TITLE: What is a symplectic form intuitively?
QUESTION [45 upvotes]: Hi,
to completely describe a classical mechanical system, you need to do three things:
-Specify a manifold $X$, the phase space. Intuitively this is the space of all possible states of your system.
-Specify a hamilton function $H:X\rightarrow \mathbb{R}$, intuitivly it assigns to each state its energy.
-Specify a symplectic form $\omega$ on $X$. What is $\omega$ intuitively? What kind of information about physics does it capture?

REPLY [15 votes]: Incidentally, I more or less disagree that symplectic geometry captures what I would consider "classical mechanics".  The reason is that in all the examples that I think deserve to be called "classical mechanics", I actually have a configuration space $N$, and your symplectic manifold is $X = {\rm T}^*N$ the cotangent bundle.  Then, of course, the symplectic form is precisely (part of) the cotangent structure.
This is not to say that symplectic geometry isn't interesting — it's led to great mathematics, and certainly captures some of "classical mechanics".  From the physics perspective, what I think makes it most interesting is that it shows that there are strange symmetries between mechanical systems, when you have a symplectomorphism ${\rm T}^*N \to {\rm T}^*N'$ that does not arise from a diffeomorphism $N \to N'$.
But physics is not invariant under all symplectomorphisms.  Otherwise, how would I know which coordinates are "position" and which are "momentum"?  And I do believe that I know this, although maybe I'm wrong.  You and I should get together and compare if our Darboux coordinates differ only by a map $N \to N'$, or by some more interesting symplectomorphism.