So, now that we have a notion of symplectic manifolds, we can talk about the Poisson bracket.   This gives a way of making the smooth functions into a Lie algebra.  The first step in the story is to use the symplectic form to associate to a function a vector field (obtained by duality from $df$).  These Hamiltonian vector fields have many nice properties: for instance, their Lie bracket is of the same type.  Moreover, they (and, locally, only they) are the vector fields whose flows preserve the symplectic form.  In mechanics, the flows of the Hamiltonian field associated to the energy function trace out the paths of a particle acted on by a conservative force.

Let ${M, \omega}$ be a symplectic manifold. Given a smooth function ${f: M \rightarrow \mathbb{R}}$, we have a 1-form ${df}$ on ${M}$. The self-duality of ${TM}$ induced by ${\omega}$ can be used to “lower indices” (kind of like how one gets a gradient on a Riemannian manifold) so that we get a vector field. Call its opposite ${H_f}$, the Hamiltonian vector field associated to ${f}$.

In other words,

$\displaystyle \sigma( H_f, V) = -df(V) = -Vf.$

By nondegeneracy, this uniquely determines ${H_f}$. (more…)