Graphs on Surfaces and Their Applications

Contents......Page 7
1.1.1 Space, Time, Motion......Page 14
1.1.2 Newton–Laplace Principle of Determinacy......Page 15
1.1.3 Principle of Relativity......Page 22
1.1.4 Principle of Relativity and Forces of Inertia......Page 25
1.1.5 Basic Dynamical Quantities. Conservation Laws......Page 28
1.2.1 Preliminary Remarks......Page 30
1.2.2 Variations and Extremals......Page 32
1.2.3 Lagrange's Equations......Page 34
1.2.4 Poincaré's Equations......Page 36
1.2.5 Motion with Constraints......Page 39
1.3.1 Symplectic Structures and Hamilton's Equations......Page 43
1.3.2 Generating Functions......Page 46
1.3.3 Symplectic Structure of the Cotangent Bundle......Page 47
1.3.4 The Problem of n Point Vortices......Page 48
1.3.5 Action in the Phase Space......Page 50
1.3.6 Integral Invariant......Page 51
1.3.7 Applications to Dynamics of Ideal Fluid......Page 53
1.4 Vakonomic Mechanics......Page 54
1.4.1 Lagrange's Problem......Page 55
1.4.2 Vakonomic Mechanics......Page 56
1.4.3 Principle of Determinacy......Page 59
1.4.4 Hamilton's Equations in Redundant Coordinates......Page 60
1.5.1 Dirac's Problem......Page 61
1.5.2 Duality......Page 63
1.6.1 Various Methods of Realization of Constraints......Page 64
1.6.2 Holonomic Constraints......Page 65
1.6.3 Anisotropic Friction......Page 67
1.6.4 Adjoint Masses......Page 68
1.6.5 Adjoint Masses and Anisotropic Friction......Page 71
1.6.6 Small Masses......Page 72
2.1.1 Orbits......Page 74
2.1.2 Anomalies......Page 80
2.1.3 Collisions and Regularization......Page 82
2.1.4 Geometry of Kepler's Problem......Page 84
2.2.1 Necessary Condition for Stability......Page 85
2.2.2 Simultaneous Collisions......Page 86
2.2.3 Binary Collisions......Page 87
2.2.4 Singularities of Solutions of the n-Body Problem......Page 91
2.3.1 Central Configurations......Page 92
2.3.2 Homographic Solutions......Page 93
2.3.4 Periodic Solutions in the Case of Bodies of Equal Masses......Page 95
2.4.1 Classification of the Final Motions According to Chazy......Page 96
2.4.2 Symmetry of the Past and Future......Page 97
2.5.1 Equations of Motion. The Jacobi Integral......Page 99
2.5.2 Relative Equilibria and Hill Regions......Page 100
2.5.3 Hill's Problem......Page 101
2.6.1 Stability in the Sense of Poisson......Page 105
2.6.2 Probability of Capture......Page 107
2.7.1 Generalized Bertrand Problem......Page 108
2.7.2 Kepler's Laws......Page 109
2.7.3 Celestial Mechanics in Spaces of Constant Curvature......Page 110
2.7.4 Potential Theory in Spaces of Constant Curvature......Page 111
3.1.1 Nöther's Theorem......Page 115
3.1.2 Symmetries in Non-Holonomic Mechanics......Page 119
3.1.3 Symmetries in Vakonomic Mechanics......Page 121
3.1.4 Symmetries in Hamiltonian Mechanics......Page 122
3.2.1 Order Reduction (Lagrangian Aspect)......Page 123
3.2.2 Order Reduction (Hamiltonian Aspect)......Page 128
3.2.3 Examples: Free Rotation of a Rigid Body and the Three-Body Problem......Page 134
3.3.1 Relative Equilibria and Effective Potential......Page 138
3.3.2 Integral Manifolds, Regions of Possible Motion, and Bifurcation Sets......Page 140
3.3.3 The Bifurcation Set in the Planar Three-Body Problem......Page 142
3.3.4 Bifurcation Sets and Integral Manifolds in the Problem of Rotation of a Heavy Rigid Body with a Fixed Point......Page 143
4 Variational Principles and Methods......Page 146
4.1.1 Principle of Stationary Abbreviated Action......Page 147
4.1.2 Geometry of a Neighbourhood of the Boundary......Page 150
4.1.3 Riemannian Geometry of Regions of Possible Motion with Boundary......Page 151
4.2.1 Rotations and Librations......Page 156
4.2.2 Librations in Non-Simply-Connected Regions of Possible Motion......Page 158
4.2.3 Librations in Simply Connected Domains and Seifert's Conjecture......Page 161
4.2.4 Periodic Oscillations of a Multi-Link Pendulum......Page 164
4.3.1 Systems with Gyroscopic Forces and Multivalued Functionals......Page 167
4.3.2 Applications of the Generalized Poincaré Geometric Theorem......Page 170
4.4 Asymptotic Solutions. Application to the Theory of Stability of Motion......Page 172
4.4.1 Existence of Asymptotic Motions......Page 173
4.4.2 Action Function in a Neighbourhood of an Unstable Equilibrium Position......Page 176
4.4.3 Instability Theorem......Page 177
4.4.4 Multi-Link Pendulum with Oscillating Point of Suspension......Page 178
4.4.5 Homoclinic Motions Close to Chains of Homoclinic Motions......Page 179
5.1.1 Quadratures......Page 182
5.1.2 Complete Integrability......Page 185
5.1.3 Normal Forms......Page 187
5.2.1 Action–Angle Variables......Page 190
5.2.2 Non-Commutative Sets of Integrals......Page 194
5.2.3 Examples of Completely Integrable Systems......Page 196
5.3.1 Method of Separation of Variables......Page 202
5.3.2 Method of L–A Pairs......Page 208
5.4.1 Differential Equations with Invariant Measure......Page 210
5.4.2 Some Solved Problems of Non-Holonomic Mechanics......Page 213
6.1.1 Averaging Principle......Page 218
6.1.2 Procedure for Eliminating Fast Variables. Non-Resonant Case......Page 222
6.1.3 Procedure for Eliminating Fast Variables. Resonant Case......Page 227
6.1.4 Averaging in Single-Frequency Systems......Page 228
6.1.5 Averaging in Systems with Constant Frequencies......Page 237
6.1.7 Effect of a Single Resonance......Page 240
6.1.8 Averaging in Two-Frequency Systems......Page 248
6.1.9 Averaging in Multi-Frequency Systems......Page 253
6.1.10 Averaging at Separatrix Crossing......Page 255
6.2.1 Application of the Averaging Principle......Page 267
6.2.2 Procedures for Eliminating Fast Variables......Page 276
6.3.1 Unperturbed Motion. Non-Degeneracy Conditions......Page 284
6.3.2 Invariant Tori of the Perturbed System......Page 285
6.3.3 Systems with Two Degrees of Freedom......Page 290
6.3.4 Diffusion of Slow Variables in Multidimensional Systems and its Exponential Estimate......Page 297
6.3.5 Diffusion without Exponentially Small Effects......Page 303
6.3.6 Variants of the Theorem on Invariant Tori......Page 305
6.3.7 KAM Theory for Lower-Dimensional Tori......Page 308
6.3.8 Variational Principle for Invariant Tori. Cantori......Page 318
6.3.9 Applications of KAM Theory......Page 322
6.4.1 Adiabatic Invariance of the Action Variable in Single-Frequency Systems......Page 325
6.4.2 Adiabatic Invariants of Multi-Frequency Hamiltonian Systems......Page 334
6.4.3 Adiabatic Phases......Page 337
6.4.4 Procedure for Eliminating Fast Variables. Conservation Time of Adiabatic Invariants......Page 343
6.4.5 Accuracy of Conservation of Adiabatic Invariants......Page 345
6.4.6 Perpetual Conservation of Adiabatic Invariants......Page 351
6.4.7 Adiabatic Invariants in Systems with Separatrix Crossings......Page 353
7.1 Nearly Integrable Hamiltonian Systems......Page 361
7.1.1 The Poincaré Method......Page 362
7.1.2 Birth of Isolated Periodic Solutions as an Obstruction to Integrability......Page 364
7.1.3 Applications of Poincaré's Method......Page 368
7.2.1 Splitting Conditions. The Poincaré Integral......Page 370
7.2.2 Splitting of Asymptotic Surfaces as an Obstruction to Integrability......Page 376
7.2.3 Some Applications......Page 380
7.3 Quasi-Random Oscillations......Page 383
7.3.1 Poincaré Return Map......Page 385
7.3.2 Symbolic Dynamics......Page 388
7.3.3 Absence of Analytic Integrals......Page 390
7.4 Non-Integrability in a Neighbourhood of an Equilibrium Position (Siegel's Method)......Page 391
7.5.1 Branching of Solutions as Obstruction to Integrability......Page 395
7.5.2 Monodromy Groups of Hamiltonian Systems with Single-Valued Integrals......Page 398
7.6 Topological and Geometrical Obstructions to Complete Integrability of Natural Systems......Page 401
7.6.1 Topology of Configuration Spaces of Integrable Systems......Page 402
7.6.2 Geometrical Obstructions to Integrability......Page 404
7.6.4 Ergodic Properties of Dynamical Systems with Multivalued Hamiltonians......Page 406
8.1 Linearization......Page 410
8.2.1 Normal Form of a Linear Natural Lagrangian System......Page 411
8.2.2 Rayleigh–Fisher–Courant Theorems on the Behaviour of Characteristic Frequencies when Rigidity Increases or Constraints are Imposed......Page 412
8.2.3 Normal Forms of Quadratic Hamiltonians......Page 413
8.3.1 Reduction to Normal Form......Page 415
8.3.2 Phase Portraits of Systems with Two Degrees of Freedom in a Neighbourhood of an Equilibrium Position at a Resonance......Page 418
8.3.3 Stability of Equilibria of Hamiltonian Systems with Two Degrees of Freedom at Resonances......Page 425
8.4.1 Reduction to Equilibrium of a System with Periodic Coefficients......Page 426
8.4.2 Reduction of a System with Periodic Coefficients to Normal Form......Page 427
8.4.3 Phase Portraits of Systems with Two Degrees of Freedom near a Closed Trajectory at a Resonance......Page 428
8.5.1 Lagrange–Dirichlet Theorem......Page 431
8.5.2 Influence of Dissipative Forces......Page 435
8.5.3 Influence of Gyroscopic Forces......Page 436
9.1.1 Frozen-in Direction Fields......Page 439
9.1.2 Integral Invariants......Page 441
9.1.3 Poincaré–Cartan Integral Invariant......Page 444
9.2.1 Liouville's Equation......Page 446
9.2.2 Condition for the Existence of an Invariant Measure......Page 447
9.2.3 Application of the Method of Small Parameter......Page 450
9.3.1 Absence of New Linear Integral Invariants and Frozen-in Direction Fields......Page 453
9.3.2 Application to Hamiltonian Systems......Page 454
9.3.3 Application to Stationary Flows of a Viscous Fluid......Page 457
9.4 Systems on Three-Dimensional Manifolds......Page 459
9.5 Integral Invariants of the Second Order and Multivalued Integrals......Page 463
9.6.1 Kovalevskaya–Lyapunov Method......Page 465
9.6.2 Conditions for the Existence of Tensor Invariants......Page 467
9.7.1 Lamb's Equation......Page 469
9.7.2 Multidimensional Hydrodynamics......Page 471
9.7.3 Invariant Volume Forms for Lamb's Equations......Page 473
Recommended Reading......Page 479
Bibliography......Page 483
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