Customer Reviews

Foundations Of Mechanics

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I don't know if this book really should be considered a book of Classical Mechanics. The reason is that its first 5 chapters (more than 500 pages!) contain almost all the global analysis you'll ever need to know, plus some quite esoteric topics such as a section on general quantization and infinite-dimensional Hamiltonian systems (with the Schroedinger and Korteweg-deVries equations as examples). Copious figures and reasonably clear notation help the reader to understand the (often hard) topological and geometrical concepts. As a book on analytic mechanics, it seems like killing a cockroach with a bazooka. If you plan to learn MECHANICS with a geometrical flavor, and not GLOBAL ANALYSIS with physical motivations, choose instead the shorter, cleaner and much more inspired book of V. I. Arnold, "Mathematical Methods of Classical Mechanics". If you belong to the second group, it's a good place anyway.

The rest of the book is dedicated to dynamical systems, including a fac-simile of a paper of Kolmogorov. However, the topics could be trated with less fuss (as, for example, in the marvellous little and sadly out-of-print book of David Ruelle).

This is a wonderful book. You could loose yourself for almost a whole career here - because this book tries to explain virtually the whole of the subject, right up to all the twentieth century contributions.

Photographs of mathematicians from Gauss and Legendre right up to the most venerable living mathematicians are included in a picture gallery at the front of the book. This is excellent.

The book requires as a beginning, all the material regarding the Hamiltonian and Lagrangian formulations of mechanics - which means you won't have a clue about what the book is saying until you have got somewhat beyond the second year at university. Then the authors start discussing topology, and the ideas which are necessary to re-formulate ideas in quite different clothing. This is very hard - the reader really needs to know about very hard mathematics. Ideas about point set topology are essential because the subject matter encompasses chaotic behaviour and the many body problem. Newtons equations (and this surprises many people) lead to large systems of non-linear equations - and the general theory of the solution of such systems leads almost inevitably to poincare point sets, winding numbers, and so forth. The theory of integral operators (see Kranoselsky, et al) has long been couched in these terms.

Get this by all means, and prepare to have a hard journey ahead.

I should mention that many parts of the book are quite readable and the authors go out of their way to reach the reader as far as possible.

It's actually a physically large book, it would be probably better to get the hard back edition if it's available.

Along with a handful of other works this book is a must for anyone interested in geometric mechanics and control. The text provides a rigorous foundation for a huge subject.

All necessary background is self-contained. However, the book is difficult and I would not recommend it as a first learning text. For that I would send you to Frankel's _The Geometry of Physics_.