Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics, No 60)

Rating:
Summary: After reading Arnold
Review: After reading Arnold, I know no other authors of classical mechanics.

Rating:
Summary: The best, but challenging for not-mathematicians
Review: 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.

Rating:
Summary: Best book on CM
Review: 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.

Rating:
Summary: Encyclopedic
Review: 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.

Rating:
Summary: Clear and to the point
Review: 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.