Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Can someone please tell me some introductory book on symplectic geometry? I have no prior idea of the subject but I do know about Lagrangian and Hamiltonian dynamics (at the level of Landau-Lifschitz Vol. 1). Thanks in advance. :-)

share|improve this question
    
Hi Debangshu Mukherjee, would you be interested in a higher-level Physics site, Physics Overflow? We are in our "public beta", during which we will finish importing high-level P.SE questions (we have already imported some, and all of TP.SE), and work on a "reviews" section, where users can review ArXiV papers and papers from journals, conferences, etc. –  Dimensio1n0 Apr 5 at 4:55
    
If you are interested in participating on Physics Overflow, you may find us here: physicsoverflow.org/trollsouthere14 Some of your questions and answers on Physics Stack Exchange have already been imported, and the others will be imported soon, so you technically already have an account on Physics Overflow, and may regain access to it here: physicsoverflow.org/trollsouthere14/regain-account-page –  Dimensio1n0 Apr 5 at 4:58
add comment

6 Answers

up vote 9 down vote accepted

If you are physically inclined, V.I.Arnold's Mathematical methods of classical mechanics provides a masterful short introduction to symplectic geometry, followed by a wealth of its applications to classical mechanics. The exposition is much more systematic than vol 1 of Landau and Lifschitz and, while mathematically sophisticated, it is also very lucid, demonstrating the interaction between physical ideas and mathematical concepts that support them. (It is also worth mentioning that Arnold was largely responsible for the reawakening of interest to symplectic geometry at the end of 1960s and pioneered the study of symplectic topology. Some of these developments were brand new when the book was first published in 1974 and are briefly discussed in the appendices).

In addition to the notes by Cannas da Silva mentioned by Dick Palais, here are further two advanced books covering somewhat different territory:

Michèle Audin, Torus actions on symplectic manifolds (2nd edition)A

Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology


A In her book, Michèle Audin herself recommends

Paulette Libermann and Charles-Michel Marle, Symplectic geometry and analytical mechanics

as a wonderful introduction to symplectic geometry.

share|improve this answer
    
It's characteristic of Arnold's expository genius that even topics tangential to the book's main line of thought (e.g. differential forms, Legendre transforms, etc) are treated more intuitively, yet more briefly, here than in more specialized books. This book is packed so full of insights that you can keep reading it for years and keep learning new things from it. –  Per Vognsen Sep 17 '10 at 6:59
    
Arnold's book is difficult to understand if you have not already mastered the subject. Audin's book is even worse (since she is very sloppy at times (the latest edition still has a lot of mistakes & sometimes important things are not fully explained) and uses non-standard notation). –  Orbicular Sep 17 '10 at 12:34
    
Orbicular, you are welcome to share your insights on the books that you do not find objectionable. My own experience with both "Mathematical methods of classical mechanics" and "Torus actions on symplectic manifolds" was diametrically opposite of yours. –  Victor Protsak Sep 17 '10 at 14:48
    
We (a group of PhD students) went through Audin's exercises. It was horrible at times, sorry! –  Orbicular Sep 17 '10 at 20:14
add comment

You can find a very nice introduction to the subject in these notes by Ana Cannas da Silva:

www.math.princeton.edu/~acannas/symplectic.pdf

share|improve this answer
    
Sir, thank you for the link. The lecture notes are pretty good and exhaustive. –  Debangshu Mukherjee Sep 18 '10 at 4:01
add comment

My favourite book on symplectic geometry is "Symplectic Invariants and Hamiltonian Dynamics" by Hofer and Zehnder. It's wonderfully written. Another lovely book (which has just been reissued as an AMS Chelsea text) is Abraham and Marsden's book "Foundations of Mechanics" which covers a lot of symplectic geometry as well as so much more...

share|improve this answer
add comment

Sternberg and Guillemin's Symplectic Techniques in Physics is one of a kind. In spite of the name it feels more like a text on mathematics than on physics, with the exception of the first motivating section of the book.

Arnold's book that Victor recommends is also one of my favorites. But much of it covers the kind of material you might find in Goldstein or Landau-Lifschitz, albeit treated from a more sophisticated and geometric point of view. If you already have that thoroughly mastered, Sternberg and Guillemin might be more what you want, especially the later parts.

share|improve this answer
add comment

You can also try the book An introduction to symplectic geometry by Rolf Berndt which should be a good fit given your prerequisites.

share|improve this answer
add comment

For a more Lie-group focused account, you can try Robert Bryant's lectures on Lie groups and symplectic geometry which are available online here. In the final lecture he describes the h-principle and others ideas of Gromov in symplectic geometry, like pseudo-holomorphic curves.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.