Mathematical Aspects of Classical and Celestial Mechanics / Edition 3

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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 n-body 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.

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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. 1989

From 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 well-selected 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 self-contained. … Its purpose is to serve as a detailed guide on the subject … ." (Ernesto A. Lacomba, Mathematical Reviews, Issue 2008 a)

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Product Details

  • ISBN-13: 9783540282464
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 11/14/2006
  • Series: Encyclopaedia of Mathematical Sciences Series , #3
  • Edition description: 3rd ed. 2006
  • Edition number: 3
  • Pages: 518
  • Product dimensions: 9.21 (w) x 6.14 (h) x 1.19 (d)

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, 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 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))

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Table of Contents


Basic Principles of Classical Mechanics     1
Newtonian Mechanics     1
Space, Time, Motion     1
Newton-Laplace 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 n-Body Problem     61
The Two-Body 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 n-Body 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 Three-Body Problem     83
Classification of the Final Motions According to Chazy     83
Symmetry of the Past and Future     84
Restricted Three-Body 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 Non-Holonomic 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 Three-Body 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 Three-Body 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 Non-Simply-Connected Regions of Possible Motion     147
Librations in Simply Connected Domains and Seifert's Conjecture     150
Periodic Oscillations of a Multi-Link Pendulum     153
Periodic Trajectories of Non-Reversible 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
Multi-Link 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
Action-Angle Variables     179
Non-Commutative 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 L-A Pairs     197
Integrable Non-Holonomic Systems     199
Differential Equations with Invariant Measure     199
Some Solved Problems of Non-Holonomic Mechanics     202
Perturbation Theory for Integrable Systems     207
Averaging of Perturbations     207
Averaging Principle     207
Procedure for Eliminating Fast Variables. Non-Resonant Case     211
Procedure for Eliminating Fast Variables. Resonant Case      216
Averaging in Single-Frequency Systems     217
Averaging in Systems with Constant Frequencies     226
Averaging in Non-Resonant Domains     229
Effect of a Single Resonance     229
Averaging in Two-Frequency Systems     237
Averaging in Multi-Frequency 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. Non-Degeneracy 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 Lower-Dimensional Tori     297
Variational Principle for Invariant Tori. Cantori     307
Applications of KAM Theory     311
Adiabatic Invariants     314
Adiabatic Invariance of the Action Variable in Single-Frequency Systems     314
Adiabatic Invariants of Multi-Frequency 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
Non-Integrable 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
Quasi-Random Oscillations     373
Poincare Return Map     375
Symbolic Dynamics     378
Absence of Analytic Integrals     380
Non-Integrability in a Neighbourhood of an Equilibrium Position (Siegel's Method)     381
Branching of Solutions and Absence of Single-Valued Integrals     385
Branching of Solutions as Obstruction to Integrability      385
Monodromy Groups of Hamiltonian Systems with Single-Valued 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
Rayleigh-Fisher-Courant 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
Lagrange-Dirichlet Theorem     422
Influence of Dissipative Forces     426
Influence of Gyroscopic Forces     427
Tensor Invariants of Equations of Dynamics     431
Tensor Invariants     431
Frozen-in Direction Fields     431
Integral Invariants     433
Poincare-Cartan 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 Frozen-in Direction Fields     445
Application to Hamiltonian Systems     446
Application to Stationary Flows of a Viscous Fluid     449
Systems on Three-Dimensional Manifolds     451
Integral Invariants of the Second Order and Multivalued Integrals     455
Tensor Invariants of Quasi-Homogeneous Systems      457
Kovalevskaya-Lyapunov 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
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