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…)