An Introduction to Geometrical Physics

Front Cover
World 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.
 

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Contents

GENERAL TOPOLOGY
7
HOMOLOGY
53
HOMOTOPY
77
MANIFOLDS CHARTS
123
DIFFERENTIABLE MANIFOLDS
135
TANGENT STRUCTURE
145
DIFFERENTIAL FORMS
195
SYMMETRIES
261
Continuum Einstein convention
450
HelmholtzVainberg theorem
451
b Kortewegde Vries equation
452
BUILDING LAGRANGIANS
453
Examples
454
b BornInfeld electrodynamics
455
d Electrodynamics
456
f Second order fermion equation
457

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 groups
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
3algebras 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
Pontryagin duality
431
Noncommutative harmonic analysis
432
TanakaKrein duality
433
VARIATIONS FUNCTIONALS
435
Variation fields
436
Path functionals
437
Functional differentials
438
Secondvariation
440
Linear functionals
442
Derivatives Fréchet and Gateaux
443
FUNCTIONAL FORMS
445
A EXTERIOR VARIATIONAL CALCULUS
446
Variations and differentials
447
The action functional
448
Euler Forms
449
SINGULAR POINTS
459
Index of a singular point
462
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
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
The algebraic structure
518
Relations between Lie algebras
521
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
ELASTICITY
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
ALPHABETIC INDEX
683
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