Poisson bracket on symplectic manifold

A special Poisson algebra

Given a symplectic manifold , we can define a Poisson bracket on the smooth functions as for smooth functions and their Hamiltonian vector fields respectively.

This can be restated (with a little bit of work) using the Lie derivative as

Poisson commuting functions

If Poisson commute, i.e. , then

• is constant on integral curves of
• is constant on integral curves of .