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Mathematical Methods of Classical Mechanics [Hardcover]

V.I. Arnol'd , K. Vogtmann , A. Weinstein
5.0 out of 5 stars  See all reviews (8 customer reviews)
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Book Description

Sept. 23 1997 0387968903 978-0387968902 2nd ed. 1989. Corr. 4th printing 1997

This book constructs the mathematical apparatus of classical mechanics from the beginning, examining basic problems in dynamics like the theory of oscillations and the Hamiltonian formalism. The author emphasizes geometrical considerations and includes phase spaces and flows, vector fields, and Lie groups. Discussion includes qualitative methods of the theory of dynamical systems and of asymptotic methods like averaging and adiabatic invariance.

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Second Edition

V.I. Arnol’d

Mathematical Methods of Classical Mechanics

"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."


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In this chapter we write down the basic experimental facts which lie at the foundation of mechanics: Galileo's principle of relativity and Newton's differential equation. Read the first page
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Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
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5.0 out of 5 stars Encyclopedic May 8 2002
Extremely stimulating, uses Galileo to motivate Newton's laws instead of postulating them. Treatment of Bertrand's theorem is beautiful, but contains one error (took me 2 years before I realized where..). However, I know of only one physicist who successully worked out all the missing steps and taught from this book. I know mathematicians who have cursed it. I used/use it for inspiration. The treatment of Liouville's integrability theorem, I found too abstract, found the old version in Whittaker's Analytical Dynamics to be clearer (Arnol'd might laugh sarcastically at this claim!)--for an interesting variation, but more from the standpoint of continuous groups, see the treatment in ch. 16 of my Classical Mechanics (Cambridge, 1997). In my text I do not restrict the discussion of integrability/nonintegrability to Hamiltonian systems but include driven dissipative systems as well. Another strength of Arnol'd: his discussion of caustics, useful for the study of galaxy formation (as I later learned while doing work in cosmology). Also, I learned from Arnol'd that Poisson brackets are not restricted to canonical systems (see also my ch. 15). I guess that every researcher in nonlinear dynamics should study Arnol'd's books, he's the 'alte Hasse' in the field.
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5.0 out of 5 stars The best, but challenging for not-mathematicians Oct. 21 2001
Arnold shines for clarity, completeness and rigour. But, at the same time, he requires a remarkable intellectual effort on the part of the reader (at least a physicist or an engineer). Some readers might see this as a book of math rather than physics, but that would not be fair: Arnold always stresses the geometrical meaning and the physical intuition of what he states or demonstrates. You can take full advantage from the effort of reading this book only if you master a wide range of mathematical topics: essentially differential geometry, ODEs and PDEs and some topology. That's not always true for engineer or physics students at the beginning graduate level. For that kind of readers, Goldstein is a much better fit. Arnold can (and maybe should) be read afterwards.
On the other hand, the exercises, although not very numberous, are very well conceived and help a lot to deepen the comprehension of the text. Also, the order of the topics is linear and very effective from a didactic point of view. The exposition is clear, concise and always goes straight to the point. Thanks to these features, it is one of the most effective books for self-teaching I ever happened to read.
From a physical point of view, the domain of applications is essentially limited to discrete systems. Furthermore, the electromagnetism and relativity are not even cited, although they can be viewed as the logical completion of classical mechanics (see, for example, Goldstein). But the extreme generality of the approach largely balance the more restricted physical domain. In my opinion, the best book you can read on the topics.
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5.0 out of 5 stars little to say Oct. 23 2000
This book is an example of great scientific writing style. Not all the epsilon and deltas are spelled out, and yet the the proofs are nowhere short of rigorous. Besides, they convey insight and intuition: the opposite of Gallavotti's "the element of mechanics" (a very competent book, but obsessed with details). As all the great mathematicians, Arnold separates what's essential from what is not, what is interesting from what is pedantic. It The result is a challenging, wonderful book. I used it (partially) as a second year undergraduate text, and the teacher stressed in the first class that "if you understand Arnold you know classical mechanics". My advice is: get a good grasp of differential geometry and topology and of the tools of the trade (mathematical analysis, ODEs, PDEs) before studying it. Otherwise it will be still readable, but will not be fully appreciated. A last note: it's interesting that Stephen Smale, a mathematician whoshare many interests with V.I.Arnold and is equally illustrious, is another master of style and clarity. You may want to check his book on dynamical systems and his essays.
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5.0 out of 5 stars Clear and to the point Dec 28 2000
By A Customer
I approached reading this book with a certain amount of trepidation. I thought that like with many mechanics books, I will be forced to put it down after page three because the struggle of continuing is too onerous. Surprising, I have gotten past chapter 5 and wish to continue. In other words Arnold does not expect too much from the reader. Contains some formal proofs but not enough dull you interest in the subject. Also unlike many mechanics books it is not filled with endless pompous writing (e.g Goldstein, and Salatan ) but gets directly to the point. Also I like the way the problems are presented. After every couple of paragraphs a problem (of not too great of difficulty) is given for the reader to try. This promotes reinforcement of the subject material. Some of the solutions are given and I only wish I had them all. In short the best advanced classical mechanics book I have come across.
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