## Symplectic Techniques in PhysicsSymplectic 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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### Contents

Introduction | 1 |

1 Gaussian optics | 7 |

2 Hamiltons method in Gaussian optics | 17 |

3 Fermats principle | 20 |

4 From Gaussian optics to linear optics | 23 |

5 Geometrical optics Hamiltons method and the theory of geometrical aberrations | 34 |

6 Fermats principle and Hamiltons principle | 42 |

7 Interference and diffraction | 47 |

30 More EulerPoisson 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 |

Motion in a YangMills field and the principle of general covariance | 272 |

36 Curvature | 283 |

37 The energymomentum tensor and the current | 296 |

8 Gaussian integrals | 51 |

9 Examples in Fresnel optics | 54 |

10 The phase factor | 60 |

11 Fresnels 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 Groenwaldvan Hove theorem | 101 |

17 Other quantizations | 104 |

18 Polarization of light | 116 |

19 The coadjoint orbit structure of a semidirect product | 124 |

20 Electromagnetism and the determination of symplectic structures | 130 |

Why symplectic geometry? | 145 |

The geometry of the moment map | 151 |

22 The DarbouxWeinstein theorem | 155 |

23 Kaehler manifolds | 160 |

24 Leftinvariant 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 |

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 |

Complete integrability | 349 |

45 Collective complete integrability | 359 |

46 Collective action variables | 367 |

47 The KostantSymes 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 higherorder calculus of variations in one variable | 407 |

Contractions of symplectic homogeneous spaces | 416 |

52 The Whitehead lemmas | 417 |

53 The HochschildSerre spectral sequence | 430 |

54 Galilean and Poincaré elementary particles | 437 |

55 Coppersmiths theory | 446 |

458 | |

467 | |

### Common terms and phrases

action action of G acts approximation associated assume axis bracket bundle called choice choose closed coadjoint commutes compact completely component condition connected consider consists constant coordinates corresponding curve defined definition denote depends derivative described determined diffeomorphism differential dimension direction discussion element equations example exists expression fact fiber Figure fixed foliation follows formula function geometrical given gives Hamiltonian hence homomorphism identify identity induces integral invariant Lie algebra light linear matrix motion multiplication neighborhood obtain operator optical orbit particle particular plane positive preceding principal projection proof properties Proposition prove quantum mechanics reduced representation restriction satisfying smooth structure submanifold Suppose symplectic form symplectic manifold tangent tangent vector Theorem theory transformation values vanishes vector field vector space write

### Popular passages

Page 458 - ATIYAH, MF and SINGER, IM The index of elliptic operators. III. Ann. of Math. 87 (1968), 546^604.