Differential Manifolds and Theoretical Physics (eBook)

Front Cover 1
Differential Manifolds and Theoretical Physics 4
Copyright Page 5
Contents 8
Preface 16
Chapter 1. Introduction 22
Mathematical Models for Physical Systems 22
Chapter 2. Classical Mechanics 26
Mechanics of Many-Particle Systems 26
Lagrangian and Hamiltonian Formulation 28
Mechanical System with Constraints 32
Exercises 57
Chapter 3. Introduction to Differential Manifolds 37
Differential Calculus in Several Variables 37
The Concept of a Differential Manifold 44
Submanifolds 47
Tangent Vectors 49
Smooth Maps of Manifolds 54
Differentials of Functions 56
Exercises 57
Chapter 4. Differential Equations on Manifolds 60
Vector Fields and Integral Curves 60
Local Existence And Uniqueness Theory 61
The Global Flow of a Vector Field 74
Complete Vector Fields 76
Exercises 78
Chapter 5. The Tangent and Cotangent Bundles 82
The Topology and Manifold Structure of the Tangent Bundle 82
The Cotangent Space and the Cotangent Bundle 87
The Canonical 1-Form on T*X 89
Exercises 91
Chapter 6. Covariant 2-Tensors and Metric Structures 93
Covariant Tensors of Degree 2 93
The Index of a Metric 95
Riemannian and Lorentzian Metrics 95
Behavior Under Mappings 98
Induced Metrics on Submanifolds 100
Raising and Lowering Indices 104
The Gradient of a Function 105
Partitions of Unity 105
Existence of Metrics on a Differential Manifold 108
Topology and Critical Points of a Fuction 111
Exercises 113
Chapter 7. Lagrangian and Hamiltonian Mechanics for Holonomic Systems 115
Introduction 115
The Total Force Mapping 116
Forces of Constraint 117
Lagrange's Equations 120
Conservative Forces 120
The Legendre Transformation 124
Conservation of Energy 126
Hamilton's Equations 127
2-Forms 131
Exterior Derivative 133
Canonical 2-Form on T*X 135
The Mappings # and b 135
Hamiltonian and Lagrangian Vector Fields 136
Time-Dependent Systems 142
Exercises 145
Chapter 8. Tensors 148
Tensors on a Vector Space 148
Tensor Fields on Manifolds 150
The Lie Derivative 153
The Bracket of Vector Fields 156
Vector Fields as Differential Operators 158
Exercises 159
Chapter 9. Differential Forms 162
Exterior Forms on a Vector Space 162
Orientation of Vector Spaces 167
Volume Element of a Metric 170
Differential Forms on a Manifold 171
Orientation of Manifolds 172
Orientation of Hypersurfaces 175
Interior Product 177
Exterior Derivative 177
Poincaré Lemma 182
De Rham Cohomology Groups 182
Manifolds with Boundary 183
Induced Orientation 184
Hodge *-Duality 186
Divergence and Laplacian Operators 189
Calculations in Three-Dimensional Euclidean Space 189
Calculations in Minkowski Spacetime 191
Geometrical Aspects of Differential Forms 192
Smooth Vector Bundles 193
Vector Subbundles 193
Kernel of a Differential Form 194
Integrable Subbundles and the Frobenius Theorem 197
Integral Manifolds 205
Maximal Integral Manifolds 206
Inaccessibility Theorem 208
Nonintegrable Subbundles 209
Vector-Valued Differential Forms 210
Exercises 212
Chapter 10. Integration of Differential Forms 217
The Integeral of a Differential Form 217
Strokes's Theorem 220
Transformation Properties of Interals 222
.-Divergence of a Vector Field 224
Other Versions of Stroke's Theorem 225
Integration of Functions 228
The Classical Integral Theorems 229
Exercises 231
Chapter 11. The Special Theory of Relativity 234
Basic Concepts and Relativity Groups 234
Relativistic Law of Velocity Addition 241
Relativity of Simultaneity 243
Relativistic Length Contraction 243
Relativistic Time Dilation 244
The Invariant Spacetime Interval 244
The Proper Lorentz Group and the Poincaré Group 245
The Spacetime Mainifold of Special Relativity 246
Reativistic Time Units 248
Accelerated MotionŒA Space Odyssey 250
Energy and Momentum 254
Relativitic Correction to Newtonian Mechanics 255
Conservation of Energy and Momentum 256
Mass and Energy 257
Changes in Rest Mass 257
Summary 258
Exercises 258
Chapter 12. Electromagnetic Theory 260
The Lorentz Force Law and the Faraday Tensor 260
The 4-Current 264
Doppler Effect 266
Maxwell's Equations 267
The Electromagnetic Plane Wave 269
The 4-Potential 271
Existence of Scalar and Vector Potentials in R3 272
Exercises 274
Chapter 13. The Mechanics of Rigid Body Motion 276
Hamiltonian Systems and Equivalent Models 276
The Rigid Body 277
O(3) and SO(3) 277
Space and Body Representations 280
The Geometry of Rigid Body Motion 282
Left-Invariant 1-Form 284
Symmetry Group 285
Adjoint Representation 285
Momentum Mapping 286
Coadjoint Representation 287
Space Motions with Specified Momentum 287
Coadjoint Orbits and Body Motions 288
Special Proerties of SO(3) 292
Stationary Rotations 295
Classical Interpretation–Inertial Tensor, Principal Axes 295
Stability of Stationary Rotations 298
Poinsot Construction 301
Euler Equations 303
Phase Plane Analysis of Stability 304
Exercises 305
Chapter 14. Lie Groups 307
Lie Groups and their Lie Algebras 307
Exponential Mapping 310
Canonical Coordinates 310
Subgroups and Homomorphisms 311
Adjoint Representation 312
Invariant Forms 313
Coset Spaces and Actions 314
Exercises 317
Chapter 15. Geometrical Models 318
Geometrical Mechanical Systems 318
Liouville's Theorem 319
Variational Principles 321
Forces 322
Fixed Energy Systems 325
Configuration Projections 326
Lorentz Force Law 327
Pseudomechanical Systems 327
Restriction Mappings 328
Rigid Body and Torque 329
Gauge Group Actions 331
Moving Frames and Geodesic Motion 332
Basic Theorem Local (Lemma 15.36) 335
Basic Theorem Global (Theorem 15.39) 337
Principal Bundle Model Using a Special Frame 340
The Souriac Equations 342
Structure of the Lie Algebra of the Lorentz Group 343
Construction of a Gauge Invariant 2-Form 343
Curvature Form 348
The Souriau Gms 349
Appendix: Conservation Laws 350
Exercises 353
Chapter 16. Principal Bundles and Connections Gauge Fields and Classical Particles
Principal Bundles 356
Reduction of the Structural Group 358
Connections on Principal Bundles 358
Horizontal Lifts of Vectors 359
Curvature Form and Integrability Theorem 360
Horizontal Lifts of Curves 362
Associated Bundles 362
Parallel Transport 363
Gauge Fields and Classical Particles 363
Natural 2-Form on Coadjoint Orbits 364
Pseudomechanical System for Particles in a Gauge Field 366
Sternberg's Theorem 367
Geometrical-Mechanical System for Particles in a Gauge Field 368
Affine Group Model 370
Exercises 372
Chapter 17. Quantum Effects, Line Bundles, and Holonomy Groups 375
Quantum Effects 375
Probability Amplitudes 376
Probability Amplitude Phase Factors 376
DeBroglie and Feynman 377
Phase Factors and 1-Forms 377
Cow and BohmƒAharanov Experiments 379
Complex Line Bundles and Holonomy 381
Integral Condition for Curvature Form 383
Bundle Description of Phase Factor Calculation 386
RemarksŒGeometric Quantization 387
Holonomy and Curvature for General Lie Groups 388
Exercises 389
Chapter 18. Physical Laws for the Gauge Fields 392
Gauss's Law in Electromagnetic Theory 392
Charge Conservation 393
Curvature and Bundle-Valued Differential Forms 394
Covariant Exterior Derivative 396
Covariant Derivative of Sections and Parallel Transport 397
The Group of Gauge Transformations 398
The Killing Form 400
The Source Equation and Currents for Gauge Fields 400
Exercises 403
Bibliography 408
Index 410