Yesterday I discussed the Poisson bracket on a symplectic manifold and some of its basic properties. Naturally enough, someone decided to axiomatize all this.

Poisson manifolds

So, a Poisson manifold is a smooth manifold ${M}$ together with a Lie algebra structure ${\{\cdot, \cdot\}}$ (the Poisson bracket) on the space of smooth functions on ${M}$ such that

$\displaystyle \{ fg, h \} = f\{g, h\} + g \{f, h \}.$

To check that a symplectic manifold is indeed a smooth manifold, we need only recall that there is a vector field ${H_h}$ with ${\{ f, h \} = -H_h f}$, so the above is just the derivation identity. All the other properties of the Poisson bracket on a symplectic manifold were established yesterday.

Now let’s switch to the general Poisson manifold case.

It turns out that the Poisson structure alone is enough to show many similarities. Indeed, the derivation identity above implies that there is a vector field, still denoted by ${H_h}$, with

$\displaystyle H_h f = - \{ f, h \}$

and in particular it follows that ${H_f f = 0}$ by antisymmetry. The set of such vector fields ${H_h}$ will be called, as in the symplectic case, Hamiltonian vector fields. I claim that the identity

$\displaystyle \boxed{ [H_f, H_g] = H_{ \{f,g\}} }$ (more…)

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…)