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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. :-)

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up vote 13 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.

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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

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

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Sir, thank you for the link. The lecture notes are pretty good and exhaustive. – Debangshu Mukherjee Sep 18 '10 at 4:01

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...

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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.

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

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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.

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A classic, physically inclined introduction is (Structure of Dynamical Systems: A Symplectic View of Physics, by J.M. Souriau).

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