Symplectic techniques in physics

Victor Guillemin, Shlomo Sternberg0521248663, 9780521248662

Symplectic geometry is very useful for formulating clearly and concisely problems in classical physics and also for understanding the link between classical problems and their quantum counterparts. It is thus a subject of interest to both mathematicians and physicists, though they have approached the subject from different viewpoints. This is the first book that attempts to reconcile these approaches. The authors use the uncluttered, coordinate-free approach to symplectic geometry and classical mechanics that has been developed by mathematicians over the course of the past thirty years, but at the same time apply the apparatus to a great number of concrete problems. Some of the themes emphasized in the book include the pivotal role of completely integrable systems, the importance of symmetries, analogies between classical dynamics and optics, the importance of symplectic tools in classical variational theory, symplectic features of classical field theories, and the principle of general covariance.

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Table of contents :
Title page ……Page 1
Date-line ……Page 2
Contents ……Page 3
Preface page ……Page 7
I Introduction ……Page 11
1 Gaussian optics ……Page 17
2 Hamilton’s method in Gaussian optics ……Page 27
3 Fermat’s principle ……Page 30
4 From Gaussian optics to linear optics ……Page 33
5 Geometrical optics, Hamilton’s method, and the theory of geometrical aberrations ……Page 44
6 Fermat’s principle and Hamilton’s principle ……Page 52
7 Interference and diffraction ……Page 57
8 Gaussian integrals ……Page 61
9 Examples in Fresnel optics ……Page 64
10 The phase factor ……Page 70
11 Fresnel’s formula ……Page 81
12 Fresnel optics and quantum mechanics ……Page 85
13 Holography ……Page 95
14 Poisson brackets ……Page 98
15 The Heisenberg group and representation ……Page 102
16 The Groenwald-van Hove theorem ……Page 111
17 Other quantizations ……Page 114
18 Polarization of light ……Page 126
19 The coadjoint orbit of a semidirect product ……Page 134
20 Electromagnetism and the determination of symplectic structures ……Page 140
Epilogue: Why symplectic geometry? ……Page 155
21 Normal forms ……Page 161
22 The Darboux-Weinstein theorem ……Page 165
23 Kaehler manifolds ……Page 170
24 Left-invariant forms and Lie algebra cohomology ……Page 179
25 Symplectic group actions ……Page 182
26 The moment map and some of its properties ……Page 193
27 Group actions and foliations ……Page 206
28 Collective motion ……Page 220
29 Cotangent bundles and the moment map for semidirect products ……Page 230
30 More Euler-Poisson equations ……Page 243
31 The choice of a collective Hamiltonian ……Page 252
32 Convexity properties of toral group actions ……Page 259
33 The lemma of stationary phase ……Page 270
34 Geometric quantization ……Page 275
35 The equations of motion of a classical particle in a Yang-Mills field ……Page 282
36 Curvature ……Page 293
37 The energy-momentum tensor and the current ……Page 306
38 The principle of general covariance ……Page 314
39 Isotropic and coisotropic embeddings ……Page 323
40 Symplectic induction ……Page 329
41 Symplectic slices and moment reconstruction ……Page 334
42 An alternative approach to the equations of motion ……Page 341
43 The moment map and kinetic theory ……Page 354
44 Fibrations by tori ……Page 359
45 Collective complete integrability ……Page 369
46 Collective action variables ……Page 377
47 The Kostant-Symes lemma and some of its variants ……Page 381
48 Systems of Calogero type ……Page 391
49 Solitons and coadjoint structures ……Page 401
50 The algebra of formal pseudodifferential operators ……Page 407
51 The higher-order calculus of variations in one variable ……Page 417
V Contractions of symplectic homogeneous spaces ……Page 426
52 The Whitehead lemmas ……Page 427
53 The Hochschild-Serre spectral sequence ……Page 440
54 Galilean and Poincare elementary particles ……Page 447
55 Coppersmith’s theory ……Page 456
References ……Page 468
Index ……Page 477