An Introduction to Geometrical PhysicsWorld Scientific, 1995 - 699 pages This book stresses the unifying power of the geometrical framework in bringing together concepts from the different areas of physics. Common underpinnings of optics, elasticity, gravitation, relativistic fields, particle mechanics and other subjects are underlined. It attempts to extricate the notion of space currently in the physical literature from the metric connotation.The book's goal is to present mathematical ideas associated with geometrical physics in a rather introductory language. Included are many examples from elementary physics and also, for those wishing to reach a higher level of understanding, a more advanced treatment of the mathematical topics. It is aimed as an elementary text, more so than most others on the market, and is intended for first year graduate students. |
Contents
SPACE AND GEOMETRY | 1 |
GENERAL TOPOLOGY | 7 |
HOMOLOGY | 53 |
HOMOTOPY | 77 |
MANIFOLDS CHARTS | 123 |
DIFFERENTIABLE MANIFOLDS | 135 |
TANGENT STRUCTURE | 145 |
DIFFERENTIAL FORMS | 195 |
Examples | 454 |
Relation to topology | 463 |
Critical points | 464 |
Morse lemma | 465 |
Morse indices and topology | 466 |
Catastrophes | 467 |
EUCLIDEAN SPACES AND SUBSPACES | 469 |
A STRUCTURE EQUATIONS | 470 |
SYMMETRIES | 261 |
FIBER BUNDLES | 289 |
NONCOMMUTATIVE GEOMETRY | 341 |
MATHEMATICAL TOPICS | 349 |
THE BASIC ALGEBRAIC STRUCTURES | 351 |
A GROUPS AND LESSER STRUCTURES | 352 |
Representations | 353 |
Subgroups | 355 |
B RINGS AND FIELDS | 356 |
Fields | 357 |
Ring of a group | 358 |
MODULES AND VECTOR SPACES | 359 |
Vector spaces | 360 |
The notion of action | 361 |
Inner product | 362 |
ALGEBRAS | 363 |
Kinds of algebras | 364 |
Lie algebra | 365 |
Enveloping algebra | 366 |
Dual algebra | 367 |
Bialgebras or Hopf algebras | 368 |
Rmatrices | 370 |
DISCRETE GROUPS BRAIDS AND KNOTS | 371 |
Words and free groups | 372 |
Presentations | 373 |
Cyclic groups | 374 |
B BRAIDS | 377 |
Braids in everyday life | 378 |
Braids presented | 381 |
Direct product representations | 382 |
The YangBaxter equation | 383 |
KNOTS AND LINKS | 384 |
Links | 386 |
Knot group | 387 |
Invariant polynomials | 388 |
SETS AND MEASURES | 391 |
Borel algebra | 393 |
Partition of identity | 394 |
B ERGODISM | 395 |
The ergodic problem | 396 |
TOPOLOGICAL LINEAR SPACES | 399 |
Norm | 400 |
Banach space | 402 |
Function spaces | 403 |
BANACH ALGEBRAS | 405 |
Banach algebras | 406 |
algebras and Calgebras | 409 |
From Geometry to Algebra | 411 |
von Neumann algebras | 413 |
The Jones polynomials | 417 |
REPRESENTATIONS | 423 |
A LINEAR REPRESENTATIONS | 424 |
Unitary representations | 425 |
Characters | 426 |
Tensor products | 427 |
B REGULAR REPRESENTATION | 428 |
Generalities | 429 |
FOURIER EXPANSIONS | 430 |
VARIATIONS FUNCTIONALS | 435 |
FUNCTIONAL FORMS | 445 |
Second quadratic form | 471 |
First quadratic form | 472 |
Gauss Ricci and Codazzi equations | 473 |
Riemann tensor | 474 |
GEOMETRY OF SURFACES | 475 |
RELATION TO TOPOLOGY | 478 |
The Chern theorem | 480 |
NONEUCLIDEAN GEOMETRIES | 481 |
The spherical case | 483 |
The BoliyaiLobachevsky case | 486 |
On the geodesic curves | 488 |
The Poincaré space | 489 |
GEODESICS | 493 |
A SELFPARALLEL CURVES | 494 |
In Optics | 495 |
The absolute derivative | 496 |
Selfparallelism | 497 |
Complete spaces | 498 |
Fermi transport | 499 |
Jacobi equation | 500 |
Vorticity shear and expansion | 504 |
LandauRaychaudhury equation | 506 |
PHYSICAL TOPICS | 507 |
HAMILTONIAN MECHANICS | 509 |
Symplectic structure | 510 |
Time evolution | 512 |
Canonical transformations | 513 |
Phase spaces as bundles | 516 |
Liouville integrability | 524 |
MORE MECHANICS | 525 |
HamiltonJacobi equation | 527 |
B THE LAGRANGE DERIVATIVE | 529 |
THE RIGID BODY | 533 |
The phase space | 534 |
The space and the body derivatives | 535 |
Moving frames | 536 |
The rotation group | 538 |
The Poinsot construction | 541 |
STATISTICS AND ELASTICITY | 543 |
A STATISTICAL MECHANICS | 544 |
B LATTICE MODELS | 548 |
Spontaneous breakdown of symmetry | 551 |
The Potts model | 553 |
Cayley tree and Bethe lattice | 557 |
CELASTICITY | 558 |
Classical elasticity | 562 |
Nematic systems | 566 |
The Franck index | 569 |
PROPAGATION OF DISCONTINUITIES | 571 |
Partial differential equations | 572 |
GEOMETRICAL OPTICS | 583 |
CLASSICAL RELATIVISTIC FIELDS | 593 |
GAUGE FIELDS | 611 |
GENERAL RELATIVITY | 629 |
DE SITTER SPACES | 641 |
SYMMETRIES ON PHASE SPACE | 651 |
GLOSSARY | 661 |
REFERENCES | 669 |
| 683 | |
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Common terms and phrases
abelian antisymmetric braid group bundle called commutative compact compact space components configuration space connection consequently consider coordinates corresponding covariant derivative curvature curve defined differentiable manifold dimension dual dx¹ dx² dx³ element equation equivalent euclidean space example exists fiber Figure finite function f fundamental group geodesic geometrical given group G hamiltonian Hilbert space homeomorphic homotopy identity integral invariant inverse isomorphic lagrangian lattice Lie algebra Lie derivative Lie group linear loop mapping f matrix metric natural basis neighbourhood notation obtained open sets operation orthogonal p-form particles path phase space Phys physical plane Poincaré polynomials principal bundle properties quantum quotient representation simple spacetime sphere structure subgroup subsets subspace Suppose symmetric symplectic tangent tensor theorem topological space topology torsion transformations trivial usual vanishing vector field vector space velocity von Neumann algebras wavefunction μν