Mathematical Aspects of Classical and Celestial Mechanics
Authors: Arnold, Vladimir I., Kozlov, Valery V., Neishtadt, Anatoly I.
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 About this book

In this book we describe the basic principles, problems, and methods of cl sical mechanics. Our main attention is devoted to the mathematical side of the subject. Although the physical background of the models considered here and the applied aspects of the phenomena studied in this book are explored to a considerably lesser extent, we have tried to set forth ?rst and foremost the “working” apparatus of classical mechanics. This apparatus is contained mainly in Chapters 1, 3, 5, 6, and 8. Chapter 1 is devoted to the basic mathematical models of classical  chanics that are usually used for describing the motion of real mechanical systems. Special attention is given to the study of motion with constraints and to the problems of realization of constraints in dynamics. In Chapter 3 we discuss symmetry groups of mechanical systems and the corresponding conservation laws. We also expound various aspects of ord reduction theory for systems with symmetries, which is often used in appli tions. Chapter 4 is devoted to variational principles and methods of classical mechanics. They allow one, in particular, to obtain nontrivial results on the existence of periodic trajectories. Special attention is given to the case where the region of possible motion has a nonempty boundary. Applications of the variational methods to the theory of stability of motion are indicated.
 About the authors

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 FoundationV.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.NeishtadtNeishtadt 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))
 Reviews

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 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)
 Table of contents (9 chapters)


Basic Principles of Classical Mechanics
Pages 160

The nBody Problem
Pages 61101

Symmetry Groups and Order Reduction
Pages 103133

Variational Principles and Methods
Pages 135170

Integrable Systems and Integration Methods
Pages 171206

Table of contents (9 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Mathematical Aspects of Classical and Celestial Mechanics
 Authors

 Vladimir I. Arnold
 Valery V. Kozlov
 Anatoly I. Neishtadt
 Translated by
 Khukhro, E.
 Series Title
 Encyclopaedia of Mathematical Sciences
 Series Volume
 3
 Copyright
 2006
 Publisher
 SpringerVerlag Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
 eBook ISBN
 9783540489269
 DOI
 10.1007/9783540489269
 Hardcover ISBN
 9783540282464
 Softcover ISBN
 9783642066474
 Series ISSN
 09380396
 Edition Number
 3
 Number of Pages
 XIII, 505
 Additional Information
 Original Russian edition published by URSS, Moscow, 2002
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