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?



What is a symplectic form intuitively?
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.


