Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics, Vol. 60)

Arguably, the applicability of a mathematical theory (or its links with other well established parts of this science) is what makes it important. This book serves to justify in this sense the study of ordinary differential equations, calculus of variations, Riemannian geometry, symplectic geometry, Lie groups and Lie algebras, manifold theory as well as other more specialized subjects such as integrable systems or catastrophe theory. There are many other books on classical mechanics, some of them stronger than this one in some respects but this is the book to read if you do not want or can't consult a whole library. Foundations of Mechanics by Abraham and Marsden is a colossal treatise that certainly seeks to be a reference work rather than a textbook, it can be useful as a place to look for details you cannot find in the appendices of Arnold's book; Introduction to Mechanics and Symmetry by Marsden and Ratiu is more accessible, the historical comments and abundance of examples are very interesting or/and enlightening, however the order and choice of material is somewhat puzzling, it is inevitable to compare it with Arnold's brilliant layout: one begins with Newtonian/Galileian approach and subsequently those methods are refined and generalized with the Lagrangian and Hamiltonian formalisms. Very worth mentioning are the appendices which constitute almost half of the current edition of Arnold's book: one can find there from an intuitive discussion of Riemannian geometry and the generalization to finite and infinite dimensional Lie groups, made by Arnold in the sixties, of Euler's equations for the rigid body, to discussions of the now so popular momentum maps, Poisson structures, Kähler manifolds, KdV equations and a bit of KAM theory. Do not expect this book to give you (as a previous reviewer wrote) all the epsilons and deltas and explicit formulae you might be used to find in a textbook, the arguments are very concise and sometimes the proofs are cryptic but very often the intuitive idea and the geometrical insight of a proposition is all that is required to produce a rigorous proof and that's exactly what this book gives you.

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.

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.

Firstly, I do not agree with some of the reviews here that say "Best place to learn classical mechanics". It is the last place to learn classical mechanics. The book is about the mathematical methods of classical mechanics, rather than the physics of classical mechanics. Other books serve that purpose admirably, and the list is a long one (Try Goldstein, Landau, Taylor, Morin, Lanczos, Thornton, FLP or even an old classic like Sommerfeld's Classical Mechanics). This work, however, is like applying an ornate gold frame to a rennaisance painting. It makes the subject look pleasing, even if sometimes inaccessible. For accessibility, I found Taylor, Kleppner & Kolenkow fantastic for UG and Goldstein still the very best for Grad school. This is best served as a reference.

There are books that teach you stuff, and there are books that open the door to a world you never knew existed. Arnold's "Mathematical Methods" is, to this day, my secret door to beautiful mathematics. If you are looking for an easy read, this is the wrong place. Arnold's writing is the very opposite of Bourbaki's style of mathematical exposition that leaves very little to your imagination. This book frees your imagination, and it forces you to ask yourself many questions, something I have experienced with very few other books. Arnold was a man with strong and vocal opinions. In particular, he was a vocal supporter of geometric thinking as opposed to algebraic thinking. This book is as eloquent an argument on the depth and beauty of geometry as you could find anywhere. Arnold has poured his mind and heart in this book, and his magic will certainly affect you. All you need is an open mind.

Lagrangian and Hamiltonian mechanics are elucidated using the appropriate mathematics, namely variational calculus and symplectic geometry. The appendices provide even more useful info, including eg. the Euler-Arnold equation and contact geometry. Eschewing Bourbaki-style theorem/proof writing, much more mathematical maturity is nevertheless expected of the reader compared to a typical mechanics textbook. Great for mathematicians looking to begin their physics education!

This is a thorough, mathematical treatment of classical mechanics as well as it's underlying mathematics. The sections on Hamiltonian dynamics and symplectic geometry are particularly impressive. I've had many "mixed" experiences with little yellow books from Springer, but this is well done and quite readable. The fact that it's translated from Russian is surprisingly not an issue (no awkward phrasing, descriptions and conversational sections still "feel conversational").

I took my First course of Symplectic Geometry and i used this book, is a perfect introduction to this branch of Geometry. The ideas were very natural and easy, so let you understood very well the real meaning of the results.

This is almost a natural continuation of the excellent book "Mechanics" by Landau. It uses more sophisticated mathematics than Landau, but covers the same topics - and some more. An absolute must-read.

For a first course in classical mechanics may be a little hard, because in most cases one does not have enough math insight at the first contact with classical mechanics. But it is not impossible either, I strongly recommend this book to be used in early courses to at least familiarize with the math. The book itself is fantastic, it presents both physical concepts and the math needed to formulate them.

The content of the book is excellent (very advanced), however the print qualities of recent Springer books are appalling. Nonetheless, I managed to get this book at really low price.

This book is a remarkably clear exposition of the subject and mathematical formalism of classical mechanics. Arnold presents the material starting from simple principles and develops the subject from there, maintaining the clarity as the mathematical complexity grows. For those of us more accustomed to the abstract approach to the mathematical foundations of the subject, this come as a breath of fresh air. The chapters on Hamiltonian mechanics and it's mathematical foundations are a pleasure to read.

Un excelente libro del Matemático Vladimir Arnold, sin embargo, posee errores de impresión en muchas de las figuras, en particular en las de las paginas 19, 107, 111, 212, 392, 395 y 477 (las que he podido notar). Estos errores, si bien no le restan merito a la obra, si hacen notar que la editorial Springer no pone cuidado en que los libros sean impresos correctamente, pues comparando con algunos colegas, los errores de impresión en está obra llevan muchos años ocurriendo.