Mathematical Aspects of Classical and Celestial Mechanics (eBook)

V.I.Arnold Famous author of various Springer books in the field of dynamical systems, differential equations, hydrodynamics, magnetohydrodynamics, classical and celestial mechanics, geometry, topology, algebraic geometry, symplectic geometry, singularity theory 1958 Award of the Mathematical Society of Moscow1965 Lenin Award of the Government of the U.S.S.R.1976 Honorary Member, London Mathematical Society1979 Honorary Doctor, University P. and M. Curie, Paris1982 Carfoord Award of the Swedish Academy1983 Foreign Member, National Academy, U.S.A.1984 Foreign Member, Academy of Sciences, Paris1987 Foreign Member, Academy of Arts and Sciences, U.S.A.1988 Honorary Doctor, Warwick University, Coventry1988 Foreign Member, Royal Soc. London, GB1988 Foreign Member, Accademia Nazionale dei Lincei, Rome, Italy1990 Member, Academy of Sciences, Russia1990 Foreign Member, American Philosophical Society1991 Honorary Doctor, Utrecht1991 Honorary Doctor, Bologna1991 Member, Academy of Natural Sciences, Russia1991 Member, Academia Europaea1992 N.V. Lobachevsky Prize of Russian Academy of Sciences1994 Harvey Prize Technion Award1994 Honorary Doctor, University of Madrid, Complutense1997 Honorary Doctor, University of Toronto, Canada2001 Wolf Prize of  Wolf Foundation V.V.Kozlov Famous Springer author working in the field of general principles of dynamics, integrability of equations of motion, variational methods in mechanics, rigid body dynamics, stability theory, non-holonomic mechanics, impact theory, symmetries and integral invariants, mathematical aspects of statistical mechanics, ergodic theory and mathematical physics. 1973 Lenin Komsomol Prize (the major prize for young scientists in USSR)1986 M.V. Lomonosov 1st Degree Prize (the major prize awarded by M.V. Lomonosov Moscow State University)1988 S. A. Chaplygin Prize of Russian Academy of Sciences1994 State Prize of the Russian Federation1995 Member,  Russian Academy of Natural Sciences2000 S.V. Kovalevskaya Prize of Russian Academy of Sciences2000 Member, Academy of Sciences, Russia2003 Foreign member of the Serbian Science Society A.I.Neishtadt Neishtadt is also Springer Author, working in the field of perturbation theory (in particular averaging of perturbations, adiabatic invariants), bifurcation theory, celestial mechanics 2001 A.M.Lyapunov Prize of  Russian Academy of Sciences (joint with D.V.Anosov))

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