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 | | Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics) by V. I. Arnold, A. Weinstein (Translator), K. Vogtmann (Translator) ISBN-10: 0387968903 ISBN-10: 0-387-96890-3 ISBN-13: 9780387968902 ISBN-13: 978-0-387-96890-2 Hardcover 1997-09-05 Springer Find Lowest Price |
Editorials | ||
Product Description In this text, the author constructs the mathematical apparatus of classical mechanics from the beginning, examining all the basic problems in dynamics, including the theory of oscillations, the theory of rigid body motion, and the Hamiltonian formalism. This modern approch, based on the theory of the geometry of manifolds, distinguishes iteself from the traditional approach of standard textbooks. Geometrical considerations are emphasized throughout and include phase spaces and flows, vector fields, and Lie groups. The work includes a detailed discussion of qualitative methods of the theory of dynamical systems and of asymptotic methods like perturbation techniques, averaging, and adiabatic invariance. | ||
Reviews | ||
Wonderful This book is an excellent introduction to the world of classical physics for NON-PHYSICISTS. While some physicists will no doubt find it accessible, there is considerable reduction of physical concepts in order to get to the heart of the ideas underlying the formalism. Also, the material goes beyond what most physicists (non-theoreticians) will find practical. He focuses largely on a geometric presentation, in the language of differential geometry, symplectic geometry, differential forms, Riemannian manifolds and includes a large amount of algebraic necessities. This is not a cookbook for learning how to solve classical mechanics, nor is it a math book per se, but it is a wonderful collection of introductions to a vast amount of useful mathematical formalism that permeates the physical literature. I would strongly recommend it to someone needing a thorough supplementary mechanics text, one that relies on very little physical insight and focuses on the geometric and algebraic structures underlying them. The chapters are very well self-contained for the most part so you can skip to topics you find more appealing without feeling lost. Also, his presentation style is very clever, in case you're a fan of quick thinking and novel presentations (who isn't?). The prerequisites are familiarity with somewhat advanced calculus and "mathematical maturity". Basic knowledge of group theory would also make it an easier read. | ||
A unique, masterful and enjoyable book for graduate student in physics The book is full of little enjoyable details (jewels). Arnold is one of the few mathematicians which approaches problems with a very geometric point of view. In his interview with S.H. Lui he mentions how algebraic picture has dominated the research in mathematics and how he has tried to counter that. One can see the trace of his ingenuity all over this book. What some may call as handwaving in math circles is indeed called as physical (or geometric) intuition in physics community and is being actively encouraged. The chapters on oscillations (chap. 5) and perturbation theory (chap. 10) are very instructive. For example, parametric resonance is discussed concisely in chapter 5 which you won't be able to find it anywhere else. where can you learn about "Arnold's tongues" better than in Arnold's book? There are so many appendices at the end of the book. They are often very specialized and I don't recommend you to read them on your first read. In conclusion, I recommend this book to any physics graduate student. In fact, I hope one day it will be used as a text book for courses in classical mechanics. | ||
I would recommend foundations of mechanics by Marsden I have to admit that I haven't thoroughly read through this text. But judging from the first 10 pages, there is a lot of mathematical handwaving. In contrast, foundations of mechanics (hereafter FOM) is far superior in that it provides all the necessary background beyond calculus and linear algebra to the reader, and is logically consistent so far in my reading. I want to mention that there are certainly complete and excellent texts out there on functional analysis, differential geometry, and topology, but many texts include way more stuff than you would want to know. In particular, it is my humble opinion that once you get to a certain point of knowledgeability of a subject like algebraic topology, you have enough of a taste for it that to learn more of the subject would only help if you were to go into research. Therefore a book like FOM provides a concise and practical treatment of those various advanced mathematics topics. | ||
Best book on CM Best book on CM (based most on symplectic formulation). Extremely clear if one has enough patience to follow exactly the author's way and to work out the proposed stimulating problems. Contains an original way of introducing differential forms, integration of differential forms and homology/De Rahm's thm.: you fully get in the subject in few pages ! The first part does not make use of symplectic formalism but is also quite original and stimulating. The level is last yr. undergr. 1st yr. graduate. Very useful if used with E. ott (Chaos in Dynamical Systems) for studying nonlinear dynamics. | ||
Encyclopedic 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. | ||