What is a symplectic form intuitively? 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?
 A: My intuition for the symplectic form in mechanics is that it tells you which coordinates are conjugate.  By Darboux's Theorem, you can always write it as $\sum dx_i\wedge dp_i$, and being able to match a "position" coordinate with a "momentum" coordinate is essential to being able to do classical mechanics and to have equations of motion.
More concretely and rigorously, Steve's answer says essentially the same thing, about turning the Hamiltonian into a vector field so that there will be a flow.
A: To elaborate on a comment of Steve Huntsman: the symplectic form turns a form $d H$ into a flow $X_H$ with a number of properties, but other types of forms can do a similar job.  Indeed, there are a number of situations in physics where the relevant $\omega$ is not symplectic, for example for the following reasons:


*

*$\omega$ might be degenerate in the sense that $i_X \omega = 0$ for certain $X \ne 0$.  This occurs for instance when you pull back $\omega$ to a constraint surface in phase space.  Or you might be working on the Lagrangian side, taking the pull-back of $\omega$ along a non-invertible Lagrangian.

*In non-holonomic mechanics, $\omega$ is sometimes not closed, with the derivative $d \omega$ being related to the non-integrability of the constraint distribution.
The point is that such forms all lead to valid generalizations of Hamilton's equations, so using a symplectic form to write down Hamilton's equations is to a large extent motivated by the fact that it "just works".  The physical properties offered by using a symplectic form instead of an arbitrary two-form are the following:


*

*Non-degeneracy: the evolution vector field $X_H$ is determined uniquely by the Hamiltonian $H$.  By contrast, if you have gauge freedom, there will typically be constraints in the phase space, hence a degenerate symplectic form (see above), leading to a non-unique evolution (which is what gauge freedom is --- several mathematically distinct evolutions being physically the same).  

*Closedness: the system preserves the symplectic form 
$$
  L_{X_H} \omega = d i_{X_H} \omega + i_{X_H} d \omega = 0 
$$
if $\omega$ is closed. In the classical literature, this gives rise to a series of conservation laws called the "Poincare invariants".  Again, non-holonomic systems typically don't exhibit this property, leading to all sort of weirdness.
A: 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.
A: $\omega$ lets you turn $H: X \rightarrow \mathbb{R}$ into a vector field and then a flow by establishing $H \rightarrow X_H$ via $dH(Y) = \omega(X_H,Y)$. The almost complex structure provided by $\omega$ always locally looks the same (viz. $\mathbb{R}^{2n}$ with $(x,p)$ coordinates) by the Darboux theorem and canonical transformations are just symplectomorphisms.
