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Introduction to the Perturbation Theory of Hamiltonian Systems

  • Book
  • © 2010

Overview

Authors:
  1. Dmitry Treschev

    1. Steklov Mathematical Inst., Dept. Theoretical Physics, Russian Aacademy of Sciences, Moscow, Russian Federation

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  2. Oleg Zubelevich

    1. Russian Federation, Budget & Treasury Academy, Ministry of Finance of the, Moskva, Russian Federation

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    You can also search for this author in PubMed   Google Scholar

  • Monograph with new results in the theory of Hamiltonian systems
  • Includes supplementary material: sn.pub/extras

Part of the book series: Springer Monographs in Mathematics (SMM)

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Table of contents (9 chapters)

  1. Front Matter

    Pages I-X
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  2. Hamiltonian Equations

    • Dmitry Treschev, Oleg Zubelevich
    Pages 1-21

  3. Introduction to the KAM Theory

    • Dmitry Treschev, Oleg Zubelevich
    Pages 23-58

  4. Splitting of Asymptotic Manifolds

    • Dmitry Treschev, Oleg Zubelevich
    Pages 59-74

  5. The Separatrix Map

    • Dmitry Treschev, Oleg Zubelevich
    Pages 75-92

  6. Width of the Stochastic Layer

    • Dmitry Treschev, Oleg Zubelevich
    Pages 93-106

  7. The Continuous Averaging Method

    • Dmitry Treschev, Oleg Zubelevich
    Pages 107-130

  8. The Anti-Integrable Limit

    • Dmitry Treschev, Oleg Zubelevich
    Pages 131-142

  9. Hill’s Formula

    • Dmitry Treschev, Oleg Zubelevich
    Pages 143-162

  10. Appendix

    • Dmitry Treschev, Oleg Zubelevich
    Pages 163-200

  11. Back Matter

    Pages 201-211
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Keywords

About this book

This book is an extended version of lectures given by the ?rst author in 1995–1996 at the Department of Mechanics and Mathematics of Moscow State University. We believe that a major part of the book can be regarded as an additional material to the standard course of Hamiltonian mechanics. In comparison with the original Russian 1 version we have included new material, simpli?ed some proofs and corrected m- prints. Hamiltonian equations ?rst appeared in connection with problems of geometric optics and celestial mechanics. Later it became clear that these equations describe a large classof systemsin classical mechanics,physics,chemistry,and otherdomains. Hamiltonian systems and their discrete analogs play a basic role in such problems as rigid body dynamics, geodesics on Riemann surfaces, quasi-classic approximation in quantum mechanics, cosmological models, dynamics of particles in an accel- ator, billiards and other systems with elastic re?ections, many in?nite-dimensional models in mathematical physics, etc. In this book we study Hamiltonian systems assuming that they depend on some parameter (usually?), where for?= 0 the dynamics is in a sense simple (as a rule, integrable). Frequently such a parameter appears naturally. For example, in celestial mechanics it is accepted to take? equal to the ratio: the mass of Jupiter over the mass of the Sun. In other cases it is possible to introduce the small parameter ar- ?cially.

Reviews

From the reviews:

“This Springer monograph, based on lectures given by the first author at Moscow State University … regarded as a textbook on ‘advanced topics in perturbative Hamiltonian mechanics’. … The style is concise and precise, and the book is suitable for graduate students and researchers. Proofs are usually complete and, if not, references are given. In conclusion, the book constitutes a precious addition to the literature concerning the dynamics of perturbation theory of Hamiltonian systems.” (Luigi Chierchia, Mathematical Reviews, Issue 2011 b)

“The present book is an excellent introduction to this subject and covers several classical topics: the KAM theory (and the Birkhoff theorem); the Poincaré-Melnikov theory of the splitting of asymptotic manifolds (in connection with chaos); the separatrix map (and generalizations). Also, special methods are used: asymptotical formulas describing quantitatively stochastic layers; averaging procedures. … In conclusion, the book will be a very good reference for beginners.” (Mircea Crâşmăreanu, Zentralblatt MATH, Vol. 1181, 2010)

“This is a very readable textbook on regular perturbation theory of Hamiltonian systems. … The appendix on diophantine properties, resonance, etc., and specific functional analytic methods is a very valuable addition rendering the text almost self-contained. Most results are given with complete proofs, so that the book may be of good service to researchers and graduate students with interest in mechanics.” (G. Hörmann, Monatshefte für Mathematik, Vol. 162 (2), February, 2011)

Authors and Affiliations

  • Steklov Mathematical Inst., Dept. Theoretical Physics, Russian Aacademy of Sciences, Moscow, Russian Federation

    Dmitry Treschev

  • Russian Federation, Budget & Treasury Academy, Ministry of Finance of the, Moskva, Russian Federation

    Oleg Zubelevich

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