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I want to learn the applications of symplectic geometry in physics. Which mathematical physics textbook will have a detailed and heuristic explanation of this aspect?

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possible duplicate of Classical mechanics without coordinates book –  Danu May 26 at 8:26

1 Answer 1

Check out V.I. Arnold's Mathematical Methods of Classical Mechanics.

This book is pretty terse and can have hard to follow notation. However, it is rigorous and contains mathematical explanations and proof of a wide array of topics in mechanics. It is also filled with very interesting examples. He introduces the concepts needed from differential geometry; however, I feel as though this wouldn't be the best introduction to the subject. I recommend reading about manifolds, the tangent/cotangent bundle, integral curves and flows, as well as differential forms from another source. Although, it may not be necessary! In terms of the physics material, he really starts everything from scratch, so no prerequisites needed there.

Also check out Ana Canna's Da Silva's Lectures on Symplectic Geometry. It's pretty mathematical but definitely discusses applications to physics. It's available online here

http://www.math.ist.utl.pt/~acannas/Books/symplectic.pdf

This book is more a book on symplectic geometry than it is physics. But it does address and give examples, when applicable, of the connection between the two. There's a whole chapter on Lagrangian Mechanics, Hamiltonian mechanics, Noether's Principle, and Gauge Theory, written in modern "symplectic geometric" language.

This book is less terse than Arnold's; however, this book assumes a mild background in differential geometry. In particular, the concepts mentioned above.

For the prerequisite material for both of these books, I'd suggest John Lee's Introduction to Smooth Manifolds or Loring Tu's An Introduction to Manifolds. Just take a look at the relevant chapters in these books. The concepts I mentioned above each have there own chapters in both these books.

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