Poisson bracket on symplectic manifold

A special Poisson algebra

Given a symplectic manifold $(M,\omega )$, we can define a Poisson bracket on the smooth functions $C^{\infty }(M)$ as $\{f,g\}=\omega (X_f,X_g)$ for smooth functions $f,g\in C^{\infty }(M)$ and their Hamiltonian vector fields $X_f,X_g$ respectively.

This can be restated (with a little bit of work) using the Lie derivative as $\{F,G\}={\mathcal{L}}_{X_f}G$

Poisson commuting functions

If $f,g\in C^{\infty }(M)$ Poisson commute, i.e. $\{f,g\}=0$, then

• $f$ is constant on integral curves of $X_g$
• $g$ is constant on integral curves of $X_f$.