Mathematical Aspects of Classical And Celestial Mechanics

by Arnold, Vladimir I.; Kozlov, Valery V.; Neishtadt, Anatoly I.

• ISBN13: 9783540282464
• ISBN10: 3540282467
• Edition: 3rd
• Format: Hardcover
• Copyright: 2006-11-30
• Publisher: Springer Verlag

Summary

This work describes the fundamental principles, problems, and methods of classical mechanics. The main attention is devoted to the mathematical side of the subject. The authors have endeavored to give an exposition stressing the working apparatus of classical mechanics.The book is significantly expanded compared to the previous edition. The authors have added two chapters on the variational principles and methods of classical mechanics as well as on tensor invariants of equations of dynamics. Moreover, various other sections have been revised, added or expanded.The main purpose of the book is to acquaint the reader with classical mechanics as a whole, in both its classical and its contemporary aspects. The book addresses all mathematicians, physicists and engineers.From the reviews of the previous editions:"... The book accomplishes the goals it has set for itself. While it is not an introduction to the field, it is an excellent overview. ..." American Mathematical Monthly, Nov. 1989

Author Biography

V.I.ArnoldFamous 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 theory1958 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 FoundationV.V.KozlovFamous 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 SocietyA.I.NeishtadtNeishtadt is also Springer Author, working in the field of perturbation theory (in particular averaging of perturbations, adiabatic invariants), bifurcation theory, celestial mechanics2001 A.M.Lyapunov Prize of  Russian Academy of Sciences (joint with D.V.Anosov))

Table of Contents

1 Basic Principles of Classical Mechanics	1	(60)
1.1 Newtonian Mechanics	1	(16)
1.1.1 Space, Time, Motion	1	(1)
1.1.2 Newton—Laplace Principle of Determinacy	2	(7)
1.1.3 Principle of Relativity	9	(3)
1.1.4 Principle of Relativity and Forces of Inertia	12	(3)
1.1.5 Basic Dynamical Quantities. Conservation Laws	15	(2)
1.2 Lagrangian Mechanics	17	(13)
1.2.1 Preliminary Remarks	17	(2)
1.2.2 Variations and Extremals	19	(2)
1.2.3 Lagrange's Equations	21	(2)
1.2.4 Poincaré's Equations	23	(3)
1.2.5 Motion with Constraints	26	(4)
1.3 Hamiltonian Mechanics	30	(11)
1.3.1 Symplectic Structures and Hamilton's Equations	30	(3)
1.3.2 Generating Functions	33	(1)
1.3.3 Symplectic Structure of the Cotangent Bundle	34	(1)
1.3.4 The Problem of n Point Vortices	35	(2)
1.3.5 Action in the Phase Space	37	(1)
1.3.6 Integral Invariant	38	(2)
1.3.7 Applications to Dynamics of Ideal Fluid	40	(1)
1.4 Vakonomic Mechanics	41	(7)
1.4.1 Lagrange's Problem	42	(1)
1.4.2 Vakonomic Mechanics	43	(3)
1.4.3 Principle of Determinacy	46	(1)
1.4.4 Hamilton's Equations in Redundant Coordinates	47	(1)
1.5 Hamiltonian Formalism with Constraints	48	(3)
1.5.1 Dirac's Problem	48	(2)
1.5.2 Duality	50	(1)
1.6 Realization of Constraints	51	(10)
1.6.1 Various Methods of Realization of Constraints	51	(1)
1.6.2 Holonomic Constraints	52	(2)
1.6.3 Anisotropic Friction	54	(1)
1.6.4 Adjoint Masses	55	(3)
1.6.5 Adjoint Masses and Anisotropic Friction	58	(1)
1.6.6 Small Masses	59	(2)
2 The n-Body Problem	61	(42)
2.1 The Two-Body Problem	61	(11)
2.1.1 Orbits	61	(6)
2.1.2 Anomalies	67	(2)
2.1.3 Collisions and Regularization	69	(2)
2.1.4 Geometry of Kepler's Problem	71	(1)
2.2 Collisions and Regularization	72	(7)
2.2.1 Necessary Condition for Stability	72	(1)
2.2.2 Simultaneous Collisions	73	(1)
2.2.3 Binary Collisions	74	(4)
2.2.4 Singularities of Solutions of the n-Body Problem	78	(1)
2.3 Particular Solutions	79	(4)
2.3.1 Central Configurations	79	(1)
2.3.2 Homographic Solutions	80	(2)
2.3.3 Effective Potential and Relative Equilibria	82	(1)
2.3.4 Periodic Solutions in the Case of Bodies of Equal Masses	82	(1)
2.4 Final Motions in the Three-Body Problem	83	(3)
2.4.1 Classification of the Final Motions According to Chazy	83	(1)
2.4.2 Symmetry of the Past and Future	84	(2)
2.5 Restricted Three-Body Problem	86	(6)
2.5.1 Equations of Motion. The Jacobi Integral	86	(1)
2.5.2 Relative Equilibria and Hill Regions	87	(1)
2.5.3 Hill's Problem	88	(4)
2.6 Ergodic Theorems of Celestial Mechanics	92	(3)
2.6.1 Stability in the Sense of Poisson	92	(2)
2.6.2 Probability of Capture	94	(1)
2.7 Dynamics in Spaces of Constant Curvature	95	(8)
2.7.1 Generalized Bertrand Problem	95	(1)
2.7.2 Kepler's Laws	96	(1)
2.7.3 Celestial Mechanics in Spaces of Constant Curvature	97	(1)
2.7.4 Potential Theory in Spaces of Constant Curvature	98	(5)
3 Symmetry Groups and Order Reduction	103	(32)
3.1 Symmetries and Linear Integrals	103	(8)
3.1.1 Nöther's Theorem	103	(4)
3.1.2 Symmetries in Non-Holonomic Mechanics	107	(2)
3.1.3 Symmetries in Vakonomic Mechanics	109	(1)
3.1.4 Symmetries in Hamiltonian Mechanics	110	(1)
3.2 Reduction of Systems with Symmetries	111	(15)
3.2.1 Order Reduction (Lagrangian Aspect)	111	(5)
3.2.2 Order Reduction (Hamiltonian Aspect)	116	(6)
3.2.3 Examples: Free Rotation of a Rigid Body and the Three-Body Problem.	122	(4)
3.3 Relative Equilibria and Bifurcation of Integral Manifolds	126	(9)
3.3.1 Relative Equilibria and Effective Potential	126	(2)
3.3.2 Integral Manifolds, Regions of Possible Motion, and Bifurcation Sets	128	(2)
3.3.3 The Bifurcation Set in the Planar Three-Body Problem	130	(1)
3.3.4 Bifurcation Sets and Integral Manifolds in the Problem of Rotation of a Heavy Rigid Body with a Fixed Point	131	(4)
4 Variational Principles and Methods	135	(36)
4.1 Geometry of Regions of Possible Motion	136	(9)
4.1.1 Principle of Stationary Abbreviated Action	136	(3)
4.1.2 Geometry of a Neighbourhood of the Boundary	139	(1)
4.1.3 Riemannian Geometry of Regions of Possible Motion with Boundary	140	(5)
4.2 Periodic Trajectories of Natural Mechanical Systems	145	(11)
4.2.1 Rotations and Librations	145	(2)
4.2.2 Librations in Non-Simply-Connected Regions of Possible Motion	147	(3)
4.2.3 Librations in Simply Connected Domains and Seifert's Conjecture	150	(3)
4.2.4 Periodic Oscillations of a Multi-Link Pendulum	153	(3)
4.3 Periodic Trajectories of Non-Reversible Systems	156	(5)
4.3.1 Systems with Gyroscopic Forces and Multivalued Functionals	156	(3)
4.3.2 Applications of the Generalized Poincaré Geometric Theorem	159	(2)
4.4 Asymptotic Solutions. Application to the Theory of Stability of Motion	161	(10)
4.4.1 Existence of Asymptotic Motions	162	(3)
4.4.2 Action Function in a Neighbourhood of an Unstable Equilibrium Position	165	(1)
4.4.3 Instability Theorem	166	(1)
4.4.4 Multi-Link Pendulum with Oscillating Point of Suspension	167	(1)
4.4.5 Homoclinic Motions Close to Chains of Homoclinic Motions	168	(3)
5 Integrable Systems and Integration Methods	171	(36)
5.1 Brief Survey of Various Approaches to Integrability of Hamiltonian Systems	171	(8)
5.1.1 Quadratures	171	(3)
5.1.2 Complete Integrability	174	(2)
5.1.3 Normal Forms	176	(3)
5.2 Completely Integrable Systems	179	(12)
5.2.1 Action–Angle Variables	179	(4)
5.2.2 Non-Commutative Sets of Integrals	183	(2)
5.2.3 Examples of Completely Integrable Systems	185	(6)
5.3 Some Methods of Integration of Hamiltonian Systems	191	(8)
5.3.1 Method of Separation of Variables	191	(6)
5.3.2 Method of L–A Pairs	197	(2)
5.4 Integrable Non-Holonomic Systems	199	(8)
5.4.1 Differential Equations with Invariant Measure	199	(3)
5.4.2 Some Solved Problems of Non-Holonomic Mechanics	202	(5)
6 Perturbation Theory for Integrable Systems	207	(144)
6.1 Averaging of Perturbations	207	(49)
6.1.1 Averaging Principle	207	(4)
6.1.2 Procedure for Eliminating Fast Variables. Non-Resonant Case	211	(5)
6.1.3 Procedure for Eliminating Fast Variables. Resonant Case	216	(1)
6.1.4 Averaging in Single-Frequency Systems	217	(9)
6.1.5 Averaging in Systems with Constant Frequencies	226	(3)
6.1.6 Averaging in Non-Resonant Domains	229	(1)
6.1.7 Effect of a Single Resonance	229	(8)
6.1.8 Averaging in Two-Frequency Systems	237	(5)
6.1.9 Averaging in Multi-Frequency Systems	242	(2)
6.1.10 Averaging at Separatrix Crossing	244	(12)
6.2 Averaging in Hamiltonian Systems	256	(17)
6.2.1 Application of the Averaging Principle	256	(9)
6.2.2 Procedures for Eliminating Fast Variables	265	(8)
6.3 KAM Theory	273	(41)
6.3.1 Unperturbed Motion. Non-Degeneracy Conditions	273	(1)
6.3.2 Invariant Tori of the Perturbed System	274	(5)
6.3.3 Systems with Two Degrees of Freedom	279	(7)
6.3.4 Diffusion of Slow Variables in Multidimensional Systems and its Exponential Estimate	286	(6)
6.3.5 Diffusion without Exponentially Small Effects	292	(2)
6.3.6 Variants of the Theorem on Invariant Tori	294	(3)
6.3.7 KAM Theory for Lower-Dimensional Tori	297	(10)
6.3.8 Variational Principle for Invariant Tori. Cantori	307	(4)
6.3.9 Applications of KAM Theory	311	(3)
6.4 Adiabatic Invariants	314	(37)
6.4.1 Adiabatic Invariance of the Action Variable in Single-Frequency Systems	314	(9)
6.4.2 Adiabatic Invariants of Multi-Frequency Hamiltonian Systems	323	(3)
6.4.3 Adiabatic Phases	326	(6)
6.4.4 Procedure for Eliminating Fast Variables. Conservation Time of Adiabatic Invariants	332	(2)
6.4.5 Accuracy of Conservation of Adiabatic Invariants	334	(6)
6.4.6 Perpetual Conservation of Adiabatic Invariants	340	(2)
6.4.7 Adiabatic Invariants in Systems with Separatrix Crossings	342	(9)
7 Non-Integrable Systems	351	(50)
7.1 Nearly Integrable Hamiltonian Systems	351	(9)
7.1.1 The Poincaré Method	352	(2)
7.1.2 Birth of Isolated Periodic Solutions as an Obstruction to Integrability	354	(4)
7.1.3 Applications of Poincaré's Method	358	(2)
7.2 Splitting of Asymptotic Surfaces	360	(13)
7.2.1 Splitting Conditions. The Poincaré Integral	360	(6)
7.2.2 Splitting of Asymptotic Surfaces as an Obstruction to Integrability	366	(4)
7.2.3 Some Applications	370	(3)
7.3 Quasi-Random Oscillations	373	(8)
7.3.1 Poincare Return Map	375	(3)
7.3.2 Symbolic Dynamics	378	(2)
7.3.3 Absence of Analytic Integrals	380	(1)
7.4 Non-Integrability in a Neighbourhood of an Equilibrium Position (Siegel's Method)	381	(4)
7.5 Branching of Solutions and Absence of Single-Valued Integrals	385	(6)
7.5.1 Branching of Solutions as Obstruction to Integrability	385	(3)
7.5.2 Monodromy Groups of Hamiltonian Systems with Single-Valued Integrals	388	(3)
7.6 Topological and Geometrical Obstructions to Complete Integrability of Natural Systems	391	(10)
7.6.1 Topology of Configuration Spaces of Integrable Systems	392	(2)
7.6.2 Geometrical Obstructions to Integrability	394	(2)
7.6.3 Multidimensional Case	396	(1)
7.6.4 Ergodic Properties of Dynamical Systems with Multivalued Hamiltonians	396	(5)
8 Theory of Small Oscillations	401	(30)
8.1 Linearization	401	(1)
8.2 Normal Forms of Linear Oscillations	402	(4)
8.2.1 Normal Form of a Linear Natural Lagrangian System	402	(1)
8.2.2 Rayleigh—Fisher—Courant Theorems on the Behaviour of Characteristic Frequencies when Rigidity Increases or Constraints are Imposed	403	(1)
8.2.3 Normal Forms of Quadratic Hamiltonians	404	(2)
8.3 Normal Forms of Hamiltonian Systems near an Equilibrium Position	406	(11)
8.3.1 Reduction to Normal Form	406	(3)
8.3.2 Phase Portraits of Systems with Two Degrees of Freedom in a Neighbourhood of an Equilibrium Position at a Resonance	409	(7)
8.3.3 Stability of Equilibria of Hamiltonian Systems with Two Degrees of Freedom at Resonances	416	(1)
8.4 Normal Forms of Hamiltonian Systems near Closed Trajectories	417	(5)
8.4.1 Reduction to Equilibrium of a System with Periodic Coefficients	417	(1)
8.4.2 Reduction of a System with Periodic Coefficients to Normal Form	418	(1)
8.4.3 Phase Portraits of Systems with Two Degrees of Freedom near a Closed Trajectory at a Resonance	419	(3)
8.5 Stability of Equilibria in Conservative Fields	422	(9)
8.5.1 Lagrange—Dirichlet Theorem	422	(4)
8.5.2 Influence of Dissipative Forces	426	(1)
8.5.3 Influence of Gyroscopic Forces	427	(4)
9 Tensor Invariants of Equations of Dynamics	431	(40)
9.1 Tensor Invariants	431	(7)
9.1.1 Frozen-in Direction Fields	431	(2)
9.1.2 Integral Invariants	433	(3)
9.1.3 Poincare—Cartan Integral Invariant	436	(2)
9.2 Invariant Volume Forms	438	(7)
9.2.1 Liouville's Equation	438	(1)
9.2.2 Condition for the Existence of an Invariant Measure	439	(3)
9.2.3 Application of the Method of Small Parameter	442	(3)
9.3 Tensor Invariants and the Problem of Small Denominators	445	(6)
9.3.1 Absence of New Linear Integral Invariants and Frozen-in Direction Fields	445	(1)
9.3.2 Application to Hamiltonian Systems	446	(3)
9.3.3 Application to Stationary Flows of a Viscous Fluid	449	(2)
9.4 Systems on Three-Dimensional Manifolds	451	(4)
9.5 Integral Invariants of the Second Order and Multivalued Integrals	455	(2)
9.6 Tensor Invariants of Quasi-Homogeneous Systems	457	(4)
9.6.1 Kovalevskaya—Lyapunov Method	457	(2)
9.6.2 Conditions for the Existence of Tensor Invariants	459	(2)
9.7 General Vortex Theory	461	(10)
9.7.1 Lamb's Equation	461	(2)
9.7.2 Multidimensional Hydrodynamics	463	(2)
9.7.3 Invariant Volume Forms for Lamb's Equations	465	(6)
Recommended Reading	471	(4)
Bibliography	475	(32)
Index of Names	507	(4)
Subject Index	511