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Progress in Mathematics Volume 149 Series Editors HymanBass Joseph Oesterle Alan Weinstein J.-M. Souriau Structure of Dynamical Systems A Symplectic View of Physics Translated by C. H. Cushman-de Vries Translation editors R. H. Cushman G. M. Tuynman Springer Science+Business Media, LLC Jean-Marie Souriau C. H. Cushman-de Vries UFR de Mathematiques Dordrecht Universite de Provence The Netherlands Marseille France Translation edited by R. H. Cushman and G. M. Tuynman Richard H. Cushman Gijs M. Tuynman Department of Mathematics UFR de Mathematiques University of Utrecht Universite des Sciences Utrecht et Technologies de Lille The Netherlands Villeneuve d'Ascq France Library of Congress Cataloging-in-Publication Data Souriau, J.-M. (Jean-Marie), [Structure des systemes dynamiques. English] Structure of dynamical systems : a symplectic view of physics I J. -M. Souriau ; translated by C. H. Cushman-de Vries ; R. H. Cushman, G. M. Tuynman, translation editors. p. cm. -- (Progress in mathematics ; v. 149) ISBN 978-1-4612-6692-1 ISBN 978-1-4612-0281-3 (eBook) DOI 10.1007/978-1-4612-0281-3 1. Mechanics. 2. Statistical mechanics. 3. Quantum theory. 4. Symplectic manifolds. 5. Mathematical physics. 1. Title. II. Series: Progress in mathematics (Boston, Mass.) : voI. 149. QCI25.2.S6813 1997 530.15' 6362--dc21 97-300 CIP Printed on acid-free paper (D !Il fl8l1 © 1997Sprlnger Science+Business Media New York Originally published by Birkhlluser Boston in 1997 Softcover reprint ofthe hardcover 1s t edition 1997 Copyright is not claimed for works of U.S. Government employees. AII rights reserved. No part ofthis publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhil.user Boston for libraries and other users registered with the Copyright Clearance Center (CCC) , provided thatthe base fee of$6.00 percopy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be addressed direct1y to Springer Science+Business Media, LLC. ISBN 978-1-4612-6692-1 Typset in @'TEX by R. H. Cushman. Drawings rendered in postscript by C. (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) 987 6 5 4 3 2 1 Editors' preface It is with great pleasure that we are able to provide the reader with a translation of Souriau's elassical text "Structure des Systemes Dy- namiques" on mechanics. We have added the subtitle "a symplectic view of physics", which is elose to the title first proposed by the author. Compared to the French version, we have made several changes in nota- tion for the English edition, all of which have the author's approval. The most important change is that the symbol V' for the exterior derivative of differential forms has been replaced by the symbol d. Consequently in order to avoid confusion, the symbols d, d', ... for vector fields have been replaced by 8, 8', .... We warn the reader that differential forms 2 are always given in terms of their action on tangent vectors and not in terms of a dual basis. We use this opportunity to thank C. Sacn§ of the University of Lille I for doing all the computer graphics. R.H. Cushman G.M. Tuynman Author's preface to the French edition I sincerely thank all those who have helped me with the design and the writing of this book by their remarks, their discussions, and their advice; in particular H. Bacry, P. Casal, R. Haag, F. Halbwachs, D. Kastler, A.A. Kirillov, 1. Michel, M. Zerner, and also J. Breuneval, J. Elhadad, and H.H. Fliche who, in the course of a seminar at the "Faculte des Sciences de Marseille", have checked a large part of the computations and arguments. I also thank P. Lelong, who has graciously accepted this book in the series "Dunod Universite", and I hope that all those whom I did not mention will forgive me. Jean-Marie Souriau Notre-Dame de Vaulx, september 1968. Table of Contents Introduction XVll I. Differential Geometry §1 . Manifolds The definition of a manifold .................................. 3 Open sets .................................................... 6 Differentiable maps ........................................... 6 The tangent space ................. : .......................... 7 Submanifolds ............................................... 10 Manifolds defined by an equation ............................ 12 Covering spaces ..................... '....................... 13 Quotient manifolds .......................................... 14 Connectedness .............................................. 14 Homotopy .................................................. 16 §2. Derivations Variables ................................................... 18 Vecbr fields and derivations ................................. 19 Derivations of linear operators ............................... 22 The image of a vector field .................................. 23 Lie brackets ................................................. 25 §3. Differential equations The exponential of a vector field ............................. 27 The image of a differential equation ......................... 28 The derivative of the exponential map ....................... 29 x Table oi Contents §4. Differential forms Covariant fields ............................................. 31 The inverse image of a covariant field ..\ ' ...................... 31 The Lie derivative ........................................... 33 Covariant tensor fields ...................................... 34 p-Forms ..................................................... 35 The exterior derivative ...................................... 36 §5. Foliated manifolds Foliations ................................................... 38 The quotient of a manifold by a foliation .................... 42 Integral invariants ........................................... 43 The characteristic foliation of a form ........................ 44 §6. Lie groups Actions of a Lie group on a manifold ........................ 46 The Lie algebra of a Lie group ............................... 48 Orbits ...................................................... 49 The adjoint representation .................................. 49 Lie subalgebras and Lie subgroups ........................... 51 The stabilizer ............................................... 53 Classical examples of Lie groups ............................. 5:3 Euclidean spaces ............................................ 55 Matrix realizations .......................................... 59 §7. The calculus of variations Classical variational problems ............................... 62 Canonical variables ......................................... 63 The Hamiltonian formalism ................................. 65 A geometrical interpretation of the canonical equations ...... 66 Transformations of a variational problem .................... 68 Noether's theorem .......................................... 68 Table oI Contents Xl II. Symplectic Geometry §8. 2-Forms Orthogonality ............................................... 73 Canonical bases ............................................. 74 The symplectic group ....................................... 77 §9. Sympleetie manifolds Symplectic and presymplectic manifolds ..................... 81 Symplectic structures arising from al-form .................. 84 Poisson brackets ............................................ 85 Induced symplectic structures ............................... 87 §10. Canonieal transformations Canonical charts ............................................ 90 Canonical transformations ................................... 93 Canonical similitudes ........................................ 94 Covering spaces of symplectic manifolds ..................... 96 Infinitesimal canonical transformations ...................... 97 §11. Dynamieal Groups The definition of a dynamical group ........................ 100 The cohomology of a dynamical group ...................... 104 The cohomology of a Lie group ............................. 107 The cohomology of a Lie algebra ........................... 108 Symplectic manifolds defined by a Lie group ................ 111 III. Mechanics §12. The geometrie strueture of classical meehanies Material points ............................................ 121 Systems of material points ................................. 122 Constraints ................................................ 122 Describing forces ........................................... 125 Xll Table oI Contents The evolution space ........................................ 126 Phase spaces and the space of motions ...................... 127 The Lagrange 2-form ....................................... 129 The Lagrange form for constrained systems ................. 131 Changing the reference frame ............................... 133 The principle of Galilean relativity ......................... 135 Maxwell's principle ........................................ 137 Potentials and the variational formalism .................... 139 Geometrie consequences of Maxwell's principle .............. 141 An application: variation of constants ...................... 142 Galilean moments .......................................... 144 Remarks ................................................... 148 Examples of dynamical groups ............................. 149 §13. The prineiples of symplectie meehanies Nonrelativistic symplectic mechanics ....................... 154 Moments, mass, and the center of mass ..................... 155 The center of mass decomposition ........ (cid:0)(cid:0) ................. 157 Minkowski space and the Poincare group ................... 163 Relativistic mechanics ...................................... 166 §14. A meehanistie deseription of elementary particles Elementary systems ........................................ 173 A particle with spin ........................................ 174 Remarks ................................................... 179 A particle without spin ..................................... 180 A massless particle ......................................... 182 Remarks ................................................... 184 Nonrelativistic particles .................................... 185 Mass and barycenter of a relativistic system ................ 188 Inversions of space and time ................................ 189 A particle with nonzero mass ............................... 191 A massless particle ......................................... 192 §15. Particle dyn ami es A material point in an electromagnetic field ................ 194 A particle with spin in an electromagnetic field ............. 197 Systems of particles without interactions ................... 200 Interactions ................................................ 202