# Why does symplectic geometry have many applications in mathematics

It is not quite intuitive , at least from its origin. Could any one can give me an intuitive explanation?Thank you!

-

Symplectic geometry is applicable in lot of areas such as Classical mechanics, Hamiltonian mechanics. While browsing through the web i have found a reason as to why SG is good for classical mechanics. You may read it here: http://research.microsoft.com/en-us/um/people/cohn/thoughts/symplectic.html

-
I think the question is more about why symplectic geometry has found applications beyond classical mechanics. –  Qiaochu Yuan Sep 18 '10 at 5:34
@Qiaochu Yuan: I answered the question, w.r.t my thinking. –  anonymous Sep 18 '10 at 7:21

Any two of (a) symplectic structure, (b) almost-complex structure, and (c) Riemannian structure determine the third. This reflects the fact that $$U(n) = O(2n) \cap GL(n,\mathbb{C}) \cap Sp(2n),$$ but the intersection of any two is already $U(n)$. This gives rise to the theory of Kähler manifolds (cf. wikipedia), which are central to mirror symmetry (and the mirror symmetry conjecture).

-

There is a simple "philosophical" answer: Symplectic forms allow you to only measure two-dimensional quantities, not one-dimensional ones (you can measure area infinitesimally). That's the basic difference with Riemannian geometry, where you can measure length infinitesimally.

Furthermore symplectic geometry pops up in connection with the cotangent bundle. For instance if you have a (scalar) first order differential operator on a manifold, then the principal symbol transforms like a covector (when changing coordinates). Note that this does not require a metric.

-