Poisson manifold

Poisson manifold

A Poisson bracket on a smooth manifold $M$ is a Lie bracket $\{~,~\}$ on the space of smooth functions $C^\infty(M)$ which, additionally, satisfies the Leibniz identity: $$ \{f,gh\}=\{f,g\}h+g\{f,h\},\qquad \forall f,g,h\in C^\infty(M).$$ The pair $(M,\{~,~\})$ is called a Poisson manifold.

On a Poisson manifold $(M,\{~,~\}), any smooth function $h\in C^\infty(M)$ determines a '''hamiltonian vector field''' $X_h$ by setting: $$ X_h(f):=\{h,f\}.$$