Table Of ContentProgress 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.
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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of differential forms has been replaced by the symbol d. Consequently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
