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This Book Constructs The Mathematical Apparatus Of Classical Mechanics From The Beginning, Examining Basic Problems In Dynamics Like The Theory Of Oscillations And The Hamiltonian Formalism The Author Emphasizes Geometrical Considerations And Includes Phase Spaces And Flows, Vector Fields, And Lie Groups Discussion Includes Qualitative Methods Of The Theory Of Dynamical Systems And Of Asymptotic Methods Like Averaging And Adiabatic Invariance This book written by the great Geometric mechanic provides a firm and stable introduction and reference for mathematicians and physicists of Geometric mechanics Arnold begins the book with a brief introduction to Newtonian mechanics and develops onto Lagrangian and Hamiltonian formalisms The descriptions and derivations are rigorous in that they provide a solid ground to advance between frameworks In addition, he incorporates several topics the hidden structures that are rarely discussed in typical core courses of classical mechanics such as group theory, topology, and differential geometry When one reaches Lagrange s equations, we are introduced to the theory of manifolds and how one can appropriate mechanics to this new formalism From then on, one reaches Hamiltonian physics with discussions on symplectic manifolds along with Lie algebras and a well written chapter on perturbation theory The problems provided ask basic questions but most rely on verifying the propositions of the author.I feel the most vital portions of this book are the appendices They provide introductions to subjects which are important for researchers, one being on contact manifolds Contact structures which are important when one has an odd dimensional manifold Other topics are dynamical system symmetries and normal forms and to many fields they are still relevant for researchers.I would suggest this book for those who seek a deeper mathematical understanding of Mechanics, so can be seen as the step beyond Goldstein for example Compared to other books of Geometric Mechanics it is docile Walter Thirring s book Classical Mathematical Physics is actually rigorous and Prop Proof based If someone wishes to dive even deeper, one can check out Souriau s Structure of Dynamical Systems or the mammoth tome Foundations of Mechanics by Ralph Abraham and John Marsden.For dynamical systems and nonlinear dynamics researchers, this is a great book to mature mathematical tools and ideas. 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. 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 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 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, K hler 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.