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More About This Textbook
Overview
The main purpose of the book is to acquaint mathematicians, physicists and engineers with classical mechanics as a whole, in both its traditional and its contemporary aspects. As such, it describes the fundamental principles, problems, and methods of classical mechanics, with the emphasis firmly laid on the working apparatus, rather than the physical foundations or applications. Chapters cover the nbody problem, symmetry groups of mechanical systems and the corresponding conservation laws, the problem of the integrability of the equations of motion, the theory of oscillations and perturbation theory.
Editorial Reviews
From the Publisher
From the reviews of the previous editions: "... As an encyclopaedia article, this book does not seek to serve as a textbook, nor to replace the original articles whose results it describes. The book's goal is to provide an overview, pointing out highlights and unsolved problems, and putting individual results into a coherent context. It is full of historical nuggets, many of them surprising. ... The examples are especially helpful; if a particular topic seems difficult, a later example frequently tames it. The writing is refreshingly direct, never degenerating into a vocabulary lesson for its own sake. 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 "This is a book to curl up with in front of a fire on a cold winter's evening. ..." SIAM Reviews, Sept. 1989From the reviews of the third edition:
"Mathematical Aspects of Classical and Celestial Mechanics is the third volume of Dynamical Systems section of Springer’s Encyclopaedia of Mathematical sciences. … if you wanted an idea of the broad scope of classical mechanics, this is a good place to visit. One advantage of the present book is that the authors are particularly skilled in balancing rigor with physical intuition. … The authors provide an extensive bibliography and a wellselected set of recommended readings. Overall, this is a thoroughly professional offering." (William J. Satzer, MathDL, January, 2007)
"The new edition is a considerable updating of the last. … it is a reference for experts that will pull them back from their narrow subarea of expertise, give them a vast overview of what other experts know, and send them to the references if they actually want to be able to use something. … In conclusion, this is a book that every mathematical library must own and that many experts will want to have on their shelves." (James Murdock, SIAM Review, Vol. 49 (4), 2007)
"This book is the third English edition of an already classical piece devoted to classical mechanics as a whole, in its traditional and contemporary aspects … . The book is significantly expanded with respect to its previous editions … enriching further its already important contribution of acquainting mathematicians, physicists and engineers with the subject. … New chapters on variational principles and tensor invariants were added, making the book more selfcontained. … Its purpose is to serve as a detailed guide on the subject … ." (Ernesto A. Lacomba, Mathematical Reviews, Issue 2008 a)
Product Details
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Meet the Author
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 Moscow
1965 Lenin Award of the Government of the U.S.S.R.
1976 Honorary Member, London Mathematical Society
1979 Honorary Doctor, University P. and M. Curie, Paris
1982 Carfoord Award of the Swedish Academy
1983 Foreign Member, National Academy, U.S.A.
1984 Foreign Member, Academy of Sciences, Paris
1987 Foreign Member, Academy of Arts and Sciences, U.S.A.
1988 Honorary Doctor, Warwick University, Coventry
1988 Foreign Member, Royal Soc. London, GB
1988 Foreign Member, Accademia Nazionale dei Lincei, Rome, Italy
1990 Member, Academy of Sciences, Russia
1990 Foreign Member, American Philosophical Society
1991 Honorary Doctor, Utrecht
1991 Honorary Doctor, Bologna
1991 Member, Academy of Natural Sciences, Russia
1991 Member, Academia Europaea
1992 N.V. Lobachevsky Prize of Russian Academy of Sciences
1994 Harvey Prize Technion Award
1994 Honorary Doctor, University of Madrid, Complutense
1997 Honorary Doctor, University of Toronto, Canada
2001 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, nonholonomic 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 Sciences
1994 State Prize of the Russian Federation
1995 Member, Russian Academy of Natural Sciences
2000 S.V. Kovalevskaya Prize of Russian Academy of Sciences
2000 Member, Academy of Sciences, Russia
2003 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))
Table of Contents
Basic Principles of Classical Mechanics 1
Newtonian Mechanics 1
Space, Time, Motion 1
NewtonLaplace Principle of Determinacy 2
Principle of Relativity 9
Principle of Relativity and Forces of Inertia 12
Basic Dynamical Quantities. Conservation Laws 15
Lagrangian Mechanics 17
Preliminary Remarks 17
Variations and Extremals 19
Lagrange's Equations 21
Poincare's Equations 23
Motion with Constraints 26
Hamiltonian Mechanics 30
Symplectic Structures and Hamilton's Equations 30
Generating Functions 33
Symplectic Structure of the Cotangent Bundle 34
The Problem of n Point Vortices 35
Action in the Phase Space 37
Integral Invariant 38
Applications to Dynamics of Ideal Fluid 40
Vakonomic Mechanics 41
Lagrange's Problem 42
Vakonomic Mechanics 43
Principle of Determinacy 46
Hamilton's Equations in Redundant Coordinates 47
Hamiltonian Formalism with Constraints 48
Dirac's Problem 48
Duality 50
Realization of Constraints 51
Various Methods of Realization of Constraints 51
Holonomic Constraints 52
Anisotropic Friction 54
Adjoint Masses 55
Adjoint Masses and Anisotropic Friction 58
Small Masses 59
The nBody Problem 61
The TwoBody Problem 61
Orbits 61
Anomalies 67
Collisions and Regularization 69
Geometry of Kepler's Problem 71
Collisions and Regularization 72
Necessary Condition for Stability 72
Simultaneous Collisions 73
Binary Collisions 74
Singularities of Solutions of the nBody Problem 78
Particular Solutions 79
Central Configurations 79
Homographic Solutions 80
Effective Potential and Relative Equilibria 82
Periodic Solutions in the Case of Bodies of Equal Masses 82
Final Motions in the ThreeBody Problem 83
Classification of the Final Motions According to Chazy 83
Symmetry of the Past and Future 84
Restricted ThreeBody Problem 86
Equations of Motion. The Jacobi Integral 86
Relative Equilibria and Hill Regions 87
Hill's Problem 88
Ergodic Theorems of Celestial Mechanics 92
Stability in the Sense of Poisson 92
Probability of Capture 94
Dynamics in Spaces of Constant Curvature 95
Generalized Bertrand Problem 95
Kepler's Laws 96
Celestial Mechanics in Spaces of Constant Curvature 97
Potential Theory in Spaces of Constant Curvature 98
Symmetry Groups and Order Reduction 103
Symmetries and Linear Integrals 103
Nother's Theorem 103
Symmetries in NonHolonomic Mechanics 107
Symmetries in Vakonomic Mechanics 109
Symmetries in Hamiltonian Mechanics 110
Reduction of Systems with Symmetries 111
Order Reduction (Lagrangian Aspect) 111
Order Reduction (Hamiltonian Aspect) 116
Examples: Free Rotation of a Rigid Body and the ThreeBody Problem 122
Relative Equilibria and Bifurcation of Integral Manifolds 126
Relative Equilibria and Effective Potential 126
Integral Manifolds, Regions of Possible Motion, and Bifurcation Sets 128
The Bifurcation Set in the Planar ThreeBody Problem 130
Bifurcation Sets and Integral Manifolds in the Problem of Rotation of a Heavy Rigid Body with a Fixed Point 131
Variational Principles and Methods 135
Geometry of Regions of Possible Motion 136
Principle of Stationary Abbreviated Action 136
Geometry of a Neighbourhood of the Boundary 139
Riemannian Geometry of Regions of Possible Motion with Boundary 140
Periodic Trajectories of Natural Mechanical Systems 145
Rotations and Librations 145
Librations in NonSimplyConnected Regions of Possible Motion 147
Librations in Simply Connected Domains and Seifert's Conjecture 150
Periodic Oscillations of a MultiLink Pendulum 153
Periodic Trajectories of NonReversible Systems 156
Systems with Gyroscopic Forces and Multivalued Functionals 156
Applications of the Generalized Poincare Geometric Theorem 159
Asymptotic Solutions. Application to the Theory of Stability of Motion 161
Existence of Asymptotic Motions 162
Action Function in a Neighbourhood of an Unstable Equilibrium Position 165
Instability Theorem 166
MultiLink Pendulum with Oscillating Point of Suspension 167
Homoclinic Motions Close to Chains of Homoclinic Motions 168
Integrable Systems and Integration Methods 171
Brief Survey of Various Approaches to Integrability of Hamiltonian Systems 171
Quadratures 171
Complete Integrability 174
Normal Forms 176
Completely Integrable Systems 179
ActionAngle Variables 179
NonCommutative Sets of Integrals 183
Examples of Completely Integrable Systems 185
Some Methods of Integration of Hamiltonian Systems 191
Method of Separation of Variables 191
Method of LA Pairs 197
Integrable NonHolonomic Systems 199
Differential Equations with Invariant Measure 199
Some Solved Problems of NonHolonomic Mechanics 202
Perturbation Theory for Integrable Systems 207
Averaging of Perturbations 207
Averaging Principle 207
Procedure for Eliminating Fast Variables. NonResonant Case 211
Procedure for Eliminating Fast Variables. Resonant Case 216
Averaging in SingleFrequency Systems 217
Averaging in Systems with Constant Frequencies 226
Averaging in NonResonant Domains 229
Effect of a Single Resonance 229
Averaging in TwoFrequency Systems 237
Averaging in MultiFrequency Systems 242
Averaging at Separatrix Crossing 244
Averaging in Hamiltonian Systems 256
Application of the Averaging Principle 256
Procedures for Eliminating Fast Variables 265
KAM Theory 273
Unperturbed Motion. NonDegeneracy Conditions 273
Invariant Tori of the Perturbed System 274
Systems with Two Degrees of Freedom 279
Diffusion of Slow Variables in Multidimensional Systems and its Exponential Estimate 286
Diffusion without Exponentially Small Effects 292
Variants of the Theorem on Invariant Tori 294
KAM Theory for LowerDimensional Tori 297
Variational Principle for Invariant Tori. Cantori 307
Applications of KAM Theory 311
Adiabatic Invariants 314
Adiabatic Invariance of the Action Variable in SingleFrequency Systems 314
Adiabatic Invariants of MultiFrequency Hamiltonian Systems 323
Adiabatic Phases 326
Procedure for Eliminating Fast Variables. Conservation Time of Adiabatic Invariants 332
Accuracy of Conservation of Adiabatic Invariants 334
Perpetual Conservation of Adiabatic Invariants 340
Adiabatic Invariants in Systems with Separatrix Crossings 342
NonIntegrable Systems 351
Nearly Integrable Hamiltonian Systems 351
The Poincare Method 352
Birth of Isolated Periodic Solutions as an Obstruction to Integrability 354
Applications of Poincare's Method 358
Splitting of Asymptotic Surfaces 360
Splitting Conditions. The Poincare Integral 360
Splitting of Asymptotic Surfaces as an Obstruction to Integrability 366
Some Applications 370
QuasiRandom Oscillations 373
Poincare Return Map 375
Symbolic Dynamics 378
Absence of Analytic Integrals 380
NonIntegrability in a Neighbourhood of an Equilibrium Position (Siegel's Method) 381
Branching of Solutions and Absence of SingleValued Integrals 385
Branching of Solutions as Obstruction to Integrability 385
Monodromy Groups of Hamiltonian Systems with SingleValued Integrals 388
Topological and Geometrical Obstructions to Complete Integrability of Natural Systems 391
Topology of Configuration Spaces of Integrable Systems 392
Geometrical Obstructions to Integrability 394
Multidimensional Case 396
Ergodic Properties of Dynamical Systems with Multivalued Hamiltonians 396
Theory of Small Oscillations 401
Linearization 401
Normal Forms of Linear Oscillations 402
Normal Form of a Linear Natural Lagrangian System 402
RayleighFisherCourant Theorems on the Behaviour of Characteristic Frequencies when Rigidity Increases or Constraints are Imposed 403
Normal Forms of Quadratic Hamiltonians 404
Normal Forms of Hamiltonian Systems near an Equilibrium Position 406
Reduction to Normal Form 406
Phase Portraits of Systems with Two Degrees of Freedom in a Neighbourhood of an Equilibrium Position at a Resonance 409
Stability of Equilibria of Hamiltonian Systems with Two Degrees of Freedom at Resonances 416
Normal Forms of Hamiltonian Systems near Closed Trajectories 417
Reduction to Equilibrium of a System with Periodic Coefficients 417
Reduction of a System with Periodic Coefficients to Normal Form 418
Phase Portraits of Systems with Two Degrees of Freedom near a Closed Trajectory at a Resonance 419
Stability of Equilibria in Conservative Fields 422
LagrangeDirichlet Theorem 422
Influence of Dissipative Forces 426
Influence of Gyroscopic Forces 427
Tensor Invariants of Equations of Dynamics 431
Tensor Invariants 431
Frozenin Direction Fields 431
Integral Invariants 433
PoincareCartan Integral Invariant 436
Invariant Volume Forms 438
Liouville's Equation 438
Condition for the Existence of an Invariant Measure 439
Application of the Method of Small Parameter 442
Tensor Invariants and the Problem of Small Denominators 445
Absence of New Linear Integral Invariants and Frozenin Direction Fields 445
Application to Hamiltonian Systems 446
Application to Stationary Flows of a Viscous Fluid 449
Systems on ThreeDimensional Manifolds 451
Integral Invariants of the Second Order and Multivalued Integrals 455
Tensor Invariants of QuasiHomogeneous Systems 457
KovalevskayaLyapunov Method 457
Conditions for the Existence of Tensor Invariants 459
General Vortex Theory 461
Lamb's Equation 461
Multidimensional Hydrodynamics 463
Invariant Volume Forms for Lamb's Equations 465
Recommended Reading 471
Bibliography 475
Index of Names 507
Subject Index 511