Symplectic Techniques in Physics

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Paperback | 15 October 1990

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Symplectic geometry is very useful for clearly and concisely formulating 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 view points. 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 results 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 1. 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 textbook 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.

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"This book is brilliant and fascinating and is probably one of the most important additions in recent years to the mathematical physics literature." Contemporary Physics

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Non-Fiction Science Physics Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Applied Mathematics Probability & Statistics

Preface	p. ix
Introduction	p. 1
Gaussian optics	p. 7
Hamilton's method in Gaussian optics	p. 17
Fermat's principle	p. 20
From Gaussian optics to linear optics	p. 23
Geometrical optics, Hamilton's method, and the theory of geometrical aberrations	p. 34
Fermat's principle and Hamilton's principle	p. 42
Interference and diffraction	p. 47
Gaussian integrals	p. 51
Examples in Fresnel optics	p. 54
The phase factor	p. 60
Fresnel's formula	p. 71
Fresnel optics and quantum mechanics	p. 75
Holography	p. 85
Poisson brackets	p. 88
The Heisenberg group and representation	p. 92
The Groenwald-van Hove theorem	p. 101
Other quantizations	p. 104
Polarization of light	p. 116
The coadjoint orbit of a semidirect product	p. 124
Electromagnetism and the determination of symplectic structures	p. 130
Epilogue: Why symplectic geometry?	p. 145
The geometry of the moment map	p. 151
Normal forms	p. 151
The Darboux-Weinstein theorem	p. 155
Kaehler manifolds	p. 160
Left-invariant forms and Lie algebra cohomology	p. 169
Symplectic group actions	p. 172
The moment map and some of its properties	p. 183
Group actions and foliations	p. 196
Collective motion	p. 210
Cotangent bundles and the moment map for semidirect products	p. 220
More Euler-Poisson equations	p. 233
The choice of a collective Hamiltonian	p. 242
Convexity properties of toral group actions	p. 249
The lemma of stationary phase	p. 260
Geometric quantization	p. 265
Motion in a Yang-Mills field and the principle of general covariance	p. 272
The equations of motion of a classical particle in a Yang-Mills field	p. 272
Curvature	p. 283
The energy-momentum tensor and the current	p. 296
The principle of general covariance	p. 304
Isotropic and coisotropic embeddings	p. 313
Symplectic induction	p. 319
Symplectic slices and moment reconstruction	p. 324
An alternative approach to the equations of motion	p. 331
The moment map and kinetic theory	p. 344
Complete integrability	p. 349
Fibrations by tori	p. 349
Collective complete integrability	p. 359
Collective action variables	p. 367
The Kostant-Symes lemma and some of its variants	p. 371
Systems of Calogero type	p. 381
Solitons and coadjoint structures	p. 391
The algebra of formal pseudodifferential operators	p. 397
The higher-order calculus of variations in one variable	p. 407
Contractions of symplectic homogeneous spaces	p. 416
The Whitehead lemmas	p. 417
The Hochschild-Serre spectral sequence	p. 430
Galilean and Poincare elementary particles	p. 437
Coppersmith's theory	p. 446
References	p. 458
Index	p. 467
Table of Contents provided by Syndetics. All Rights Reserved.

Symplectic Techniques in Physics

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