Mathematical Aspects of Classical and Celestial Mechanics

Mathematical Aspects of Classical and Celestial Mechanics

von: Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt

Springer-Verlag, 2007

ISBN: 9783540489269 , 505 Seiten

3. Auflage

Format: PDF

Kopierschutz: Wasserzeichen

Windows PC,Mac OSX Apple iPad, Android Tablet PC's

Preis: 166,59 EUR

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Mathematical Aspects of Classical and Celestial Mechanics


 

Preface

5

Contents

7

1 Basic Principles of Classical Mechanics

14

1.1 Newtonian Mechanics

14

1.2 Lagrangian Mechanics

30

1.3 Hamiltonian Mechanics

43

1.4 Vakonomic Mechanics

54

1.5 Hamiltonian Formalism with Constraints

61

1.6 Realization of Constraints

64

2 The n-Body Problem

74

2.1 The Two-Body Problem

74

2.2 Collisions and Regularization

85

2.3 Particular Solutions

92

2.4 Final Motions in the Three-Body Problem

96

2.5 Restricted Three-Body Problem

99

2.6 Ergodic Theorems of Celestial Mechanics

105

2.7 Dynamics in Spaces of Constant Curvature

108

3 Symmetry Groups and Order Reduction

115

3.1 Symmetries and Linear Integrals

115

3.2 Reduction of Systems with Symmetries

123

3.3 Relative Equilibria and Bifurcation of Integral Manifolds

138

4 Variational Principles and Methods

146

4.1 Geometry of Regions of Possible Motion

147

4.2 Periodic Trajectories of Natural Mechanical Systems

156

4.3 Periodic Trajectories of Non-Reversible Systems

167

4.4 Asymptotic Solutions. Application to the Theory of Stability of Motion

172

5 Integrable Systems and Integration Methods

182

5.1 Brief Survey of Various Approaches to Integrability of Hamiltonian Systems

182

5.2 Completely Integrable Systems

190

5.3 Some Methods of Integration of Hamiltonian Systems

202

5.4 Integrable Non-Holonomic Systems

210

6 Perturbation Theory for Integrable Systems

218

6.1 Averaging of Perturbations

218

6.2 Averaging in Hamiltonian Systems

267

6.3 KAM Theory

284

6.4 Adiabatic Invariants

325

7 Non-Integrable Systems

361

7.1 Nearly Integrable Hamiltonian Systems

361

7.2 Splitting of Asymptotic Surfaces

370

7.3 Quasi-Random Oscillations

383

7.4 Non-Integrability in a Neighbourhood of an Equilibrium Position ( Siegel’s Method)

391

7.5 Branching of Solutions and Absence of Single- Valued Integrals

395

7.6 Topological and Geometrical Obstructions to Complete Integrability of Natural Systems

401

8 Theory of Small Oscillations

410

8.1 Linearization

410

8.2 Normal Forms of Linear Oscillations

411

8.3 Normal Forms of Hamiltonian Systems near an Equilibrium Position

415

8.4 Normal Forms of Hamiltonian Systems near Closed Trajectories

426

8.5 Stability of Equilibria in Conservative Fields

431

9 Tensor Invariants of Equations of Dynamics

439

9.1 Tensor Invariants

439

9.2 Invariant Volume Forms

446

9.3 Tensor Invariants and the Problem of Small Denominators

453

9.4 Systems on Three-Dimensional Manifolds

459

9.5 Integral Invariants of the Second Order and Multivalued Integrals

463

9.6 Tensor Invariants of Quasi-Homogeneous Systems

465

9.7 General Vortex Theory

469

Comments on the Bibliography

477

Recommended Reading

479

Bibliography

483

Index of Names

515

Subject Index

519