## Mathematical Aspects of Classical and Celestial MechanicsIn this book we describe the basic principles, problems, and methods of cl- sical mechanics. Our main attention is devoted to the mathematical side of the subject. Although the physical background of the models considered here and the applied aspects of the phenomena studied in this book are explored to a considerably lesser extent, we have tried to set forth ?rst and foremost the “working” apparatus of classical mechanics. This apparatus is contained mainly in Chapters 1, 3, 5, 6, and 8. Chapter 1 is devoted to the basic mathematical models of classical - chanics that are usually used for describing the motion of real mechanical systems. Special attention is given to the study of motion with constraints and to the problems of realization of constraints in dynamics. In Chapter 3 we discuss symmetry groups of mechanical systems and the corresponding conservation laws. We also expound various aspects of ord- reduction theory for systems with symmetries, which is often used in appli- tions. Chapter 4 is devoted to variational principles and methods of classical mechanics. They allow one, in particular, to obtain non-trivial results on the existence of periodic trajectories. Special attention is given to the case where the region of possible motion has a non-empty boundary. Applications of the variational methods to the theory of stability of motion are indicated. |

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### Contents

1 | |

2 | |

9 | |

12 | |

15 | |

17 | |

122 Variations and Extremals | 19 |

123 Lagranges Equations | 21 |

523 Examples of Completely Integrable Systems | 185 |

53 Some Methods of Integration of Hamiltonian Systems | 191 |

532 Method of LA Pairs | 197 |

54 Integrable NonHolonomic Systems | 199 |

542 Some Solved Problems of NonHolonomic Mechanics | 202 |

Perturbation Theory for Integrable Systems | 207 |

612 Procedure for Eliminating Fast Variables NonResonant Case | 211 |

613 Procedure for Eliminating Fast Variables Resonant Case | 216 |

124 Poincares Equations | 23 |

125 Motion with Constraints | 26 |

13 Hamiltonian Mechanics | 30 |

132 Generating Functions | 33 |

133 Symplectic Structure of the Cotangent Bundle | 34 |

134 The Problem of n Point Vortices | 35 |

135 Action in the Phase Space | 37 |

136 Integral Invariant | 38 |

137 Applications to Dynamics of Ideal Fluid | 40 |

14 Vakonomic Mechanics | 41 |

141 Lagranges Problem | 42 |

142 Vakonomic Mechanics | 43 |

143 Principle of Determinacy | 46 |

144 Hamiltons Equations in Redundant Coordinates | 47 |

15 Hamiltonian Formalism with Constraints | 48 |

152 Duality | 50 |

16 Realization of Constraints | 51 |

162 Holonomic Constraints | 52 |

163 Anisotropic Friction | 54 |

164 Adjoint Masses | 55 |

165 Adjoint Masses and Anisotropic Friction | 58 |

166 Small Masses | 59 |

The nBody Problem | 61 |

212 Anomalies | 67 |

213 Collisions and Regularization | 69 |

214 Geometry of Keplers Problem | 71 |

22 Collisions and Regularization | 72 |

222 Simultaneous Collisions | 73 |

223 Binary Collisions | 74 |

224 Singularities of Solutions of the nBody Problem | 78 |

23 Particular Solutions | 79 |

232 Homographic Solutions | 80 |

233 Effective Potential and Relative Equilibria | 82 |

24 Final Motions in the ThreeBody Problem | 83 |

242 Symmetry of the Past and Future | 84 |

25 Restricted ThreeBody Problem | 86 |

252 Relative Equilibria and Hill Regions | 87 |

253 Hills Problem | 88 |

26 Ergodic Theorems of Celestial Mechanics | 92 |

262 Probability of Capture | 94 |

27 Dynamics in Spaces of Constant Curvature | 95 |

272 Keplers Laws 177 343 | 96 |

273 Celestial Mechanics in Spaces of Constant Curvature | 97 |

274 Potential Theory in Spaces of Constant Curvature | 98 |

Symmetry Groups and Order Reduction | 103 |

312 Symmetries in NonHolonomic Mechanics | 107 |

313 Symmetries in Vakonomic Mechanics | 109 |

314 Symmetries in Hamiltonian Mechanics | 110 |

32 Reduction of Systems with Symmetries | 111 |

322 Order Reduction Hamiltonian Aspect | 116 |

Free Rotation of a Rigid Body and the ThreeBody Problem | 122 |

33 Relative Equilibria and Bifurcation of Integral Manifolds | 126 |

332 Integral Manifolds Regions of Possible Motion and Bifurcation Sets | 128 |

333 The Bifurcation Set in the Planar ThreeBody Problem | 130 |

334 Bifurcation Sets and Integral Manifolds in the Problem of Rotation of a Heavy Rigid Body with a Fixed Point | 131 |

Variational Principles and Methods | 134 |

41 Geometry of Regions of Possible Motion | 136 |

412 Geometry of a Neighbourhood of the Boundary | 139 |

413 Riemannian Geometry of Regions of Possible Motion with Boundary | 140 |

42 Periodic Trajectories of Natural Mechanical Systems | 145 |

422 Librations in NonSimplyConnected Regions of Possible Motion | 147 |

423 Librations in Simply Connected Domains and Seiferts Conjecture | 150 |

424 Periodic Oscillations of a MultiLink Pendulum 325 | 153 |

43 Periodic Trajectories of NonReversible Systems | 156 |

432 Applications of the Generalized Poincare Geometric Theorem | 159 |

44 Asymptotic Solutions Application to the Theory of Stability of Motion | 161 |

441 Existence of Asymptotic Motions | 162 |

442 Action Function in a Neighbourhood of an Unstable Equilibrium Position | 165 |

443 Instability Theorem | 166 |

444 MultiLink Pendulum with Oscillating Point of Suspension | 167 |

445 Homoclinic Motions Close to Chains of Homoclinic Motions | 168 |

Integrable Systems and Integration Methods | 171 |

512 Complete Integrability | 174 |

513 Normal Forms | 176 |

52 Completely Integrable Systems | 179 |

522 NonCommutative Sets of Integrals | 183 |

614 Averaging in SingleFrequency Systems | 217 |

615 Averaging in Systems with Constant Frequencies | 226 |

616 Averaging in NonResonant Domains | 229 |

618 Averaging in TwoFrequency Systems | 237 |

619 Averaging in MultiFrequency Systems | 242 |

6110 Averaging at Separatrix Crossing | 244 |

62 Averaging in Hamiltonian Systems | 256 |

622 Procedures for Eliminating Fast Variables | 265 |

63 KAM Theory | 273 |

632 Invariant Tori of the Perturbed System | 274 |

633 Systems with Two Degrees of Freedom | 279 |

634 Diffusion of Slow Variables in Multidimensional Systems and its Exponential Estimate | 286 |

635 Diffusion without Exponentially Small Eﬀects | 292 |

636 Variants of the Theorem on Invariant Tori | 294 |

637 KAM Theory for LowerDimensional Tori | 297 |

638 Variational Principle for Invariant Tori Cantori | 307 |

639 Applications of KAM Theory | 311 |

64 Adiabatic Invariants | 314 |

642 Adiabatic Invariants of MultiFrequency Hamiltonian Systems | 323 |

643 Adiabatic Phases | 326 |

644 Procedure for Eliminating Fast Variables Conservation Time of Adiabatic Invariants | 332 |

645 Accuracy of Conservation of Adiabatic Invariants | 334 |

646 Perpetual Conservation of Adiabatic Invariants | 340 |

647 Adiabatic Invariants in Systems with Separatrix Crossings | 342 |

NonIntegrable Systems | 351 |

711 The Poincare Method | 352 |

712 Birth of Isolated Periodic Solutions as an Obstruction to Integrability | 354 |

713 Applications of Poincares Method | 358 |

72 Splitting of Asymptotic Surfaces | 360 |

722 Splitting of Asymptotic Surfaces as an Obstruction to Integrability | 366 |

723 Some Applications | 370 |

73 QuasiRandom Oscillations | 373 |

731 Poincare Return Map | 375 |

732 Symbolic Dynamics | 378 |

733 Absence of Analytic Integrals | 380 |

74 NonIntegrability in a Neighbourhood of an Equilibrium Position Siegels Method | 381 |

75 Branching of Solutions and Absence of SingleValued Integrals | 385 |

752 Monodromy Groups of Hamiltonian Systems with SingleValued Integrals | 388 |

76 Topological and Geometrical Obstructions to Complete Integrability of Natural Systems | 391 |

761 Topology of Conﬁguration Spaces of Integrable Systems | 392 |

762 Geometrical Obstructions to Integrability | 394 |

763 Multidimensional Case | 396 |

Theory of Small Oscillations | 400 |

82 Normal Forms of Linear Oscillations | 402 |

822 RayleighFisherCourant Theorems on the Behaviour of Characteristic Frequencies when Rigidity Increases or Constraints are Imposed | 403 |

823 Normal Forms of Quadratic Hamiltonians | 404 |

83 Normal Forms of Hamiltonian Systems near an Equilibrium Position | 406 |

832 Phase Portraits of Systems with Two Degrees of Freedom in a Neighbourhood of an Equilibrium Position at a Resonance | 409 |

833 Stability of Equilibria of Hamiltonian Systems with Two Degrees of Freedom at Resonances | 416 |

84 Normal Forms of Hamiltonian Systems near Closed Trajectories | 417 |

842 Reduction of a System with Periodic Coefficients to Normal Form | 418 |

843 Phase Portraits of Systems with Two Degrees of Freedom near a Closed Trajectory at a Resonance | 419 |

85 Stability of Equilibria in Conservative Fields | 422 |

852 Inﬂuence of Dissipative Forces | 426 |

853 Inﬂuence of Gyroscopic Forces | 427 |

Tensor Invariants of Equations of Dynamics | 431 |

912 Integral Invariants | 433 |

913 PoincareCartan Integral Invariant | 436 |

92 Invariant Volume Forms | 438 |

922 Condition for the Existence of an Invariant Measure | 439 |

923 Application of the Method of Small Parameter | 442 |

93 Tensor Invariants and the Problem of Small Denominators | 445 |

932 Application to Hamiltonian Systems | 446 |

933 Application to Stationary Flows of a Viscous Fluid | 449 |

94 Systems on ThreeDimensional Manifolds | 451 |

95 Integral Invariants of the Second Order and Multivalued Integrals | 455 |

96 Tensor Invariants of QuasiHomogeneous Systems | 457 |

962 Conditions for the Existence of Tensor Invariants | 459 |

97 General Vortex Theory | 461 |

972 Multidimensional Hydrodynamics | 463 |

973 Invariant Volume Forms for Lambs Equations | 465 |

471 | |

475 | |

507 | |

511 | |

### Other editions - View all

Mathematical Aspects of Classical and Celestial Mechanics Vladimir I. Arnold,Valery V. Kozlov,Anatoly I. Neishtadt No preview available - 2009 |

Mathematical Aspects of Classical and Celestial Mechanics Vladimir I. Arnold,Valery V. Kozlov,Anatoly I. Neishtadt No preview available - 2006 |

### Common terms and phrases

adiabatic invariant algebra analytic approximation asymptotic averaged system canonical change of variables closed coefficients conditionally periodic consider const constant constraints coordinates corresponding curve defined degrees of freedom diffeomorphic differential equations domain dynamics eigenvalues energy equal equations of motion equilibrium position example existence formula frequencies functional F geodesic group G Hamilton’s equations Hamiltonian function Hamiltonian system independent integral invariant invariant tori Lagrange Lagrange’s Lagrangian Lagrangian system libration linear manifold matrix mechanics momentum map n-body problem neighbourhood non-degenerate non-resonant normal form obtain orbits oscillations parameter pendulum periodic solutions perturbed system phase portrait phase space plane Poincaré Poisson bracket potential Proposition resonance rigid body rotation separatrix slow variables smooth function stability Suppose surface symmetry system with Hamiltonian Theorem theory three-body problem tion tonian torus transl unperturbed velocity zero

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