Poisson manifold

A Poisson manifold is a smooth manifold with a bilinear bracket on smooth functions that is antisymmetric, obeys the Jacobi identity, and is a derivation in each argument. Its Poisson bivector may have nonconstant rank, unlike the inverse of a symplectic form.

A Casimir function $C$ Poisson-commutes with every smooth function, $\{C,f\}=0$. Its regular level sets are unions of symplectic leaves.

A symplectic leaf is a maximal connected immersed submanifold on which the Poisson bivector has constant full rank and therefore inverts to a symplectic form.

The rotational Lie-Poisson bracket on ${\mathbb{R}}^3$ isIts Casimir is $x_1^2+x_2^2+x_3^2$, and every sphere centered at the origin is a two-dimensional symplectic leaf on which $SO(3)$ acts symplectically.