© 2006

Mathematical Aspects of Classical and Celestial Mechanics

Third Edition


Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 3)

Table of contents

  1. Front Matter
    Pages I-XIII
  2. Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt
    Pages 1-60
  3. Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt
    Pages 61-101
  4. Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt
    Pages 103-133
  5. Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt
    Pages 135-170
  6. Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt
    Pages 171-206
  7. Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt
    Pages 207-349
  8. Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt
    Pages 351-399
  9. Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt
    Pages 401-429
  10. Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt
    Pages 431-468
  11. Back Matter
    Pages 469-518

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 non-trivial results on the existence of periodic trajectories. Special attention is given to the case where the region of possible motion has a non-empty boundary. Applications of the variational methods to the theory of stability of motion are indicated.


celestial mechanics classical mechanics classsical mechanics integrability nonintegrability perturbation theory

Authors and affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia
  2. 2.CEREMADEUniversité Paris 9 - DauphineFrance
  3. 3.Steklov Mathematical InstituteMoscowRussia
  4. 4.Department of Mathematics and MechanicsLomonosov Moscow State UniversityMoscowRussia
  5. 5.Space Research InstituteMoscowRussia
  6. 6.Department of Mathematics and MechanicsLomonosov Moscow State UniversityMoscowRussia

About the authors


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


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


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))

Bibliographic information

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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)