Symplectic Techniques in PhysicsCambridge University Press, 25 May 1990 - 468 sayfa 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. |
İçindekiler
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 | |
Sık kullanılan terimler ve kelime öbekleri
action of G acts transitively adjoint representation axis coadjoint action coadjoint orbit cohomology coisotropic commutes component conjugate connected constant coordinates corresponding curve defined denote determined diffeomorphism differential dimension element equations Euclidean example fiber formula function F G acts G orbit G-invariant Gaussian Gaussian optics geometrical given gives group G Hamiltonian action hence homomorphism identify integral invariant isomorphism isotropy group Lagrangian Lie algebra Lie group light linear matrix moment map multiplication neighborhood nondegenerate null foliation P₁ particle plane Poincaré Poisson bracket polynomials principal bundle projection proof Proposition prove quadratic representation satisfying scalar product semidirect product smooth Sternberg subalgebra subgroup submanifold subspace Suppose symplectic form symplectic manifold symplectic structure symplectic vector space tangent space tangent vector Theorem theory transformation V₁ vanishes vector field vector space write z₂
Popüler pasajlar
Sayfa 458 - ATIYAH, MF and SINGER, IM The index of elliptic operators. III. Ann. of Math. 87 (1968), 546^604.