

where j = 1, 2, · · · , N − 1, and where

N0 =

N 

α=1

⎛

−1/2 mα

,

⎛

Nj = ⎝mj +1 ⎝

j 

α=1

⎞⎛ mα ⎠ ⎝

j +1 

⎞⎞−1/2 mα ⎠ ⎠

.

α=1

Note also that ck , k = 1, 2, 3, and f ,  = 1, 2, · · · , 3(N −1), form an orthonormal basis of Q⊥ and of Q, respectively.

36

1 Geometry of Many-Body Systems

It is straightforward to verify that K(ci , cj ) = δij ,

i, j = 1, 2, 3,

K(ci , fk ) = 0,

i = 1, 2, 3,

K(fk , f ) = δk ,

k = 1, 2, . . . , 3(N − 1),

(1.111)

k,  = 1, 2, . . . , 3(N − 1).

We denote the components of x ∈ Q with respect to the orthonormal basis fk by qk = K(x, fk ),

k = 1, 2, . . . , 3(N − 1).

(1.112)

Then, the (qk ) serve as the Cartesian coordinates of Q ∼ = R3(N−1) , which serve to define the Jacobi vectors through r j := q3j −2 e1 + q3j −1 e2 + q3j e3 ,

j = 1, 2, . . . , N − 1.

(1.113)

It is straightforward to verify that r j is written out as r j : = K(x, f3j −2 )e1 + K(x, f3j −1 )e2 + K(x, f3j )e3 = Nj

j 

mα (−mj +1 )(x α · e1 )e1 + mj +1 Nj

α=1

+ Nj + Nj

α=1 j 

= −Nj mj +1

mα (−mj +1 )(x α · e2 )e2 + mj +1 Nj mα (−mj +1 )(x α · e3 )e3 + mj +1 Nj j ! 

+ Nj mj +1

j 

mα

On introducing Mj =

 mα (x j +1 · e3 )e3

α=1

α=1

α=1

! " mα (x j +1 · e1 )e1 + (x j +1 · e2 )e2 + (x j +1 · e3 )e3

α=1 j 

j 

 mα (x j +1 · e2 )e2

j j    "   mα x α · e1 e1 + mα x α · e2 e2 + mα x α · e3 e3

+1 1/2  j

α=1

rj =

j  α=1

α=1

= mj +1

 mα (x j +1 · e1 )e1

α=1

j 

α=1



j 

mα

j j −1/2  −1    x j +1 − mα mα x α .

α=1

j

α=1 mα ,

α=1

α=1

the Jacobi vectors r j are put in a compact form,

j   1 1 −1/2  1  x j +1 − + mα x α , Mj mj +1 Mj α=1

Mj =

j  α=1

mα .

(1.114)

1.6 Spatial Many-Body Systems

37

In particular, for N = 3, the Jacobi vectors are expressed as  r 1 = q1 e 1 + q2 e 2 + q3 e 3 =  r 2 = q4 e 1 + q5 e 2 + q6 e 3 =

m1 m2 (x 2 − x 1 ), m1 + m2 m3 (m1 + m2 )  m1 x 1 + m2 x 2  x3 − . m1 + m2 + m3 m1 + m2

So far we have obtained the isomorphisms, Q ∼ = R3(N−1) ∼ = R3×(N−1) , given by x=



qk fk → (qk ) → (r 1 , · · · , r N−1 ),

where R3×(N−1) denotes the linear space of the real 3 × (N − 1) matrices. Thus, the center-of-mass system Q for the spatial N-body system is identified with the space of the (N − 1)-tuple of the Jacobi vectors (r 1 , · · · , r N−1 ): Q∼ = {(r 1 , · · · , r N−1 ); r j ∈ R3 , j = 1, . . . , N − 1}.

(1.115)

For confirmation, we give a procedure to form a configuration (x 1 , · · · , x N ) ∈ Q from a given set of Jacobi vectors (r 1 , · · · , r N−1 ). To this end, we recall the fact that the Jacobi vector r j is a constant multiple of the vector whose tail is at the center-of-mass of the particles consisting of the first to the j -th and whose head is at the position of the (j + 1)-th particle, and further that the r N−1 passes the centerof-mass for the whole particle system. Then, the position of each particle is shown to be expressed as

x N −1



1 1/2 MN −1 r N −1 , MN −1 mN MN −1 + mN  1  1 1 1/2 1 1/2 MN −2 = xN − + r N −1 + + r N −2 , MN −1 mN MN −2 mN −1 MN −2 + mN −1

xN =

1

+

.. . x2 = x3 − x1 = −

 1  1 1 1/2 1 1/2 M1 + r2 + + r 1, M2 m3 M1 m2 M1 + m 2

N 1  mα x α . m1 α=2

38

1 Geometry of Many-Body Systems

From these equations, the position vectors x N , x N−1 , · · · , x 1 are found to be expressed as linear combinations of the Jacobi vectors r N−1 , · · · , r 1 . In particular, for N = 3, one obtains 

m1 + m2 r 2, (m1 + m2 + m3 )m3   m3 m1 r2 + r 1, x2 = − (m1 + m2 + m3 )(m1 + m2 ) (m1 + m2 )m2   m3 m2 r2 − r 1. x1 = − (m1 + m2 + m3 )(m1 + m2 ) m1 (m1 + m2 ) x3 =

We are now interested in the bulk of the many-body configurations. Among the configurations, there is a special configuration of small bulk. For example, if all the particles lie on a line, the bulk of the configuration is viewed as slender. To describe the bulk or the distribution of particles, we consider the subspace spanned by x 1 , x 2 , · · · , x N , Wx = span{x 1 , x 2 , · · · , x N }.

(1.116)

If all the particles are put together at a point, then one has dim Wx = 0. If the particles are sitting on a line, one has dim Wx = 1. For planar and spatial configurations of particles, we have dim Wx = 2 and dim Wx = 3, respectively. Then, the whole space Q is broken up into subsets, according to dim Wx ; Qk = {x ∈ Q; dim Wx = k},

k = 0, 1, 2, 3.

(1.117)

In particular, for a reason to be explained soon (see Lemma 1.6.1), we set Q˙ = Q2 ∪ Q3 .

(1.118)

Then, the center-of-mass system Q is broken up into ˙ Q = Q0 ∪ Q1 ∪ Q.

(1.119)

We note that this decomposition holds true if we describe the configuration space in terms of the Jacobi vectors (see (1.115)). In fact, if we regard (r 1 , · · · , r N−1 ) as a 3 × (N − 1) matrix, then we see that dim Wx = rank(r 1 , · · · , r N−1 ).

(1.120)

Here we consider the action of the rotation group SO(3) on the center-of-mass system Q. The SO(3) acts on Q in such a manner that g ∈ SO(3) rotates all the particles simultaneously. Put another way, the action g of g is defined by g (x) = gx = (gx 1 , gx 2 , · · · , gx N ),

x ∈ Q.

(1.121)

1.6 Spatial Many-Body Systems

39

Here,  we have to note that  the Q is invariant under the action of SO(3). In fact, if mα x α = 0, then mα gx α = 0. If we consider the system of the Jacobi vectors (1.115) as the configuration space, the SO(3) action is described as (r 1 , r 2 , · · · , r N−1 ) −→ (gr 1 , gr 2 , · · · , gr N−1 ).

(1.122)

We note further that since dim Wgx = dim Wx , each of the subsets Q0 , Q1 , Q˙ is invariant by the action of SO(3) (g Q˙ ⊂ Q˙ etc.). The action of SO(3) is then dealt ˙ In particular, as for the SO(3) action on Q, ˙ with on each of the subsets Q0 , Q1 , Q. we can prove the following lemma. ˙ that is, if a point x ∈ Q ˙ is fixed by Lemma 1.6.1 The SO(3) acts freely on Q, g ∈ SO(3), then g must be the identity. ˙ and g ∈ SO(3) such that gx = x. Since Proof Suppose that there exist x ∈ Q ˙ dim Wx ≥ 2 for x ∈ Q, there are at least two linearly independent vectors, which we take as x 1 , x 2 after rearranging the numbering of the position vectors if necessary. From gx = x, we have gx 1 = x 1 , gx 2 = x 2 , which implies that any vector in the plane spanned by x 1 and x 2 are left invariant under the action of g. Further, since g is an orthogonal transformation, it preserves the inner product, i.e., for any x, y ∈ R3 , one has gx · gy = x · y. It then follows that any vector in the line perpendicular to the plane spanned by x 1 and x 2 is kept sitting in the same line, if transformed by g. Hence, for any vector n in this line, either of the relations gn = ±n holds true. Incidentally, the condition det g = 1 requires that gn = n. On account of linearity, the transformation g leaves invariant all the vectors in R3 , which means that g = I (the identity). Thus, we have shown that the action of SO(3) is free. This ends the proof. We now consider the cases other than dim Wx ≥ 2. To this end, we introduce the notion of isotropy subgroup. For x ∈ Q, the isotropy subgroup at x is defined to be Gx = {g ∈ SO(3); gx = x}.

(1.123)

˙ the Gx is Lemma 1.6.2 According to whether x belongs to Q0 , Q1 or Q, isomorphic to one of the following; ⎧ SO(3) if x ∈ Q0 , ⎪ ⎪ ⎨ Gx ∼ = SO(2) if x ∈ Q1 , ⎪ ⎪ ⎩ ˙ {I } if x ∈ Q.

(1.124)

Proof In terms of isotropy subgroups, Lemma 1.6.1 states that if x ∈ Q˙ then Gx = {I }. For x ∈ Q0 , one has clearly Gx ∼ = SO(3). If x ∈ Q1 , all the particles lie on a line. We denote a unit vector on this line by a. Then the condition gx = x implies that g is a rotation about the a axis, that is, one has g = et R(a) , t ∈ R. Since

40

1 Geometry of Many-Body Systems

there exists an h ∈ SO(3) such that a = he3 , the rotation about a is expressed as et R(a) = het R(e3 ) h−1 , which shows that Gx ∼ = SO(2). This ends the proof. We now introduce the notion of equivalence relation on Q in such a manner that two configurations are said to be equivalent to each other if they are related by a rotation. In other words, for x, y ∈ Q, the x and y are said to be equivalent if there exists a g ∈ SO(3) such that y = gx. We denote by x ∼ y the equivalence of x, y ∈ Q. Then, we can easily verify that (i) x ∼ x, (ii) x ∼ y ⇒ y ∼ x, (iii) x ∼ y, y ∼ z ⇒ x ∼ z. The quotient space by this equivalence relation, which is denoted by M := Q/SO(3), is called the interior space or the shape space. We denote by π the natural projection from Q to M. Put another way, the π maps x ∈ Q to its equivalence class [x], π : Q −→ M := Q/SO(3);

π(x) = [x].

(1.125)

We further introduce the notion of the orbit by the group action. The subset of Q defined by Ox := {gx ∈ Q| g ∈ SO(3)}

(1.126)

is called the orbit of SO(3) through x ∈ Q. By definition, the Ox and [x] coincide with each other as subsets. In this sense, the quotient space Q/SO(3) is referred to as the orbit space, which is not a manifold in general. However, if we restrict the center-of-mass system to those x ∈ Q satisfying dim Wx ≥ 2, the quotient space ˙ M˙ := Q/SO(3) becomes a manifold. Here we give an intuitive explanation of this ˙ the equivalence class of x ∈ Q˙ is homeomorphic fact. Since SO(3) acts freely on Q, ˙ that is, Ox ∼ with SO(3) for any x ∈ Q, = SO(3). Hence, the quotient space ˙ ˙ Q/SO(3) is of constant dimension, dim Q/SO(3) = 3(N − 1) − 3 = 3N − 6. It is a rather difficult problem to identify topologically the shape space. However, we can explicitly find it for a spatial three-body system. ˙ Lemma 1.6.3 With the condition dim Wx ≥ 2, the shape space Q/SO(3) for the center-of-mass system of spatial three bodies is homeomorphic to the upper half space R3+ , where R3+ = {(ξ, η, ζ ) ∈ R3 ; ζ > 0}. Proof From (1.115) with N = 3, we view the center-of-mass system in question as Q = {(r 1 , r 2 )| r i ∈ R3 , i = 1, 2}. From (1.120) with dim Wx ≥ 2, the r 1 and r 2 are linearly independent, so that one has r 1 × r 2 = 0. In view of this fact, we define a map Q˙ → R3+ by (r 1 , r 2 ) −→ (ξ, η, ζ ) = (|r 1 |2 − |r 2 |2 , 2r 1 · r 2 , 2|r 1 × r 2 |).

(1.127)

1.6 Spatial Many-Body Systems

41

We show that this map gives rise to a map from the restricted shape space M˙ = ˙ Q/SO(3) to R3+ . According to (1.122), the SO(3) action on (r 1 , r 2 ) is given by (r 1 , r 2 ) → (gr 1 , gr 2 ). Since the three quantities given in the right-hand side of (1.127) are all invariant under the action of SO(3), they have the same values ˙ so that a map from M˙ = Q/SO(3) ˙ for equivalent pairs (r 1 , r 2 ), (r 1 , r 2 ) ∈ Q, to 3 R+ can be defined by [(r 1 , r 2 )] → (ξ, η, ζ ). To show that this map is bijective, we refer to the equality (|r 1 |2 − |r 2 |2 )2 + 4(r 1 · r 2 )2 + 4|r 1 × r 2 |2 = (|r 1 |2 + |r 2 |2 )2 . For a given a =

3

i=1 ai e i

(1.128)

∈ R3+ with a3 > 0, we can find a solution to

a1 = |r 1 |2 − |r 2 |2 ,

a2 = 2r 1 · r 2 ,

a3 = 2|r 1 × r 2 |,

by using (1.128). This is because from |a| = |r 1 |2 + |r 2 |2 and a1 = |r 1 |2 − |r 2 |2 , the quantities |r 1 |2 and |r 2 |2 are found to be expressed in terms of ai , i = 1, 2, 3, and then the second and the third equations determine the angle made by r 1 , r 2 . Thus, we have found a solution (r 1 , r 2 ). The other solutions are put in the form (gr 1 , gr 2 ), g ∈ SO(3). In fact, if we have another solution (r 1 , r 2 ), the respective magnitudes of (r 1 , r 2 ) and the angle made by them are the same as those for (r 1 , r 2 ). Then, one can transfer r 1 to r 1 by a rotation g1 and then by a rotation g2 about r 1 one can transfer g1 r 2 to r 2 , that is, g2 g1 r 2 = r 2 . Thus, we have (g2 g1 r 1 , g2 g1 r 2 ) = (r 1 , r 2 ). This implies that the map of M˙ to R3+ defined by [(r 1 , r 2 )] → (ξ, η, ζ ) proves to be a bijection. (If we take into account the topology of the quotient space, we can show that the map M˙ → R3+ is actually a homeomorphism.) This ends the proof. ˙ we consider the whole shape space Q/SO(3) for the Without restriction to Q, three-body system. On account of (1.120), for a three-body system, r 1 and r 2 are linearly dependent if dim Wx = 1, and hence one has r 1 × r 2 = 0. If we apply the map (1.127) to Q1 , we have (r 1 , r 2 ) −→ (|r 1 |2 − |r 2 |2 , 2r 1 · r 2 , 0),

(r 1 , r 2 ) ∈ Q1 .

This map gives rise to a bijection from Q1 /SO(3), the quotient of Q1 by SO(3), to ˙ 2 := R2 − {0}, where R ˙ 2 denotes the plane ζ = 0 without the origin. The plane R 2 R itself forms the boundary of the upper half space R3+ . In a similar manner, we see that Q0 /SO(3) is nothing but the origin (ξ, η, ζ ) = 0, which has been deleted from the plane mentioned above. Thus, we have shown the following: Proposition 1.6.2 The shape space for the three-body system, Q/SO(3) = ˙ Q/SO(3) ∪ Q1 /SO(3) ∪ Q0 /SO(3), is topologically a closed upper half space, which is a disjoint union of three subsets of different dimension: ˙ 2 ∪ {0}. Q/SO(3) ∼ = {(ξ, η, ζ ) ∈ R3 : ζ ≥ 0} = R3+ ∪ R

(1.129)

42

1 Geometry of Many-Body Systems

Shape spaces for spatial N-body systems with N ≥ 4 are discussed in Appendix 5.3.

1.7 Rotation and Vibration for Spatial Many-Body Systems ˙ and refer to it as the center-of-mass system, and In what follows, we treat mainly Q, ˙ ˙ to M = Q/SO(3) as the shape space accordingly. The inner product defined on Q˙ ˙ at x ∈ Q. ˙ For tangent vectors induces the inner product on the tangent space Tx (Q) ˙ u = (u1 , u2 , · · · , uN ) and v = (v 1 , v 2 , · · · , v N ) at x ∈ Q, the inner product of them is defined to be Kx (u, v) =

N 

˙ u, v ∈ Tx (Q),

mα u α · v α ,

(1.130)

α=1

 mα u α = where uα · v α denote the standard inner products on R3 and where  ˙ The Kx is closely related with the kinetic energy. In mα v α = 0 for u, v ∈ Tx (Q). fact, for u = v, the Kx (u, u) is twice the kinetic energy. The Kx defined by (1.130) is usually described as ds 2 =

N 

mα dx α · dx α ,

(1.131)

α=1

˙ which is called a mass-weighted metric (or metric, for short) on Q. The first task for us to do is to distinguish rotational and vibrational motions. The rotation of a many-body system is realized by the action of SO(3), which we have already treated. The infinitesimal rotation is called a rotational vector, which is defined as follows: We denote by R(φ) the skew symmetric matrix corresponding to a vector φ ∈ R3 . Then the one-parameter group g(t) = exp tR(φ) determines ˙ Differentiating g(t)x with respect to t at t = 0, we obtain the a curve g(t)x in Q. infinitesimal rotation,   d g(t)x  = R(φ)x = (R(φ)x 1 , R(φ)x 2 , · · · , R(φ)x N ). (1.132) dt t =0 In terms of vector fields, the rotational vector is expressed as N  α=1

 ∂ ∂  , =φ· xα × ∂x α ∂x α N

R(φ)x α ·

(1.133)

α=1

where ∂/∂x α denotes the gradient operator. The above expression implies that the infinitesimal rotation is closely related with the total angular momentum operator.

1.7 Rotation and Vibration for Spatial Many-Body Systems

43

We turn to vibrational vectors. A tangent vector v = (v 1 , v 2 , · · · , v N ) at x is called a vibrational vector if it is orthogonal to any rotational vector R(φ)x at x ∈ Q˙ with respect to the metric Kx , that is, if v satisfies Kx (R(φ)x, v) = 0 for any φ ∈ R3 . Written out, this condition is expressed as Kx (R(φ)x, v) =

N 

mα (φ × x α ) · v α = φ ·

α=1

N 

mα x α × v α = 0,

α=1

which implies that v = (v 1 , v 2 , · · · , v N ) is a vibrational vector if and only if N 

mα x α × v α = 0.

(1.134)

α=1

This condition means that the total angular momentum of the N-body system vanishes, which is quite natural as a condition for  vibrational vectors from a physical point of view. We note here that the condition mα v α = 0 is tacitly assumed, ˙ A vibrational vector v is which is a condition for v to be a tangent vector to Q. called also an infinitesimal vibration. Proposition 1.7.1 Let Vx,rot and Vx,vib denote the space of rotational and vibra˙ respectively: tional vectors at x ∈ Q, Vx,rot = {R(φ)x; φ ∈ R3 }, ˙ Vx,vib = {v ∈ Tx (Q);

N 

⊥ mα x α × v α = 0} = Vx,rot

α=1

˙ is decomposed into the orthogonal direct sum of Then, the tangent space Tx (Q) Vx,rot and Vx,vib: ˙ = Vx,rot ⊕ Vx,vib. Tx (Q)

(1.135)

In terms of differential geometry, the above equation implies that a connection is defined on Q˙ and the Vx,rot and Vx,vib are called a vertical and a horizontal subspace, respectively. We call this connection the Guichardet connection [22]. ˙ one has We note also that for u = (u1 , · · · , uN ) ∈ Tgx (Q), 

mα gx α × uα = g



mα x α × v α

with

v α = g −1 uα ,

which implies that Vgx,vib = gVx,vib ,

g ∈ SO(3).

(1.136)

44

1 Geometry of Many-Body Systems

(1)

(2)

(3)

(4)

(5)

(6)

Fig. 1.8 Canonical modes of an X4 molecule

Here we give a simple example of vibrational vectors. Suppose that there are four particles which are located at x 1 = e1 , x 2 = e2 , x 3 = −e1 , x 4 = −e2 . We assume that all particles have an equal mass mα = 1 for simplicity. Then, we can find the following vibrational vectors ek , k = 1, . . . , 6, satisfying (1.134) with x α given above: e1 = 12 (e1 , e2 , −e1 , −e2 ),

e2 = 12 (−e1 , e2 , e1 , −e2 ),

e3 = 12 (−e3 , e3 , −e3 , e3 ),

e4 = 12 (−e2 , −e1 , e2 , e1 ),

e5 = 12 (−e2 , e2 , −e2 , e2 ),

e6 = 12 (e1 , −e1 , e1 , −e1 ).

These vectors form an orthonormal basis of Vx,rot with respect to Kx , and are known also as normal modes for the planar square X4 molecule (Fig. 1.8). Any infinitesimal vibration gives rise to an infinitesimal deformation of the shape ˙ of the many-body system. This can be seen from (1.125) with Q restricted to Q. ˙ → M˙ is surjective, the differential of π at x ∈ Q˙ Since the projection π : Q ˙ → Tπ(x)(M) ˙ π∗ : Tx (Q)

(1.137)

is also surjective, whose kernel is shown to be given by ker π∗ = Vx,rot.

(1.138)

˙ are put together to imply This and the direct sum decomposition (1.135) of Tx (Q) that ˙ Vx,vib ∼ = Tπ(x)(M),

(1.139)

1.7 Rotation and Vibration for Spatial Many-Body Systems

45

the proof of which will be given later. The meaning of this equation is that the infinitesimal vibrations at x and the infinitesimal deformation of the shape π(x) ˙ q ∈ are in one-to-one correspondence, so that it follows that for any X ∈ Tq (M), −1 ˙ M and x ∈ π (q), there exists a unique vibrational vector u ∈ Vx,vib such that π∗ (u) = X. While this book does not give a precise definition of a manifold or the definition of the differential of a map either (for those definitions, see [1] and [15], for examples), the differential π∗ can be understood in an elementary manner as ˙ we take a curve x(t) which passes x at t = 0 with follows: For any u ∈ Tx (Q), ˙ the tangent vector dx/dt|t =0 = u at x. Since π(x(t)) is a curve passing π(x) on M, ˙ dπ(x(t))/dt|t =0 is a tangent vector to M at π(x), which is denoted by π∗ (u): π∗ (u) =

  d ˙ π(x(t)) ∈ Tπ(x)(M), dt t =0

u=

 dx  . dt t =0

˙ → Tπ(x)(M) ˙ is thus determined. The map π∗ : Tx (Q) With this understanding of π∗ in mind, we can verify Eq. (1.138) in the following ˙ we denote the orbit of SO(3) through the point x by manner. For x ∈ Q, Ox = {gx| g ∈ SO(3)}, which consists of all the configurations equivalent to x. Then, from the definition of Vx,rot, one has Vx,rot = Tx (Ox ), which implies that the Vx,rot is the tangent space to the orbit Ox . For any u ∈ Vx,rot, there exists a vector φ ∈ R3 such that u = R(φ)x (by the definition of Vx,rot). Then, R(φ) is exponentiated to give a one-parameter group, exp(tR(φ)), whose orbit through x is expressed as x(t) = exp(tR(φ))x. Since x and x(t) are equivalent with respect to SO(3), we have π(x(t)) = π(x). Differentiated with respect to t at t = 0, this equation provides π∗ (R(φ)x) = π∗ (u) = 0. This implies that u ∈ ker π∗ . Conversely, for any v ∈ ker π∗ , there exist a curve x(t) such that dx(t)/dt|t =0 = v, x(0) = x. Differentiation of π(x(t)) with respect to t results in ˙ dπ(x(t))/dt|t =0 = π∗ (v) = 0. If v ∈ / Vx,rot, then v is not tangent to Ox at x ∈ Q. Then for an infinitesimal parameter ε, the x(ε) does not stay on Ox . Put another way, it leaves Ox . This implies that x(ε) is not equivalent to x. (If x(ε) stays in Ox , one has x(ε) ∈ Ox , so that π(x(ε)) = π(x).) Hence, π(x(ε)) = π(x), if ε = 0. This ˙ Then, one has implies that the curve π(x(t)) has no stationary point at π(x) ∈ M. dπ(x(t))/dt|t =0 = 0, which is a contradiction. Therefore, it holds that v ∈ Vx,rot. This ends the proof of ker π∗ = Vx,rot.

46

1 Geometry of Many-Body Systems

So far we have introduced the connection by using the decomposition (1.135) of the tangent space. There is a dual way to define the connection in terms of an so(3)-valued differential form. To this end, we need to introduce an inertia tensor. For each x ∈ Q, the inertia tensor Ax : R3 → R3 is defined through N 

Ax (φ) =

mα x α × (φ × x α ),

φ ∈ R3 .

(1.140)

α=1

This definition is an extension of the inertia tensor for a rigid body. For a rigid body, each x α is relatively fixed, so that one has only to define Ax for a fixed configuration x, but in the present case, the configuration x ∈ Q can vary without restriction. Under the condition dim Wx ≥ 2, the Ax is shown to be symmetric positivedefinite. As the symmetric property is easy to prove, we give here the proof of positive-definiteness only. Since φ · Ax (φ) =

N 

mα (φ × x α ) · (φ × x α ) ≥ 0,

α=1

the equality of this equation holds if and only if φ × x α = 0 for all α = 1, 2, . . . , N. If φ = 0, the condition φ × x α = 0 implies that there exist Cα ∈ R such that x α = Cα φ. This means that dim Wx ≤ 1, which contradicts the assumption dim Wx ≥ 2. Then it turns out that φ = 0, as is wanted. Because of the positive-definiteness, the ˙ In addition, the Ax is shown to transform according to Ax has its inverse, if x ∈ Q. Agx (φ) = gAx (g −1 φ),

g ∈ SO(3).

(1.141)

In fact, we easily verify that Agx (φ) =

N 

mα gx α × (gg −1 φ × gx α )

α=1

=g

N 

mα x α × (g −1 φ × x α ) = gAx (g −1 φ).

α=1

We can put Ax in a matrix form. In fact, the defining equation of Ax (φ) is arranged as Ax (φ) =

N  α=1

mα |x α |2 I −

N  α=1

 mα x α x Tα φ,

(1.142)

1.7 Rotation and Vibration for Spatial Many-Body Systems

47

where I denotes the 3 × 3 identity matrix. Then, the matrix elements of Ax are expressed as ea · Ax (eb ) =

N 

mα |x α |2 δab −

α=1

N 

mα (ea · x α )(x α · eb ),

a, b = 1, 2, 3,

α=1

where ea are the standard basis vectors of R3 . See (2.9) and (2.10) for the usual definition of inertia tensor for rigid bodies. We are now in a position to define the connection form. The connection form ω ˙ is the so(3)-valued one-form given by on Q ωx =

R(A−1 x

N 

mα x α × dx α ).

(1.143)

mα x α × v α = 0.

(1.144)

α=1

By definition, we see that ωx (v) = 0 ⇔

N  α=1

Further, for a rotational vector R(φ)x, the ωx takes the value ωx (R(φ)x) = R(A−1 x



mα x α × (φ × x α )) = R(φ).

(1.145)

We will show that the ω transforms according to (∗g ω)(v) = Adg ω(v),

˙ v ∈ Tx (Q),

(1.146)

where the superscript asterisk denotes the pull-back and the v is viewed as a vector ˙ Before the proof of (1.146), we give a brief review of the pull-back fields on Q. ˙ the action of SO(3) and the differential map. From the action of SO(3) on Q, on the tangent vector fields on Q˙ is induced in the following manner: For u = ˙ we consider a curve x(t) which passes the point x ∈ Q˙ (u1 , u2 , · · · , uN ) ∈ Tx (Q), at t = 0 with u as its tangent vector, dx/dt|t =0 = u. By the action of g , this curve is transformed into gx(t). The tangent vector to this curve at gx is given by g dx/dt|t =0 . This provides the definition of the differential map g∗ of g . Namely, ˙ → Tgx (Q) ˙ is given by g∗ : Tx (Q) g ∗ (u) = (gu1 , gu2 , · · · , guN ). We have here to note that Eq. (1.136) should be expressed as Vgx,vib = g∗ Vx,vib. The pull-back ∗g of g is now defined in terms of the differential map through (∗g ω)x (v) := ωgx (g ∗ (v)),

˙ v ∈ Tx (Q).

48

1 Geometry of Many-Body Systems

We are now in a position to prove (1.146). The right-hand side of the above equation is arranged by using Agx = gAx g −1 , and rewritten as ωgx (g ∗ (v)) = R(A−1 gx



mα gx α × gv α )  mα x α × v α )g −1 = gR(A−1 x

= Adg ωx (v). Proposition 1.7.2 The connection form ω is defined on the center-of-mass system Q˙ by (1.143). The ω has the properties described as (1.144) and (1.145), and transforms according to (1.146) under the action of SO(3). In order to see a further meaning of the connection form, we give the decom˙ according to the direct sum decomposiposition of a tangent vector v ∈ Tx (Q) tion (1.135) by using the connection form. Since ωx (v) ∈ so(3), the quantity ωx (v)x is a rotational vector (see (1.132)). We show that ωx (v)x is exactly the rotational component of v. To this end, we introduce the operator Px by Px (v) := ωx (v)x,

Px (v) = (Px (v)1 , · · · , Px (v)N ).

Then, from the definition of ωx (v) and that of R, we have Px (v)α = (A−1 x

N 

mβ x β × v β ) × x α .

(1.147)

β=1

By using the operator Px , the tangent vector v is naturally decomposed into v = Px (v) + (v − Px (v)). Then, a straightforward calculation provides N  α=1

mα x α × (v α − Px (v)α ) = 0,

N 

mα Px (v)α · (v α − Px (v)α ) = 0,

α=1

which implies that the Px (v) and v − Px (v) are the rotational and the vibrational components of v, respectively. Thus we find that the Px is a projection map, ˙ −→ Vx,rot; Px : Tx (Q)

Px (v) = ωx (v)x.

According to the orthogonal decomposition v = Px (v) + (v − Px (v)), the metric ˙ is also decomposed into on Tx (Q) Kx (v, v) = Kx (Px (v), Px (v)) + Kx (v − Px (v), v − Px (v)).

(1.148)

1.7 Rotation and Vibration for Spatial Many-Body Systems

49

The decomposition (1.148) of the metric and the vector space isomor˙ On account phism (1.139) are put together to define a Riemannian metric on M. ˙ ˙ with of (1.139), for X, Y ∈ Tp (M), there exist vectors u, v ∈ Vx,vib ⊂ Tx (Q)  π(x) = p such that π∗ u = X, π∗ v = Y . Then, a Riemannian metric Kp is defined on M˙ through Kp (X, Y ) = Kx (u, v),

u, v ∈ Vx,vib.

(1.149)

In fact, since Kx is invariant under the action of SO(3), that is (∗g K)x = Kx or Kgx (gu, gv) = Kx (u, v),

˙ u, v ∈ Tx (Q),

and since the invariance Vgx,vib = g∗ Vx,vib holds, the right-hand side of (1.149) depends on π(x) ∈ M˙ only. So far we have described the geometric quantities such as the metric, the inertia tensor, and the connection form in terms of position vectors. In the rest of this section, we describe those quantities in terms of Jacobi vectors. We will show that the metric, the inertia tensor, and the connection form can be expressed in terms of the Jacobi vectors (1.114) as follows: ds 2 =

N−1 

dr k · dr k ,

(1.150)

k=1

Ax (φ) =

N−1 

r k × (φ × r k ),

(1.151)

k=1 N−1    r k × dr k . ω = R A−1 x

(1.152)

k=1

We prove these equations by induction with respect to the number N(≥2) of particles with attention to the fact that the center-of-mass of the N-particle system and that of the (N +1)-particle system are different from each other. Hence, we have to take into account the shift of the center-of-mass when we apply the assumption of induction. As a matter of fact, since the center-of-mass of the N particles N in the m x /M with M = (N + 1)-body system is given by XN = N α α N N α=1 α=1 mα , we have to translate the position of each particle by x α → x α − XN in order to apply the assumption of induction.

50

1 Geometry of Many-Body Systems

We start with the proof of (1.150). For N = 2, one has m1 dx 1 · dx 1 + m2 dx 2 · dx 2 = dr 1 · dr 1 , as is easily seen. We now assume that Eq. (1.150) holds for N, which is put in the form N 

mα d(x α − XN ) · d(x α − XN ) =

α=1

N−1 

dr k · dr k .

k=1

We note here that r k with k = 1, . . . , N − 1 are invariant under the parallel translation x α → x α − XN . Since N α=1 mα dx α = MN dX N , the above equation is rewritten as N 

mα dx α · dx α = MN dXN · dXN +

N−1 

α=1

dr k · dr k .

k=1

We proceed to describe the metric for the (N +1)-particle system. A straightforward calculation with this equation provides N+1 

mα dx α · dx α = MN dXN · dXN +

α=1

N−1 

dr k · dr k + mN+1 dx N+1 · dx N+1

k=1

=

N−1 

dr k · dr k + dr N · dr N ,

k=1

where use has been made of x N+1 =



1/2 MN rN , mN+1 (MN + mN+1 )

XN = −



1/2 mN+1 rN . MN (MN + mN+1 ) (1.153)

Thus, we have shown that Eq. (1.150) holds for N + 1 as well. We turn to the proof of (1.151). For N = 2, one has m1 x 1 × (φ × x 1 ) + m2 x 2 × (φ × x 2 ) = r 1 × (φ × r 1 ), as is easily seen. The assumption of induction in the present case is put in the form N 

mα (x α − X N ) × (φ × (x α − XN )) =

α=1

N−1 

r k × (φ × r k ).

k=1

Expanding this equation, we obtain N  α=1

mα x α × (φ × x α ) = MN XN × (φ × X N ) +

N−1  k=1

r k × (φ × r k ).

1.8 Local Description of Spatial Many-Body Systems

51

On account of this, the inertia tensor for the (N + 1)-particle system is expressed and arranged as N+1 

mα x α × (φ × x α ) = MN XN × (φ × X N ) +

α=1

N−1 

r k × (φ × r k )

k=1

+ mN+1 x N+1 × (φ × x N+1 ) =

N−1 

r k × (φ × r k ) + r N × (φ × r N ),

k=1

where use has been made of (1.153). This ends the proof of (1.151). In conclusion, we prove (1.152). To this end, we have only to show N 

mα x α × dx α =

α=1

N−1 

r k × dr k .

(1.154)

k=1

This can be proved in the same manner as that used in the proof of (1.150) and (1.151).

1.8 Local Description of Spatial Many-Body Systems 1.8.1 Local Product Structure In order to make a further study of many-body systems, we need to introduce local ˙ To this end, we use the fact coordinate systems on the center-of-mass system Q. that π : Q˙ → M˙ is a principal fiber bundle. In place of stating the definition of the principal fiber bundle (see Appendix 5.2), we explain here the local product structure of Q˙ as a principal fiber bundle. Let U be an open subset of the shape space ˙ that is, the inverse image π −1 (U ) of U is an open subset of Q. ˙ By definition, a M, given point q ∈ U determines the shape of the many-body configuration. Then, one can set those particles in the space R3 . A way to do so is for example as follows: For any x ∈ π −1 (q), there are at least two linearly independent position vectors, which we denote by x 1 and x 2 without loss of generality. We can put the x 1 on the positive side of the axis e1 by a rotation, keeping the shape invariant, and then the second particle on the +e2 side of the e1 –e2 plane, by a rotation about x 1 . The other particles take their due positions accordingly. If we place the particles in the space

52

1 Geometry of Many-Body Systems

R3 in this manner, we have a configuration of the many particles for each q ∈ U . ˙ This procedure gives us a mapping σ : U → Q, σ (q) = (σ 1 (q), σ 2 (q), · · · , σ N (q)).

(1.155)

Though the choice of σ is not unique, once σ is given, we can obtain other configurations by rotating σ (q). If the subset U is not so large, every configuration x with the shape π(x) ∈ U is realized in this manner: x = gσ (q),

g ∈ SO(3),

i.e., (x 1 , · · · , x N ) = (gσ 1 (q), · · · , gσ N (q)).

(1.156)

This equation implies that π −1 (U ) has the local product structure π −1 (U ) ∼ = U × SO(3). We denote by (q i ), i = 1, . . . , 3N − 6, a local coordinate system on U , where we note that the q i defined here are different from those defined in (1.112). Natural candidates for local coordinates of SO(3) are the Euler angles. Thus we have obtained a local coordinate system on π −1 (U ) ∼ = U × SO(3). If we take the Jacobi vectors to describe the configuration of the center-of-mass system, the lower equation of (1.156) should be replaced by (r 1 , · · · , r N−1 ) = (gσ 1 (q), · · · , gσ N−1 (q)),

(1.157)

where σ (q) describes a way to place the Jacobi vectors in the space R3 . It is worth mentioning that the expression (1.156) is closely related to that ˙ adopted in a paper  by Eckart [17] without reference to the bundle structure of Q. Let σ α (q) = a ea yaα (q). Then, one has, from (1.156), xα = g

3  a=1

ea yaα (q) =



εa yaα (q),

εa := gea ,

(1.158)

a

where εa , a = 1, 2, 3, are called the unit vectors along a moving system of axes in [17]. Eckart tried to find a local section suitable for small vibrations of polyatomic molecules, which is precisely realized as the Eckart section in [54] by means of Riemann normal coordinates in a vicinity of an equilibrium point in the shape space and their horizontal lifts to the center-of-mass system. In what follows, we call the standard basis {ea } the space frame, and {εa } the rotated frame.

1.8 Local Description of Spatial Many-Body Systems

53

1.8.2 Local Description in the Space Frame We start with a brief review of rotational vectors and the connection form. Rotational vectors are defined to be infinitesimal generators of the SO(3) action, which are expressed in terms of differential operators (see (1.133)) as N 

(φ × x α ) ·

α=1

∂ = φ · J, ∂x α

φ ∈ R3 ,

J =



xα ×

α

∂ . ∂x α

(1.159)

The inertia tensor is defined to be Ax (v) =

N 

mα x α × (v × x α ),

(1.160)

α=1

and the connection form ωx is defined to be N    ωx = R A−1 mα x α × dx α , x

(1.161)

α=1

which satisfies (see (1.145))  ω φ · J ) = R(φ),

(1.162)

and is subject to the transformation (see (1.146)) ωhx = Adh ωx ,

h ∈ SO(3).

(1.163)

We proceed to describe locally-defined connection forms together with rotational and vibrational vectors in local coordinates. Let ω˜ a and Ja be components of ω and J with respect to the standard basis vectors ea (or the space frame), respectively: ω= J =

 

R(ea )ω˜ a , ea Ja ,

Ja =

(1.164a)  ∂ (ea × x α ) · . ∂x α α

(1.164b)

We note that from (1.163) the components ω˜ a transform according to a = ω˜ hx



hab ω˜ xb ,

h = (hab ) ∈ SO(3).

(1.165)

On account of the local description x = gσ (q), the local expressions of the d t R(ea ) rotational vectors (Ja )x = dt e gσ (q)|t =0 are viewed as right-invariant vector

54

1 Geometry of Many-Body Systems

fields already obtained in (1.99). As for the connection form, from (1.162) it follows that ω(Ja ) = R(ea ), which implies that ω˜ a are dual to Ja . The forms ω˜ a together ˙ which satisfies with dq i constitute a local basis of the space of one-forms on Q, ω˜ a (Jb ) = δba ,

dq i (Ja ) = 0,

a, b = 1, 2, 3,

i = 1, . . . , 3N − 6.

(1.166)

To be precise in notation, we have to use the pull-back π ∗ dq i for dq i , but we use dq i for notational simplicity. We now set out to obtain a local expression of ω˜ a . By differentiating x α = gσ α (q) and arranging the resultant equation, we obtain 

mα x α × dx α =



α

mα x α × dgg −1 x α +



α

mα x α ×

 ∂x α

α

i

∂q i

dq i .

(1.167)

In view of this, we choose to use the components, a , of the right-invariant form dgg −1 (see (1.93)) and introduce the quantities   ∂x α  mα x α × · ea . γia = A−1 x ∂q i α

(1.168)

Then, the connection form (1.161) is put in the form ω=



ω˜ a R(ea ),

ω˜ a = a +



γia dq i ,

(1.169)

where the local expression of the right-invariant forms a are given in (1.94). By using ω˜ a and dq i , we can determine vibrational vector fields Xj through ω˜ a (Xj ) = 0,

dq i (Xj ) = δji .

(1.170)

˙ The Xj and Ja are put together to form a local basis of tangent vector fields on Q. The local expression of Xj is now found explicitly through (1.170) to be Xj =

 ∂ − γja Ja , ∂q j

j = 1, 2, · · · , 3N − 6.

(1.171)

This is a unique vibrational (or horizontal) vector field which is in one-to-one ˙ and is called the horizontal lift of ∂/∂q j . correspondence with ∂/∂q j on U ⊂ M, We notice that the transformation rule for ω˜ a by g implies that γia transforms in a similar manner,  γia (hx) = hab γib (x), h = (hab ) ∈ SO(3). (1.172)

1.8 Local Description of Spatial Many-Body Systems

55

For h = et R(ec ) , the above equation is differentiated with respect to t at t = 0 to give Jc (γia ) = −



εcab γib ,

(1.173)

where εabc are the antisymmetric symbols with ε123 = 1. In addition, we have to touch upon the transformation property of the components of the inertia tensor. Let A˜ ab (x) = ea · Ax (eb ). Then, from Ahx = hAx h−1 (see (1.141)), we obtain A˜ ab (hx) =



hac A˜ cd (x)hbd ,

h = (hab ) ∈ SO(3).

(1.174)

c,d

The infinitesimal transformation of this equation for h = et R(ec ) proves to be given by Jc (A˜ ab ) = ea · [R(ec ), Ax ](eb ) =



εcda A˜ db +



εcdb A˜ ad .

(1.175)

The basis, Ja and Xi , of the tangent vector fields is shown to satisfy the following commutation relations:  [Ja , Jb ] = − εabc Jc , (1.176a)  [Xi , Xj ] = − Fijc Jc , (1.176b) [Xi , Ja ] = 0,

(1.176c)

where Fijc is defined to be Fijc :=

∂γjc ∂q i

−

∂γic  − εabc γja γjb , ∂q j

(1.177)

a,b

and where in the course of calculation, Eq. (1.173) has been effectively used. Equation (1.176b) means that two independent vibrational vector fields, Xi and Xj , are coupled together to give rise to an infinitesimal rotation, if Fijc = 0. The quantities Fijc are called the curvature tensor and will be revisited in Sect. 1.10 (see (1.237)). Put another way, molecular vibrations cannot be separated from rotation, if Fijc = 0. Another implication is that the distribution spanned by {Xi } is not completely integrable in the sense of Frobenius [75], so that there are no submanifolds to which Xi are tangent.  We proceed to the metric ds 2 = mα dx α · dx α , which is decomposed into the sum of vibrational and rotational terms (see (1.148)). We start with the decomposition of the infinitesimal displacement, dx = (dx 1 , . . . , dx N ), by using

56

1 Geometry of Many-Body Systems

the local basis, ω˜ a and dq i , of the space of one-forms. A calculation results in dx =

3 

Ba ω˜ a +

3N−6 

a=1

Bi dq i ,

(1.178)

B αi dq i ,

(1.179)

i=1

or dx α =



B αa ω˜ a +



a

i

1 N where Ba = (B 1a , . . . , B N a ) and Bi = (B i , . . . , B i ) are defined to be

B αa = Ja x α = ea × x α , B αi = Xi x α =

(1.180a)

 ∂x α ∂x β  × xα , − A−1 mβ x β × x i ∂q ∂q i

(1.180b)

β

respectively. The system {Ba , Bi } forms a moving frame on the center-of-mass  ˙ From (1.179), the metric ds 2 = system Q. m dx α α · dx α is written out and α arranged as ds 2 =

 α

mα B αa · B αb ω˜ a ω˜ b +

 α

a,b

mα B αi · B αj dq i dq j ,

(1.181)

i,j

where use has been made of the fact that rotational vectors and vibrational vectors are orthogonal. As for the first term of the right-hand side of the above equation, one easily verifies that 

mα B αa · B αb = ea · Ax (eb ) = A˜ ab .

(1.182)

α

The second side of (1.181) is invariant under the SO(3)  term of the right-hand  action, α mα gB αi · gB αj = α mα B αi · B αj , so that it projects to the shape space ˙ which we denote by aij , M˙ to define a metric tensor on M, aij :=



mα B αi · B αj .

(1.183)

α

Thus, the metric ds 2 on Q˙ proves to be expressed as ds 2 =

 a,b

A˜ ab ω˜ a ω˜ b +

 i,j

aij dq i dq j .

(1.184)

1.8 Local Description of Spatial Many-Body Systems

57

Riemannian geometry of many-particle systems in terms of the moving frame Ba , Bi is found in [32].

1.8.3 Local Description in the Rotated Frame So far we have obtained the local description of the inertia tensor, the connection, and the metric in the space frame, in which the right-invariant vector fields and the right-invariant forms on SO(3) have been used. We can obtain another local description in the rotated frame, in which the left-invariant vector fields and the leftinvariant forms on SO(3) will be used. We choose here to describe the configuration x ∈ Q˙ in terms of the Jacobi vectors. The inertia tensor, Ax : R3 → R3 , is defined through Ax (v) =

N−1 

r k × (v × r k ) ,

v ∈ R3 ,

(1.185)

k=1

and the connection form ω is defined for x ∈ Q˙ to be    N−1  ωx = R A−1 r k × dr k . x

(1.186)

k=1

Needless to say, the transformation properties of Ax and ωx under the SO(3) action are the same as in the last section. Local coordinates are introduced through (1.157). On account of r k = gσ k (q), the connection form ωx given above with x = gσ (q) is written out and arranged as    gσ k (q) × d(gσ k (q)) ωgσ (q) = R A−1 gσ (q)    σ k (q) × (g −1 dgσ k (q) + dσ k (q)) g −1 = gR A−1 σ (q)       σ k (q)×( ×σ k (q)) g −1 + gR A−1 σ k (q)×dσ k (q) g −1 = gR A−1 σ (q) σ (q) = g(g −1 dg + ωσ (q) )g −1 ,

(1.187)

where use has been made of R() = g −1 dg and where  N−1    ωσ (q) := R A−1 σ k (q) × dσ k (q) . σ (q) k=1

(1.188)

58

1 Geometry of Many-Body Systems

We define ai (q) to be ai (q) = A−1 σ (q)



σk×

k

∂σ k  ·ea , ∂q i

a = 1, 2, 3, i = 1, . . . , 3N −6,

(1.189)

and use the components  a of the left-invariant one-form g −1 dg (see (1.93)) to put the connection form ω given by (1.187) in the form ωgσ (q) =

3 

a R(gea ),

a :=  a +

3N−6 

a=1

ai (q)dq i ,

(1.190)

i=1

where the rotated frame {gea } has been adopted in contrast to (1.169). We notice here that from the definitions (1.169) and (1.190), ω˜ a and a are related by ω˜ a =



gab b .

(1.191)

The horizontal (or vibrational) vector fields Yj are defined through a (Yj ) = 0,

dq i (Yj ) = δji ,

(1.192)

and shown to be expressed as  ∂ − aj (q)Ka , j ∂q 3

Yj =

j = 1, 2, · · · , 3N − 6,

(1.193)

a=1

where Ka , a = 1, 2, 3, denote the left-invariant vector fields on SO(3), which are dual to  a (see (1.98) and (1.95)). Like Xj given in (1.171), the Yj is called also the horizontal lift of ∂/∂q j , where Xj and Yj are equal to each other in spite of their different expressions. The dq i , a and the Yj , Ka form local basis of one-forms and of vector fields on π −1 (U ) ∼ = U × SO(3), respectively. They are dual to each other. The commutation relations among Yj are shown to be given by [Yj , Yj ] = −



κijc Kc ,

κijc :=

∂cj

c

∂q i

−

 ∂ci − εabc ai bj , ∂q j

(1.194)

where κijc are called also the curvature tensor and related to Fijc given in (1.177), like (1.191), by Fija =



gab κijb .

(1.195)

1.8 Local Description of Spatial Many-Body Systems

59

N−1

Like (1.184), we can express the metric ds 2 = dq i , a as ds 2 =



aij dq i dq j +

i,j



k=1

dr k · dr k in terms of

Aab a b ,

(1.196)

a,b

where the quantities aij and Aab are defined to be aij := ds 2 (Yi , Yj ),

(1.197)

Aab := ds 2 (Ka , Kb ) ,

(1.198)

ds 2 (Yi , Ka ) = 0,

(1.199)

respectively, and where

as will be verified shortly. We now find explicit expressions of aij and Aab by writing out the right-hand side of the above equations. Since g −1 dg(Ka ) = R(ea ), one has Ka (g) = gR(e a ), so that Ka r k = Ka (gσ k (q)) = Ka (g)σ k (q) = g(ea × σ k (q)). Hence, we find that the components of the metric tensor on M˙ are expressed and arranged as aij =

 (Yi r k ) · (Yj r k ) k

=

  ∂σ k k

∂q i

−

 a

  ∂σ   k b ai (ea × σ k ) · −  (e × σ ) . b k j ∂q j

(1.200)

b

The components of the inertial tensor are given by Aab =

  (Ka r k ) · (Kb r k ) = (ea × σ k ) · (eb × σ k ). k

(1.201)

k

Equation (1.199) is a consequence of the fact that vibrational vectors are orthogonal to rotational vectors. In fact, since Yj are vibrational vectors, they are orthogonal to the rotational vectors Ja (see (1.164b)) and since Ka are expressed as linear combinations of Jb (see (1.101)), Yj arealso orthogonal to Ka . Of course, Eq. (1.199) can be verified by calculating k dr k (Yi ) · dr k (Ka ) along with the   k ) · ea and (1.201). relations b Aab bi = ( σ k × ∂σ ∂q i In conclusion of this section, we give the transformation law for local expressions of the connection form and of the inertia tensor. Let σ  : U  → Q˙ be another local section with U  ∩ U = ∅. Then, there exists an SO(3)-valued function h(q) ∈ SO(3) such that σ  (q) = h(q)σ (q), q ∈ U  ∩ U . From (1.187), it then follows that ωσ  (q) = dhh−1 + hωσ (q) h−1 ,

(1.202)

60

1 Geometry of Many-Body Systems

which provides the transformation law ∂h −1 h + hi (q)h−1 , ∂q i

i (q) =

(1.203)

where ωσ  (q) =



i (q)dq i ,

ωσ (q) =



i

i (q)dq i .

(1.204)

i

Moreover, the transformation law for the inertia tensor A = (Aab ) is given by A = hAh−1 ,

(1.205)

where   A = Aab ,

Aab :=

 (ea × σ k ) · (eb × σ k ).

(1.206)

k

1.9 Spatial Three-Body Systems We introduce local coordinates for a spatial three-body system and express geometric quantities to show that vibrations gives rise to rotations, in particular. In terms of the Jacobi vectors, the center-of-mass system is expressed as Q = {(r 1 , r 2 )}. We now define a local section σ : U → Q˙ to be σ (q) = (σ 1 (q), σ 2 (q)),

σ 1 (q) = q1 e3 ,

σ 2 (q) = q2 e3 + q3 e1 ,

(1.207)

where we have put the first Jacobi vector σ 1 on the positive side of the e3 axis and the second Jacobi vector σ 2 on the +e1 side of the e3 –e1 plane. We have to restrict the range of (q1 , q2 , q3 ) to q1 > 0, q3 > 0 so that the Jacobi vectors σ 1 and σ 2 may be linearly independent. The triple (q1 , q2 , q3 ) forms a local coordinate system on ˙ A generic point of π −1 (U ) is described an open subset U of the shape space M. as gσ (q) with g ∈ SO(3) expressed as g = eφR(e3 ) eθR(e2 ) eψR(e3 ) in terms of the Euler angles (φ, θ, ψ), so that a local coordinate system on π −1 (U ) is formed by (φ, θ, ψ, q1 , q2 , q3 ). For N − 1 = 2, Eqs. (1.201) and (1.207) are put together to result in ⎛ ⎜ Aσ (q) = ⎜ ⎝

q12 + q22

0

−q2 q3

0

q12 + q22 + q32

0

−q2 q3

0

q32

⎞ ⎟ ⎟. ⎠

(1.208)

1.9 Spatial Three-Body Systems

61

The inverse of Aσ (q) is easily found to be ⎛

A−1 σ (q)

q2 1 0 2 ⎜ q2 q ⎜ 1 1 q3 ⎜ 1 ⎜ 0 =⎜ 0 2 q1 + q22 + q32 ⎜ ⎜ q12 + q22 ⎝ q2 0 q12 q3 q12 q32

⎞ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

(1.209)

The connection form at σ (q) is then found from (1.188) to be 2   q dq − q dq  2 3 3 2 ωσ (q) = R A−1 σ (q) × dσ (q) = 2 R(e2 ). k k σ (q) 2 + q2 q + q 1 2 3 k=1

(1.210)

The connection form at a generic point x = gσ (q) is evaluated by using (1.190). We are now in a position to write down the differential equation for a vibrational motion of the three-body system in terms of the local coordinates. Since a necessary and sufficient condition for a curve x(t) = g(t)σ (q(t)) to be vibrational is ωx(t )(x(t)) ˙ = 0, Eqs. (1.187) and (1.210) are put together to provide the differential equation for a vibrational motion in the form g −1 g˙ + ωσ (q) (q) ˙ = g −1

q2 q˙3 − q3 q˙2 dg + 2 R(e2 ) = 0. dt q1 + q22 + q32

(1.211)

If q(t) is given, the above differential equation linear in g is easy to integrate at least theoretically. We choose a simple closed curve q(t) given by q1 (t) = c,

q2 (t) = a cos t,

q3 (t) = a sin t,

(1.212)

where a, b are constants. Then, Eq. (1.211) becomes   a2 dg , = −gR 2 e 2 dt c + a2 which is easily integrated to give   g(t) = g(0) exp − tR

 a2 . e 2 c2 + a 2

(1.213)

From this, it follows that the change in g associated with the closed curve q(t) given   2πa 2  in (1.212) is evaluated as g(0)−1 g(2π) = exp − R 2 e 2 . This gives the c + a2 −2πa 2 angle, 2 , gained by the vibrational motion of the three-body system along c + a2

62

1 Geometry of Many-Body Systems

with a deformation of shape with the same initial and final shapes. Thus, any rotation angle about the e2 -axis can be realized, if the parameters a, c are chosen suitably and if t is replaced by −t. We now inquire why the rotation is made about the e2 -axis by vibrational motions of a three-body system in R3 . To this end, we have only to show that the plane spanned by three particles is constant during vibrational motion. Let r 1 and r 2 be the linearly independent Jacobi vectors, r k = gσ k , k = 1, 2. Then, the unit normal to the plane is given by n=

r1 × r2 q1 q3 ge2 = = ±ge2 , r 1 × r 2  |q1 q3 |

where the three particles take no collinear configuration, i.e., q1 q3 = 0. By using (1.211), the time-derivative of n is evaluated as dg dn q2 q˙3 − q3 q˙2 = ± e2 = ∓ 2 gR(e 2 )e2 = 0. dt dt q1 + q22 + q32 This implies that n is constant in t, and thereby the vibrational motion takes place on a fixed plane. So far we have obtained the local expression of the inertia tensor, its inverse, and the connection form for the spatial three-body system. We now find a local expression of the curvature and the metric. From (1.210), the vectors λi =   ai ea , i = 1, 2, 3, associated with the connection form ωσ (q) = R(λi )dq i are given by λ1 = 0,

λ2 = −

q12

q3 e2 , + q22 + q32

λ3 =

q12

q2 e2 , + q22 + q32

respectively. From (1.194) and (1.214) with i = R(λi ), the vectors κ ij = associated with the curvature tensor are calculated as κ 11 = κ 22 = κ 33 = 0, κ 23 = −κ 32 =

2q12 (q12

+ q22

+

e2 , q32 )2

κ 12 = −κ 21 = κ 31 = −κ 13 =

(q12 (q12

(1.214) 

κija ea

2q1q3 e2 , + q22 + q32 )2 2q1 q2 e2 . + q22 + q32 )2 (1.215)

Now that we have verified that κijc = 0, we turn about to look at (1.194), which means that the interference between infinitesimal vibrations results in an infinitesimal rotations. Hence, for the spatial three-body system, infinitesimal vibrations indeed interfere together to give rise to an infinitesimal rotation. In fact, we have already obtained such an example on the level of finite rotations (see (1.213)).

1.9 Spatial Three-Body Systems

63

As for the metric tensor on the shape space, by the use of (1.200), we find the expression of the metric tensor and further of its inverse, respectively, as ⎛

1

0 q12 + q22

⎞ 0 ⎟ q2 q3 ⎟ ⎟ q12 + q22 + q32 ⎟ , ⎟ q12 + q32 ⎠

⎜   ⎜ ⎜0 aij = ⎜ q12 + q22 + q32 ⎜ ⎝ q2 q3 0 2 q1 + q22 + q32 q12 + q22 + q32 ⎞ ⎛ 1 0 0 2 ⎟ ⎜ 2   ⎜0 q1 + q3 − q2 q3 ⎟ ⎟ ⎜ ij a =⎜ q12 q12 ⎟ . ⎟ ⎜ ⎝ q2 q3 q12 + q22 ⎠ 0 − 2 q1 q12

(1.216)

(1.217)

In the rest of this section, we give another local section, with respect to which the inertia tensor takes a diagonal matrix form. We choose the local section defined to be  χ χ  ψ ψ σ 1 (q) = ρ cos cos e1 − sin sin e2 , 2 2 2 2  ψ ψ χ χ  σ 2 (q) = ρ sin cos e1 + cos sin e2 , 2 2 2 2

(1.218a) (1.218b)

where (q i ) = (ρ, ψ, χ) are local coordinates of the shape space R3+ , as is shown in (1.219). The Jacobi vectors r 1 = gσ 1 (q), r 2 = gσ 2 (q) were used in a paper [51] with ψ/2 replaced by ψ. From r k = gσ k , k = 1, 2, together with (1.218), we easily verify that the SO(3) invariants given in (1.127) and (1.128) are written out as |r 1 |2 + |r 2 |2 = ρ 2 ,

(1.219a)

|r 1 |2 − |r 2 |2 = ρ 2 cos ψ cos χ,

(1.219b)

2r 1 · r 2 = ρ 2 sin ψ cos χ,

(1.219c)

2|r 1 × r 2 | = ρ 2 sin χ.

(1.219d)

where ρ > 0,

0 ≤ ψ ≤ 2π,

0 0. From this, one has dθ/dt = L/(mr 2 ). Then, the time derivatives (5.6) of er and eθ are put in the form L der = eθ , dt mr 2

L deθ = − 2 er , dt mr

(5.9)

respectively. Since dθ/dt = L/(mr 2 ), Eq. (5.7) is also rewritten as dr dr L = er + eθ . dt dt mr

(5.10)

Differentiating further this equation with respect to t, and using the formula (5.9) together with the fact that L is conserved, we obtain  d 2r d 2r L2  = − er . dt 2 dt 2 m2 r 3

(5.11)

This equation and the equation of motion (5.1) with (5.4) are put together to provide the reduced equation of motion, m

 d 2r dt 2

−

L2  = f (r). m2 r 3

(5.12)

140

5 Appendices

We again refer to Kepler’s first law which states that the orbit is an ellipse. We wish to derive a second-order differential equation to which r(t) is subject along the orbit. As is well known, an ellipse with the semi-major axis a and the semi-minor axis b is expressed, in terms of the planar polar coordinates (r, θ ) with one of its foci at the origin, as b2 /a r= , 1 + ε cos θ

 ε=

1−

b2 , a2

(5.13)

where ε is the eccentricity of the ellipse. Differentiating (1 + ε cos θ )r = b2 /a with respect to t, multiplying the resultant equation by r, and using (5.8), we obtain −ε sin θ

L b2 dr dr dθ 2 r + (1 + ε cos θ )r = −ε sin θ + = 0. dt dt m a dt

Further, differentiating this with respect to t, we obtain −ε cos θ

b2 d 2 r dθ L + = 0. dt m a dt 2

Multiplying this equation by r 2 and arranging the resultant equation, we obtain  L2  b2 L2 dθ L b2 2 d 2 r 2 r = ε cos θ − 1 = εr cos θ = . a dt 2 dt m m2 ar m2 From this, it follows that the trajectory r(t) along the ellipse is subject to the differential equation d 2r L2 aL2 = − . dt 2 m2 r 3 b 2 m2 r 2

(5.14)

Comparison of (5.12) and (5.14) results in f (r) = −

aL2 1 . b2m r 2

(5.15)

Though we have obtained the above result on the assumption that the orbits in question are ellipses in accordance with Kepler’s first law, we have not imposed the condition ε < 1 for the conic section to be an ellipse, in the manipulation of (5.13) into (5.14). This means that the result (5.15) holds true not only for ε < 1 but also for ε = 1, ε > 1. Put another way, the result (5.15) holds true for all conic sections as orbits of a celestial body.

5.1 Newton’s Law of Gravitation, Revisited

141

What the third law means Equation (5.15) seems to state that the force acting on the planet is an attractive force and its magnitude is proportional to r −2 . However, this expression of f depends on the parameters a, b and the constant of motion L, which depends on initial conditions for the motion. We have to express the force independently of such constants, in order to obtain a force law. To this end, we apply Kepler’s third law, which is concerned with the orbital period in the case of ε < 1. Then, we have to describe the period in terms of the orbit parameters depending on initial conditions. Since the area swept by the radius vector during the period T is equal to the area enclosed by the ellipse, we have 

T

πab =

L dS dt = T, dt 2m

0

where we have used (5.2). Thus, the period is expressed as T =

2mπab . L

(5.16)

Hence, Kepler’s third law is put in the form T =

2mπab = γ a 3/2, L

γ = const.,

where γ is independent of orbits or takes the same value for all planets. Then, the 2 4mπ 2 coefficient of −r −2 in the right-hand side of Eq. (5.15) is written as baL , 2m = γ2 which is a constant independent of orbits. Thus, the force F is found to be described as F =−

4mπ 2 r . γ 2 r3

A similar statement to that given in the paragraph after Eq. (5.15) is valid, i.e., the present result holds true even if the orbit of a celestial body is an ellipse, a parabola, or a hyperbola. The universal law of gravitation According to Newton’s third law (action–reaction law), the Sun is attracted to the planet by a force of the same magnitude. Since the force F is proportional to the mass of the planet, under the symmetric consideration, it should also be proportional to the mass, M, of the Sun. Hence, there exists a universal constant G such that 4π 2 = GM. It then turns out that F is put in the form of the universal law of γ2 gravitation, F = −G

mM r . r2 r

(5.17)

142

5 Appendices

Once the force of this form is given, the equation of motion (5.1) can be easily integrated to provide conic sections (5.13) as orbits, by using the well-known constants of motion, the angular momentum and the Laplace–Runge–Lenz vector as well as the total energy [21].

5.2 Principal Fiber Bundles The definition of a principal fiber bundle is given in [48], in which the group action is “to the right”. In contrast to this, the group action adopted in the present book is “to the left”. The definition given below is modified according to the left action of the group. Let P and M be manifolds and G a Lie group. The P is called a principal fiber bundle over M with G a structure group (or for short, a G-bundle), if the following conditions are satisfied: (i) G acts on P to the left without fixed point. Put another way, a differentiable map is given G × P −→ P ,

(g, p) → g · p

(5.18)

in such a manner that (gh)p = g(hp),

g, h ∈ G, p ∈ P ,

(5.19a)

if a · p = p, then a = e, the identity of G.

(5.19b)

(ii) There exists a surjective map (or a projection) π : P → M satisfying π(gp) = π(p); 

(5.20a) 

if π(p) = π(p ), then there exists uniquely g ∈ G such that p = gp. (5.20b) Put another way, M is the quotient space of P by the equivalent relation induced by G, and the projection π is differentiable. (iii) P is locally trivial. In other words, for each point x of M, there exists an open subset U such that π −1 (U ) is isomorphic with U × G in the sense that p ∈ π −1 (U ) → (π(p), φ(p)) ∈ U × G is a differentiable isomorphism such that φ(ap) = aφ(p) for every a ∈ G. The condition (iii) of the definition of the principal fiber bundle P leads to the existence of an open covering {Uα } of M such that p ∈ π −1 (Uα ) → (π(p), φα (p)) ∈ Uα × G

5.3 Spatial N-Body Systems with N ≥ 4

143

is an isomorphism. In this setting, a map σα : Uα → π −1 (Uα ) is called a local section if it satisfies π(σα (x)) = x,

φα (σα (x)) = e.

Let Gp be the subspace of the tangent space Tp (Q) consisting of vectors tangent to the fiber through p. According to [48] with modification of the group action from right action to left action, a connection in P is an assignment of a subspace Qp of Tp (P ) to each p ∈ P such that: (i) Tp (P ) = Gp ⊕ Qp , (ii) Qgp = (Lg )∗ Qp , p ∈ P , g ∈ G, where Lg p = gp, (iii) Qp depends differentiably on p, where (Lg )∗ denotes the differential map of Lg . In a dual manner, a connection form ω is defined to be a one-form on P which satisfies: (i) ω(ξP ) = ξ, ξ ∈ g, (ii) (Lg )∗ ω = Adg ω.

ξP :=

d tξ dt e g|t =0 ,

If G acts on P to the right, the above conditions (i) and (ii) are replaced by d get ξ |t =0 and (Rg )∗ ω = Adg −1 ω, respectively. ξP = dt We now make a comment on curvature. According to [48], the curvature form  on the principal bundle P is defined to be 1 (X, Y ) = dω(X, Y ) + [ω(X), ω(Y )], 2

X, Y ∈ Tp (P ),

p ∈ P.

In contrast to this, the definition given in [64] is expressed as (X, Y ) = dω(X, Y ) + [ω(X), ω(Y )]. The definitions given above differ in the second terms of the respective right-hand sides, which is due to the convention of the exterior product of differential forms. Alternatively, the curvature form is expressed as  = dω + ω ∧ ω. However, in this book, the curvature form is defined to be  = dω − ω ∧ ω, as is given in (1.234). The two definitions differ in the respective second terms of the right-hand sides. The difference depends on whether the group action is to the right or to the left.

5.3 Spatial N -Body Systems with N ≥ 4 The space of O(3) invariants A way to study the shape space for the spatial N-body system with N ≥ 4 is to look for invariants under the SO(3) action. To this end, we start by finding invariants

144

5 Appendices

under the O(3) action. Let X = (r 1 , · · · , r N−1 ) ∈ R3×(N−1) be a matrix consisting of Jacobi vectors. Then, as is easily seen, the matrix ⎞ |r 1 |2 · · · r 1 · r N−1 ⎜ r 2 · r 1 · · · r 2 · r N−1 ⎟ ⎟ ⎜ Z = XT X = ⎜ ⎟ .. .. ⎠ ⎝ . . 2 r N−1 · r 1 · · · |r N−1 | ⎛

is O(3) invariant. The first thing to note is the fact that rankX = rankZ. In fact, since rankZ = rank(XT X) = dim(Im(XT |ImX ) ≤ dim(ImX) = rankX, we have the inequality rankZ ≤ rankX. On the other hand, for X ∈ R3×(N−1) and Z ∈ R(N−1)×(N−1) , one has rankX + dim(KerX) = rankZ + dim(KerZ) = N − 1. We show that KerZ ⊂ KerX. For x ∈ KerZ, one has Zx = XT Xx = 0, which leads to Xx · Xx = XT Xx · x = 0. Thus, one has Xx = 0, or equivalently, x ∈ KerX. It then follows that rankX ≤ rankZ. Thus, we come to the conclusion that rankX = rankZ. The matrix Z is a symmetric and positive-semi-definite matrix, of course. Since X is of rank less than or equal to three, so is Z. Now, we set S = {S ∈ R(N−1)×(N−1) ; S T = S, S ≥ 0, rankS ≤ 3},

(5.21)

where S ≥ 0 is a symbol meaning that S is positive-semi-definite. The relations among the center-of-mass system Q, the shape space M, and S are shown by the diagram Q | | ↓ S

π " M = Q/SO(3),

(5.22)

$ φ

where the map φ : M → S is defined to be φ(π(X)) = XT X.

(5.23)

5.3 Spatial N-Body Systems with N ≥ 4

145

Note that the map φ is well-defined, since φ(π(gX)) = XT X for g ∈ SO(3). The parity operator P on M is defined to be P π(X) = π(−X),

X ∈ Q.

Proposition 5.3.1 The map φ : M → S is a two-sheeted covering, in general. To be precise, if φ(π(X)) = φ(π(X )) then π(X ) = P π(X) or π(X ) = π(X) for X and X with rankX = rankX = 3, but π(X ) = π(X) for X and X with r = rankX = rankX and 0 ≤ r ≤ 2. Proof Suppose π(X) and π(X ) have the same target, φ(π(X)) = φ(π(X )). Let rankX = rankX = r ≤ 3. Let λj , j = 1, . . . , N − 1, be the eigenvalues of XT X = X T X with λ1 ≥ · · · ≥ λr > λr+1 = · · · = λN−1 = 0, and v 1 , · · · , v r , v r+1 , · · · , v N−1 the orthonormal eigenvectors associated with the eigenvalues λj , j = 1, . . . , N − 1, respectively. We denote by vjα the components of v j ∈ RN−1 , α, j = 1, . . . , N − 1, and define the vectors x k ∈ R3 , by using X = (r 1 , · · · , r N−1 ), to be xk =

N−1 

vkα r α ,

k = 1, . . . , N − 1.

α=1

The inner products among these vectors are xk · x =

N−1 

β

vkα v r α · r β =

α,β=1

N−1 

β

vkα Zαβ v =

α,β=1

N−1 

vkα λ vα = λ δk .

α=1

In particular, one has x  · x  = λ . Since λ = 0 for  ≥ r + 1, we find that x  = 0 for  ≥ r + 1. For k = 1, . . . , r, we introduce the vectors eˆ k = √1λ x k , k

which form an orthonormal system in R3 . In a similar manner, by using r α from X , we can define another orthonormal system eˆ k , k = 1, . . . , r. For both orthonormal systems, there exists h ∈ O(3) such that hˆek = eˆ k ,

k = 1, . . . , r.

If 0 ≤ r ≤ 2, then h can be chosen from SO(3), but if r = 3, h cannot be restricted to SO(3). Incidentally, the Jacobi vectors r α (resp. r α ) can be described in terms of eˆ k (resp. eˆ k ). In fact, we define the following vectors and arrange them to obtain r 4  k=1

β

λk vk eˆ k =

N−1 r  α=1 k=1

β

vk vkα r α =

N−1  N−1  α=1 k=1

β

vk vkα r α =

N−1  α=1

δβα r α = r β ,

146

5 Appendices

where we have used the fact that x k = 0 for k > r and the fact that {v j } form an orthonormal system in RN−1 , which is alternatively expressed as V T V = I and V V T = I in terms of V = (v 1 , · · · , v N−1 ). In the same manner, we obtain r √ β  ˆ k = r β . It then follows that k=1 λk vk e hr α = r α ,

α = 1, . . . , N − 1,

h ∈ O(3),

/ SO(3), then π(hX) = π(−X) = or equivalently, hX = X . If h ∈ O(3) and h ∈ P π(X). If h ∈ SO(3), then π(hX) = π(X), of course. The last task to do is to show that φ is surjective. Let S ∈ S be a positivesemi-definite symmetric matrix of rank less than or equal to three. We denote the eigenvalues of S by λ1 ≥ · · · ≥ λr > λr+1 = · · · = λN−1 = 0 and the associated orthonormal eigenvectors by u1 , · · · , ur , ur+1 , · · · , uN−1 , respectively. Then, S is expressed as S = (u1 , · · · , ur , ur+1 , · · · , uN−1 )(u1 , · · · , ur , ur+1 , · · · , uN−1 )T , where  is the diagonal matrix with entries λi δij , i, j = 1, . . . , N − 1. On denoting the components of uk by uαk , the components of S are expressed as Sαβ =

N−1 

β

uαk λk uk =

k=1

r 

β

uαk λk uk .

k=1

We now define the Jacobi vectors to be rα =

r 4 

λk uαk eˆ k ,

α = 1, . . . , N − 1.

k=1

Then, the inner products among r α are calculated as rα · rβ =

r 

β

uαk λk uk = Sαβ .

k=1

This means that there exists a system of Jacobi vectors X = (r 1 , · · · , r N−1 ) such that XT X = S. Hence, φ proves to be surjective. This completes the proof. The shape space for the four-body system According to the rank of the configuration of the Jacobi vectors, the center-of-mass system Q is decomposed into Q = %3k=0 Qk ,

Qk := {X = (r 1 , · · · , r N−1 )| rank(X) = k}.

(5.24)

5.3 Spatial N-Body Systems with N ≥ 4

147

On account of Proposition 5.3.1, the factor spaces Qk /SO(3) for k = 0, 1, 2, are homeomorphic to the subspace, Sk , of S consisting of positive-semi-definite matrices of rank k, but the factor space Q3 /SO(3) is a double cover of the space, S3 , of positive-definite matrices of rank three. Clearly, S0 = {0}. We now consider the S1 . Any positive-semi-definite symmetric matrix S ∈ R(N−1)×(N−1) of rank one is put in the form v 1 λ1 v T1 , where λ1 is a positive eigenvalue of S, and v 1 ∈ RN−1 is the associated eigenvector. Since v 1 λ1 v T1 = (−v 1 )λ1 (−v 1 )T , one has S1 = Q1 /SO(3) ∼ = R+ × RP N−2 . Before attending to S2 and S3 , we recall the fact that the space of positive-semidefinite symmetric matrices forms a convex cone in the space of symmetric matrices. In fact, for any positive-semi-definite symmetric matrix A and any positive number μ, μA is also a positive-semi-definite symmetric matrix and further for positivesemi-definite symmetric matrices A, B ∈ Rn×n , one verifies that x · (λA + (1 − λ)B)x ≥ 0 for 0 ≤ λ ≤ 1, x ∈ Rn , which means that λA + (1 − λ)B is positivesemi-definite as well. In what follows, we confine ourselves to the center-of-mass system Q of spatial four bodies. Then, the S given in (5.21) with N = 4 is the set of positive-semi˜ of definite symmetric 3 × 3 matrices. Hence, S is a convex cone in the space, S, 3 × 3 symmetric matrices, where S˜ is homeomorphic to R6 . Let ˜ trS = 1}. B = {S ∈ S; Then, the intersection, S ∩ B, of the convex cone and a hyperplane in R6 is home5 omorphic to D , the five-dimensional closed disk. Thus, S − {0} is homeomorphic 5 to R+ × D . The subspace R+ × D 5 corresponds to the set of positive-definite symmetric 3 × 3 matrices, and the boundary R+ × S 4 to the set of non-positivedefinite symmetric 3 × 3 matrices except for {0} [65]. It then turns out that the S3 and the union S1 ∪ S2 are homeomorphic to R+ × D 5 and to R+ × S 4 , respectively. The double of S3 is homeomorphic to Z2 × R+ × D 5 ∼ = R+ × (S 5 − S 4 ), where D 5 is doubled to be the northern and the southern hemispheres of S 5 and where S 4 denotes the equator of S 5 . In the boundary of R+ × (S 5 − S 4 ), S2 ∪ S1 is realized so as to be S2 ∼ = R+ × (S 4 − RP 2 ) and S1 ∼ = R+ × RP 2 . The boundary of S1 is S0 = {0}. In particular, the subspace of positive-semi-definite symmetric matrices of rank less than or equal to two, S2 ∪ S1 ∪ S0 , is the linear space homeomorphic R5 . Proposition 5.3.2 The shape space for the spatial four-body system is stratified into Q3 /SO(3) ∼ = R+ × (S 5 − S 4 ), Q2 /SO(3) ∼ = R+ × (S 4 − RP 2 ), Q1 /SO(3) ∼ = R+ × RP 2 , Q0 /SO(3) ∼ = {0}.

148

5 Appendices

In particular, the shape space of non-singular configurations and the total shape space are diffeomorphic, respectively, to ˙ Q/SO(3) = Q3 /SO(3) ∪ Q2 /SO(3) ∼ = R+ × (S 5 − RP 2 ), ˙ Q/SO(3) = Q/SO(3) ∪ Q1 /SO(3) ∪ Q0 /SO(3) ∼ = R6 . Now that we have identified the shape space for the spatial four-body system, we proceed to SO(3)-invariants in order to describe the shape space. The six quantities r α · r β with 1 ≤ α ≤ β ≤ 3 are O(3)-invariants. As an SO(3) invariant, we take r 1 · (r 2 × r 3 ). According to Littlejohn–Reinsch [52], the following quantities (v, w1 , · · · , w5 ) are adopted as coordinates of the shape space Q/SO(3) ∼ = R6 , √ √ 3 (|r 1 |2 − |r 2 |2 ), w2 = 3r 1 · r 2 , v = r 1 · (r 2 × r 3 ), w1 = 2 √ √ 1 w3 = 3r 2 · r 3 , w4 = 3r 3 · r 1 , w5 = (−|r 1 |2 − |r 2 |2 + 2|r 3 |2 ). 2 (5.25) In the rest of this section, we will touch on the shape space of the N-body system in general. We begin with a review of the singular-value decomposition of matrices. The singular-value decomposition of matrices Lemma 5.3.1 For a matrix A ∈ Rm×n of rank r, there exist orthogonal matrices U ∈ O(m) and V ∈ O(n) such that A = U #V T ,

#=

  S0 ∈ Rm×n , 00

(5.26)

where S = diag(σ1 , · · · , σr ) with σ1 ≥ σ2 ≥ · · · ≥ σr > 0. Proof Since AT A is a positive-semi-definite symmetric matrix, it has non-negative eigenvalues, which we denote by σ12 , · · · , σn2 , where σj ≥ 0 are called the singular values of A and arranged in descending order, σ1 ≥ σ2 ≥ · · · ≥ σr > 0 = σr+1 = · · · = σn . We denote by v 1 , · · · , v r , v r+1 , · · · , v n the normalized eigenvectors associated with σj2 , respectively. Here, we note that the latter orthonormal vectors v r+1 , · · · , v n are subject to freedom of choice. Put another way, any orthonormal vectors v r+j = n−r k=1 v r+k hkj with (hkj ) ∈ O(n − r) can be adopted in place of v r+j , j = 1, · · · , n − r. Let V1 = (v 1 , · · · , v r ),

V2 = (v r+1 · · · , v n ),

V = (V1 , V2 ),

5.3 Spatial N-Body Systems with N ≥ 4

149

where V1 and V2 satisfy V1T V1 = Ir ,

V2T V2 = In−r ,

respectively, and where Ir and In−r denote the r ×r and the (n−r)×(n−r) identity matrices, respectively. Then, by the definition of eigenvectors, we have AT AV1 = V1 S 2 ,

AT AV2 = 0,

S = diag(σ1 , · · · , σr ).

By multiplying V1T and V2T to the left of the first and the second of the above equations, respectively, and by arranging the resulting equations, we obtain (AV1 S −1 )T AV1 S −1 = Ir ,

(AV2 )T AV2 = 0,

respectively. The first of the above equations means that U1 := AV1 S −1 is a collection of orthonormal vectors in Rm , and the second implies that AV2 = 0. Let U2 be a collection of orthonormal vectors in Rm such that U = (U1 , U2 ) ∈ O(m), where the choice of U2 is not unique, and subject to the transformation O(m − r). Now, by multiplying U T and V to the left and the right of A, respectively, we obtain  T    U1 AV1 U1T AV2 S0 T U AV = = = #, T T U2 AV1 U2 AV2 00 where we have used the fact that AV1 = U1 S,

AV2 = 0,

U2T U1 = 0.

Thus, A is decomposed into A = U #V T . This completes the proof. Stratification of the shape space Before attending to Eq. (5.26), we refer to Steifel manifolds. The space of r-frames in Rm is called a Stiefel manifold and denoted by Vr (Rm ) or V (m, r) [64], and further realized as Vr (Rm ) = O(m)/O(m − r) = SO(m)/SO(m − r). Eq. (5.26) is rewritten as A = U1 SV1T ,

U1T U1 = Ir ,

S > 0,

V1T V1 = Ir ,

where U1 and V1 belong to the Stiefel manifolds Vr (Rm ) and Vr (Rn ), respectively, and S is a positive-definite symmetric matrix. However, such a decomposition is not unique. In fact, one obtains another decomposition A = (U1 h−1 )hSh−1 (V1 h−1 )T ,

h ∈ O(r),

150

5 Appendices

where hSh−1 is an r × r positive-definite symmetric matrix, and where U1 h−1 and (V1 h−1 )T are subject to the conditions (U1 h−1 )T U1 h−1 = Ir and (V1 h−1 )T (V1 h−1 ) = Ir , respectively. It then turns out that in general, any linear map φ : Rn → Rm of rank r is decomposed into the composition φ = ι ◦ σ ◦ π of linear maps, π

σ

ι

Rn −→ Rr −→ Rr −→ Rm , where π : Rn → Rr and ι : Rr → Rm are subject to the conditions π ◦ π T = idRr and ιT ◦ ι = idRr , respectively, and σ : Rr → Rr is a positive-definite symmetric map. We denote by Rm×n (r) and S + (r) the set of m × n real matrices of rank r and the set of r × r positive-definite real symmetric matrices. Then, the above decomposition of φ implies that Rm×n (r) ∼ =

Vr (Rm ) × S + (r) × Vr (Rn ) , O(r)

where the equivalence relation by O(r) is defined to be (ι, σ, π) ∼ (ιh−1 , hσ h−1 , hπ),

h ∈ O(r).

We apply the decomposition of Rm×n (r) in order to stratify the center-of-mass system. The center-of-mass system Q of the spatial N-body system is viewed as the set of 3 × (N − 1) real matrices consisting of Jacobi column vectors and stratified according to the rank of such matrices, so that we obtain the stratification R3×(N−1) =

5 0≤r≤min(3,N−1)

Vr (R3 ) × S + (r) × Vr (RN−1 ) . O(r)

Accordingly, the shape space M = Q/SO(3) is stratified into the disjoint union of the factor spaces of respective strata by SO(3) [80]. SO(3) invariants For the four-body system, we have already found coordinates (5.25) of the shape space. In the rest of this section, without reference to the detailed topology of the shape space, we will give coordinates of the shape space for the N-body system with N ≥ 4, which will amount to finding invariants of SO(3). According to (1.119) or (5.24), the shape space is stratified into ˙ M = M0 ∪ M1 ∪ M, where Mk = Qk /SO(3), k = 0, 1, 2, 3,

M˙ = M2 ∪ M3 .

5.3 Spatial N-Body Systems with N ≥ 4

151

The space M0 is a singleton. We turn to M1 = Q1 /SO(3). Let X = (r 1 , · · · , r N−1 ) ∈ Q1 . Since rankX = 1, there are numbers ξk and a unit vector u such that r k = ξk u,

k = 1, . . . , N − 1,

where there is a number 0 ≤  ≤ N − 1 such that ξ > 0 with u being replaced by −u if necessary. Then, X is arranged as X = (r 1 , · · · , r N−1 ) = uξ

ξ

1

ξ

,··· ,

ξ−1 ξ+1 ξN−1  , , 1, ,..., ξ ξ ξ

where u ∈ R3×1 and (ξ1 /ξ , · · · , ξN−1 /ξ ) ∈ R1×(N−1) and where ξ = |r  | > 0, ξk = (r k · r  )/|r  |, which are SO(3) invariant. Thus, we have found the local coordinates ξk /ξ , k =  of M1 ∼ = R+ × RP N−2 . ˙ Since rank(r 1 , · · · , r N−1 ) ≥ 2, there We proceed to find local coordinates on M. exist linearly independent vectors r i , r j with i = j . We may take i = 1, j = 2 without loss of generality. Then, there exist orthonormal vectors u1 , u2 such that r 1 = ξ1 u1 ,

r 2 = ξ2 u1 + ξ3 u2 ,

ξ1 > 0, ξ3 > 0.

(5.27)

We introduce u3 by u3 = u1 × u2 . Then, each r k can be expressed as a linear combination of ua , a = 1, 2, 3, so that there exist a matrix Y ∈ R3×(N−1) such that the configuration X is expressed as ⎞ ξ1 ξ2 ξ4 ξ3N−8 Y = ⎝ 0 ξ3 ξ5 · · · ξ3N−7 ⎠ , 0 0 ξ6 ξ3N−6 ⎛

X = (r 1 , · · · , r N−1 ) = (u1 , u2 , u3 )Y,

(5.28)

where ξj ∈ R, j = 1, 2, . . . , 3N − 6. As is easily seen, the quantities ξj are SO(3) invariant. These quantities can be expressed as functions of the Jacobi vectors, as will be shown below. First, from (5.27), we easily obtain ξ1 = |r 1 |,

ξ2 =

r1 · r2 , |r 1 |

ξ3 =

|r 1 × r 2 | . |r 1 |

Further calculation shows that the unit vectors uk are described as u1 =

r1 , |r 1 |

u2 =

r 1 × (r 2 × r 1 ) , |r 1 × (r 2 × r 1 )|

u3 =

r1 × r2 , |r 1 × r 2 |

respectively. To deal with ξj , j ≥ 4, we refer to the definition of Y , according to which, the Jacobi vectors r  , 3 ≤  ≤ N − 1, are expressed as r  = ξ3−5 u1 + ξ3−4 u2 + ξ3−3 u3 ,

3 ≤  ≤ N − 1.

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5 Appendices

Now it is straightforward to verify that ξ3−5 =

r1 · r , |r 1 |

ξ3−4 =

(r  × r 1 ) · (r 2 × r 1 ) , |r 1 ||r 2 × r 1 |

ξ3−3 =

(r 1 × r 2 ) · r  . |r 1 × r 2 |

Our last task is to show that the map ξ : M → R3N−6 ,

π(X) −→ (ξ1 , . . . , ξ3N−6 )

is injective. Suppose that ξ(π(X)) = ξ(π(X )). Then, one has X = (r 1 , · · · , r N−1 ) = (u1 , u2 , u3 )Y, X = (r 1 , · · · , r N−1 ) = (v 1 , v 2 , v 3 )Y, where Y is the matrix with entries ξj , as is given in (5.28), and where uk and v k are both orthonormal systems with u3 = u1 × u2 and v 3 = v 1 × v 2 . Since the orientation of both orthonormal systems are the same, there exist g ∈ SO(3) such that (v 1 , v 2 , v 3 ) = g(u1 , u2 , u3 ). Hence, one obtains π(X) = π(X ), which shows that the map ξ is injective. It then ˙ turns out that (ξ1 , · · · , ξ3N−6 ) serve as local coordinates of the shape space M. So far we have studied the shape space for many-body systems with SO(3) action. See [76] for shape spaces under O(3) action. In addition, it is worth mentioning that apart from many-body mechanics, shape spaces form an interesting subject in statistical science [47].

5.4 The Orthogonal Group O(n) Before studying many-body systems in Rn , we present a review of the orthogonal group O(n). The groups defined to be O(n) = {X ∈ Rn×n ; XT X = In }, SO(n) = {X ∈ Rn×n ; XT X = In , det X = 1} are called the orthogonal group and the rotation group, respectively. The condition XT X = In implies that X preserves the inner product on Rn . Put another way, for any x, y ∈ Rn , the equation Xx · Xy = x · y holds true, where the dot · indicates the standard inner product on Rn . It is easily seen that both O(n) and SO(n) form groups. These groups are also known as Lie groups, i.e., they are algebraically groups and topologically manifolds. We touch here on the difference between O(n)

5.4 The Orthogonal Group O(n)

153

and SO(n). From XT X = In , it follows that det X = ±1. This implies that O(n) consists of two subsets, one of which is determined by det X = 1 and the other by det X = −1 and no intersection of them exists. The subset determined by det X = 1 is refereed to as SO(n). 2 We now explain how O(n) can be viewed as a submanifold of Rn×n = Rn . Let X = (xij ). Then, the defining equations of O(n) are expressed as fij (X) =

n 

xki xkj − δij = 0,

1 ≤ i ≤ j ≤ n,

k=1

which impose n(n + 1)/2 conditions on Rn×n . We wish to show that the functions fij are indeed independent of one another. If they are independent, on account of the following proposition (see [15] for proof), O(n) is a submanifold of Rn×n . Proposition 5.4.1 Let f : Rm → Rk be a smooth map, and v0 ∈ Rk . If f −1 (v0 ) = ∅ and if Df (x) : Rm → Rk is surjective for all x ∈ f −1 (v0 ), then f −1 (v0 ) is an (m − k)-dimensional submanifold of Rm , where Df (x) denotes the derivative of f at x. In order to apply Proposition 5.4.1 with m = n2 and k = n(n + 1)/2, we proceed to the proof of the mutual independence of ∇fij on O(n), where ∇fij denotes the gradient of fij (X). To show the independence of the gradient vectors, we apply a lemma on Grammian matrices: Lemma 5.4.1 For r (r ≤ m) vectors a j = (akj ) and bj = (bkj ) of Rm with 1 ≤ k ≤ m, 1 ≤ j ≤ r, the determinant of the r × r matrix (a i · bj ) is given by  a ···   i.1 1 det(a i · bj ) =  .. · · ·  i1