منifoldهای دیفرانسیل و فیزیک نظری (ریاضیات خالص و کاربردی، جلد ۱۱۶)

This work shows how the concepts of manifold theory can be used to describe the physical world. The concepts of modern differential geometry are presented in this comprehensive study of classical mechanics, field theory, and simple quantum effects. 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