Differential manifolds and theoretical physics, Volume 116

0122002318......Page 1
Differential Manifolds and Theoretical Physics (Pure and Applied Mathematics, Vol 116)......Page 4
Copyright Page......Page 5
Contents......Page 8
Preface......Page 16
Mathematical Models for Physical Systems......Page 22
Mechanics of Many-Particle Systems......Page 26
Lagrangian and Hamiltonian Formulation......Page 28
Mechanical System with Constraints......Page 32
Exercises......Page 57
Differential Calculus in Several Variables......Page 37
The Concept of a Differential Manifold......Page 44
Submanifolds......Page 47
Tangent Vectors......Page 49
Smooth Maps of Manifolds......Page 54
Differentials of Functions......Page 56
Vector Fields and Integral Curves......Page 60
Local Existence And Uniqueness Theory......Page 61
The Global Flow of a Vector Field......Page 74
Complete Vector Fields......Page 76
Exercises......Page 78
The Topology and Manifold Structure of the Tangent Bundle......Page 82
The Cotangent Space and the Cotangent Bundle......Page 87
The Canonical 1-Form on T*X......Page 89
Exercises......Page 91
Covariant Tensors of Degree 2......Page 93
Riemannian and Lorentzian Metrics......Page 95
Behavior Under Mappings......Page 98
Induced Metrics on Submanifolds......Page 100
Raising and Lowering Indices......Page 104
Partitions of Unity......Page 105
Existence of Metrics on a Differential Manifold......Page 108
Topology and Critical Points of a Fuction......Page 111
Exercises......Page 113
Introduction......Page 115
The Total Force Mapping......Page 116
Forces of Constraint......Page 117
Conservative Forces......Page 120
The Legendre Transformation......Page 124
Conservation of Energy......Page 126
Hamilton\'s Equations......Page 127
2-Forms......Page 131
Exterior Derivative......Page 133
The Mappings # and b......Page 135
Hamiltonian and Lagrangian Vector Fields......Page 136
Time-Dependent Systems......Page 142
Exercises......Page 145
Tensors on a Vector Space......Page 148
Tensor Fields on Manifolds......Page 150
The Lie Derivative......Page 153
The Bracket of Vector Fields......Page 156
Vector Fields as Differential Operators......Page 158
Exercises......Page 159
Exterior Forms on a Vector Space......Page 162
Orientation of Vector Spaces......Page 167
Volume Element of a Metric......Page 170
Differential Forms on a Manifold......Page 171
Orientation of Manifolds......Page 172
Orientation of Hypersurfaces......Page 175
Exterior Derivative......Page 177
De Rham Cohomology Groups......Page 182
Manifolds with Boundary......Page 183
Induced Orientation......Page 184
Hodge *-Duality......Page 186
Calculations in Three-Dimensional Euclidean Space......Page 189
Calculations in Minkowski Spacetime......Page 191
Geometrical Aspects of Differential Forms......Page 192
Vector Subbundles......Page 193
Kernel of a Differential Form......Page 194
Integrable Subbundles and the Frobenius Theorem......Page 197
Integral Manifolds......Page 205
Maximal Integral Manifolds......Page 206
Inaccessibility Theorem......Page 208
Nonintegrable Subbundles......Page 209
Vector-Valued Differential Forms......Page 210
Exercises......Page 212
The Integeral of a Differential Form......Page 217
Strokes\'s Theorem......Page 220
Transformation Properties of Interals......Page 222
ω-Divergence of a Vector Field......Page 224
Other Versions of Stroke\'s Theorem......Page 225
Integration of Functions......Page 228
The Classical Integral Theorems......Page 229
Exercises......Page 231
Basic Concepts and Relativity Groups......Page 234
Relativistic Law of Velocity Addition......Page 241
Relativistic Length Contraction......Page 243
The Invariant Spacetime Interval......Page 244
The Proper Lorentz Group and the Poincaré Group......Page 245
The Spacetime Mainifold of Special Relativity......Page 246
Reativistic Time Units......Page 248
Accelerated MotionŒA Space Odyssey......Page 250
Energy and Momentum......Page 254
Relativitic Correction to Newtonian Mechanics......Page 255
Conservation of Energy and Momentum......Page 256
Changes in Rest Mass......Page 257
Exercises......Page 258
The Lorentz Force Law and the Faraday Tensor......Page 260
The 4-Current......Page 264
Doppler Effect......Page 266
Maxwell\'s Equations......Page 267
The Electromagnetic Plane Wave......Page 269
The 4-Potential......Page 271
Existence of Scalar and Vector Potentials in R3......Page 272
Exercises......Page 274
Hamiltonian Systems and Equivalent Models......Page 276
O(3) and SO(3)......Page 277
Space and Body Representations......Page 280
The Geometry of Rigid Body Motion......Page 282
Left-Invariant 1-Form......Page 284
Adjoint Representation......Page 285
Momentum Mapping......Page 286
Space Motions with Specified Momentum......Page 287
Coadjoint Orbits and Body Motions......Page 288
Special Proerties of SO(3)......Page 292
Classical Interpretation–Inertial Tensor, Principal Axes......Page 295
Stability of Stationary Rotations......Page 298
Poinsot Construction......Page 301
Euler Equations......Page 303
Phase Plane Analysis of Stability......Page 304
Exercises......Page 305
Lie Groups and their Lie Algebras......Page 307
Canonical Coordinates......Page 310
Subgroups and Homomorphisms......Page 311
Adjoint Representation......Page 312
Invariant Forms......Page 313
Coset Spaces and Actions......Page 314
Exercises......Page 317
Geometrical Mechanical Systems......Page 318
Liouville\'s Theorem......Page 319
Variational Principles......Page 321
Forces......Page 322
Fixed Energy Systems......Page 325
Configuration Projections......Page 326
Pseudomechanical Systems......Page 327
Restriction Mappings......Page 328
Rigid Body and Torque......Page 329
Gauge Group Actions......Page 331
Moving Frames and Geodesic Motion......Page 332
Basic Theorem Local (Lemma 15.36)......Page 335
Basic Theorem Global (Theorem 15.39)......Page 337
Principal Bundle Model Using a Special Frame......Page 340
The Souriac Equations......Page 342
Construction of a Gauge Invariant 2-Form......Page 343
Curvature Form......Page 348
The Souriau Gms......Page 349
Appendix: Conservation Laws......Page 350
Exercises......Page 353
Principal Bundles......Page 356
Connections on Principal Bundles......Page 358
Horizontal Lifts of Vectors......Page 359
Curvature Form and Integrability Theorem......Page 360
Associated Bundles......Page 362
Gauge Fields and Classical Particles......Page 363
Natural 2-Form on Coadjoint Orbits......Page 364
Pseudomechanical System for Particles in a Gauge Field......Page 366
Sternberg\'s Theorem......Page 367
Geometrical-Mechanical System for Particles in a Gauge Field......Page 368
Affine Group Model......Page 370
Exercises......Page 372
Quantum Effects......Page 375
Probability Amplitude Phase Factors......Page 376
Phase Factors and 1-Forms......Page 377
Cow and BohmƒAharanov Experiments......Page 379
Complex Line Bundles and Holonomy......Page 381
Integral Condition for Curvature Form......Page 383
Bundle Description of Phase Factor Calculation......Page 386
RemarksŒGeometric Quantization......Page 387
Holonomy and Curvature for General Lie Groups......Page 388
Exercises......Page 389
Gauss\'s Law in Electromagnetic Theory......Page 392
Charge Conservation......Page 393
Curvature and Bundle-Valued Differential Forms......Page 394
Covariant Exterior Derivative......Page 396
Covariant Derivative of Sections and Parallel Transport......Page 397
The Group of Gauge Transformations......Page 398
The Source Equation and Currents for Gauge Fields......Page 400
Exercises......Page 403
Bibliography......Page 408
Index......Page 410