Ordinary Differential Equations

Rating:
Summary: excellent, 1st of 2 english versions
Review: Be aware there are 2 versions of this book
available in English; this one from MIT press
is (contrary to one of the reviews above) is
translated from the *first* Russian edition;
there is another version from Springer-Verlag
translated from the *third* Russian edition.
They're translated by different people so
some wording etc is different but otherwise
they're similar, though not identical. The
later edition has some reworked passages
and modest amount of new material, but it's
not a hugely different book.

Both are excellent, are are all the other
books & papers I've seen by V.I. Arnol'd.

Rating:
Summary: excellent, 1st of 2 english versions
Review: It is hardly needed to add words to the existing positive reviews of the book. In the line of previous comments, I just mention that it is an enjoyable book on a basic subject of great interest also for engineers and physicists. The matter is treated with the evident purpose to make the reader fully aware of the interesting geometrical and dynamic implications of the conclusions reached at each step. It is a nice counterexample for those who believe that, to be rigorous, a mathematical book needs to be very hard to read.

Rating:
Summary: Stimulating
Review: Like his books on classical mechanics, a book that theoretical physicists should read. Unfortunately, the discussion of local integrability is too abstract and there is no distinction made with global integrability. Also irritating: because of a singularity at the origin the damped harmonic oscillator is not recognized as integrable in spite of the existence of a global conservation law, excepting one point in phase space. Integrability is an extremely difficult subject and maybe Arnol'd could have taught us more about it. I've discussed integrability/nonintegrability from a physicist's perspective in my Classical Mechanics (Cambridge, 1997).

Rating:
Summary: awesome!
Review: This book is WAY better than Hirsch's Dynamical Systems fiasco. Once again Arnold has dissapointed Hirsch's fans by this masterpiece. Just read it and you'll see what I'm talking about. Arnold's ODE is simply the best there is.

Rating:
Summary: A beauty; a struggle
Review: This has to be one of the most amazing math books I've ever read. Arnol'd seems to do the impossible here - he blends abstract theory with an intuitive exposition while avoiding any tendency to become verbose. By the end of Arnol'd, it's hard not to have a deep understanding of the way that ODEs and their solutions behave.

Arnol'd accomplishes this feat through an intense parsimony of words and topics. Everything in this book builds on the central theme of the relations between vector fields and one-parameter groups of diffeomorphisms, and the topics are illustrated (and often motivated) almost exclusively through problems in classical mechanics, most notably the plane pendulum. Almost no solution techniques are given in this book - expect no mention of integrating factors or Bessel functions. One of the main reasons that the book does so much without bogging down is that the mathematical formalism is minimal and terse - proofs are often one or two lines long, merely mentioning the conceptual justification of a result without detailed, formal constructions.

But the result of this parsimony is that Arnol'd is a very difficult book. To understand every detail and to be able to attempt every problem, I think, basically requires a math degree - lots of linear algebra (for his monumental 116-page chapter on linear systems), a solid background in analysis and topology, and a bit of differential geometry and abstract algebra are prerequsites for a full understanding. (I found the section on the "topological classification of singular points," in particular, nearly incomprehensible with my thin chemistry-major math background.) There are foibles, too, including proofs that satisfy the requirements for some theorem or definition without actually stating what theorem or definition is now applicable. One can detect some mild arrogance in places (after an arduous two-page proof, he mentions "As always in proving obvious theorems, it is easier to carry out the proof of the extension theorem than to read through it.") Also, a few typos can be found here and there, which sometimes result in confusion.

One very curious thing about Arnol'd is that my most brilliant math-major friends find it impenetrable, whereas I know biologists who got through it with no problem. So I guess that, for a mere mortal, reading Arnol'd demands a willingness to have a feel for a big picture without worrying about every epsilon and delta.

So grab a copy of this book, let it flow, and learn about ODEs. It's well worth the effort.