Mathematical Aspects of Classical and Celestial Mechanics (Encyclopaedia of Mathematical Sciences Book 3) 3rd Edition by Vladimir I. Arnold (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is not a beginners guide and it is not intended to be one. It is a very sophisticated discussion of advanced topics covering classical mechanics completely (not only the mathematical methods as the title suggests, but also its physical background). It is not centered around integrable systems like most monographs on this topics but on exact results beyond integrability, which have been mainly obtained by the methods of KAM-Theorie (Kolmogorov, Arnold (one of the authors) and Moser), although e.g. the discussion of the two-body-problem is very detailed and complete. The focus is more on methods than on providing details of particular systemn this – the title was obviously chosen with care, but every method is explained on well selected physical examples illustrating not only the straight forward application but also the difficulties and subtleties, without digressing to unphysical systems or pure mathematics. All three authors are well known for their scientific merrits, but in this book they also show themselves as masters of excellent and comprehensive presentation of advanced topics, including non-integrable systems and an excellent and up to date coverage of all known aspects of the n-body-problem in cellestial mechanics. The book is not suitable as a beginners guide, but knowledge of Landau-Lifshitz Vol. I and/or Arnolds excellent book on the mathematical aspects of classical mechanics together with some background on Lie-Groups, differential forms and manifolds is sufficient to read the book, although presumably not to fully appreciate its excellence in exactness and precision as well as elegance in presentation. Who wants to know what the variational principle works, what are the differences between Newton’s, Langrange’s and Hamilton’s equations, what integrabilty really is, how modern integration techniques work, how symmetries can be used to find integrals of motion in the most general sense, what convergent perturbation expansions are and on what principles they are based etc. should read this book and he should better read it twice. Topics rarely found in advanced texts like several approaches to nonholonomic mechanics and a complete discussion of small oscillations. Very likely even skilled experts will find subtleties they did not know in this book that will stand second to none for many years to come.

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