Symplectic techniques in physics / Victor Guillemin, Shlomo Sternberg.

Format:

Book

Author/Creator:

Guillemin, Victor, 1937-

Contributor:

Sternberg, Shlomo.

Language:

English

Subjects (All):

Geometry, Differential.

Mathematical physics.

Transformations (Mathematics).

Physical Description:

xi, 468 pages : illustrations ; 24 cm

Edition:

First paperback edition.

Place of Publication:

Cambridge [Cambridgeshire] ; New York : Cambridge University Press, 1990, c1984.

Summary:

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 last thirty years, but at the same time apply the apparatus to a great number of concrete problems. In the first chapter, the authors provide an elementary introduction to symplectic geometry and explain the key concepts and resuits in a way accessible to physicists and mathematicians. The remainder of the book is devoted to the detailed analysis and study of the ideas discussed in Chapter I. 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. This work can be used as a text book for graduate courses, but the depth of coverage and the wealth of information and application means that it will be of continuing interest to and of lasting significance for mathematicians and mathematically minded physicists.

Contents:

1 Gaussian optics 7

2 Hamilton's method in Gaussian optics 17

3 Fermat's principle 20

4 From Gaussian optics to linear optics 23

5 Geometrical optics, Hamilton's method, and the theory of geometrical aberrations 34

6 Fermat's principle and Hamilton's principle 42

7 Interference and diffraction 47

8 Gaussian integrals 51

9 Examples in Fresnel optics 54

10 The phase factor 60

11 Fresnel's formula 71

12 Fresnel optics and quantum mechanics 75

13 Holography 85

14 Poisson brackets 88

15 The Heisenberg group and representation 92

16 The Groenwald-van Hove theorem 101

17 Other quantizations 104

18 Polarization of light 116

19 The coadjoint orbit of a semidirect product 124

20 Electromagnetism and the determination of symplectic structures 130

Epilogue: Why symplectic geometry? 145

II The geometry of the moment map 151

21 Normal forms 151

22 The Darboux-Weinstein theorem 155

23 Kaehler manifolds 160

24 Left-invariant forms and Lie algebra cohomology 169

25 Symplectic group actions 172

26 The moment map and some of its properties 183

27 Group actions and foliations 196

28 Collective motion 210

29 Cotangent bundles and the moment map for semidirect products 220

30 More Euler-Poisson equations 233

31 The choice of a collective Hamiltonian 242

32 Convexity properties of toral group actions 249

33 The lemma of stationary phase 260

34 Geometric quantization 265

III Motion in a Yang-Mills field and the principle of general covariance 272

35 The equations of motion of a classical particle in a Yang-Mills field 272

36 Curvature 283

37 The energy-momentum tensor and the current 296

38 The principle of general covariance 304

39 Isotropic and coisotropic embeddings 313

40 Symplectic induction 319

41 Symplectic slices and moment reconstruction 324

42 An alternative approach to the equations of motion 331

43 The moment map and kinetic theory 344

IV Complete integrability 349

44 Fibrations by tori 349

45 Collective complete integrability 359

46 Collective action variables 367

47 The Kostant-Symes lemma and some of its variants 371

48 Systems of Calogero type 381

49 Solitons and coadjoint structures 391

50 The algebra of formal pseudodifferential operators 397

51 The higher-order calculus of variations in one variable 407

V Contractions of symplectic homogeneous spaces 416

52 The Whitehead lemmas 417

53 The Hochschild-Serre spectral sequence 430

54 Galilean and Poincare elementary particles 437

55 Coppersmith's theory 446.

Notes:

Bibliography: pages 458-465.

Includes index.

ISBN:

0521389909

OCLC:

23664289