Introduction to the Perturbation Theory of Hamiltonian Systems (Springer Monographs in Mathematics)

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9783642261046: Introduction to the Perturbation Theory of Hamiltonian Systems (Springer Monographs in Mathematics)

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.

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From the Back Cover:

This book presents the basic methods of regular perturbation theory of Hamiltonian systems, including KAM-theory, splitting of asymptotic manifolds, the separatrix map, averaging, anti-integrable limit, etc. in a readable way. Although concise, it discusses all main aspects of the basic modern theory of perturbed Hamiltonian systems and most results are given with complete proofs.

It will be a valuable reference for Hamiltonian systems, and of special interest to researchers and graduate students of the KAM community.


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)

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