nLab
Poisson manifold

Poisson manifolds are a mathematical setup for classical mechanics with finitely many degrees of freedom.

A Poisson algebra? is a commutative unital associative algebra $A$, in this case over the field of real or complex numbers, equipped with a Lie bracket $\{,\}:A\otimes A\to A$ such that, for any $f\in A$, $\{f,-\}:A\to A$ is a derivation of $A$ as an associative algebra.

A Poisson manifold is a real smooth manifold $M$ together with a Lie algebra bracket $\{,\}:C^{\mathrm{\infty }}(M)\times C^{\mathrm{\infty }}(M)\to C^{\mathrm{\infty }}(M)$ on the vector space of smooth functions on $M$ which together with the pointwise multiplication of functions makes it a Poisson algebra. As derivations of $C^{\mathrm{\infty }}(M)$ correspond to smooth tangent vector fields, for each $f\in C^{\mathrm{\infty }}(M)$ there is a vector $X_f$ given by $X_f(g)=\{f,g\}$ and called the Hamiltonian vector field corresponding to the function $f$, which is viewed as a classical hamiltonian function.

Alternatively a Poisson structure on a manifold is given by a choice of smooth antisymmetric bivector called a Poisson bivector $P\in {\Lambda }^{2}TM$; then $\{f,g\}:=⟨\mathrm{df}\otimes \mathrm{dg},P⟩$.

Every symplectic manifold carries a natural Poisson structure; however, such Poisson manifolds are very special. It is a basic theorem that Poisson structures on a manifold are equivalent to the smooth foliations of the underlying manifold such that each leaf is a symplectic manifold.

The dual to a finite-dimensional Lie algebra has a natural structure of a Poisson manifold due to Kirillov. Its leaves are called coadjoint orbits.

A morphism $h:M\to N$ of Poisson manifolds is a morphism of smooth manifolds such that, for all $f,g\in C^{\mathrm{\infty }}(N)$, $\{f\circ h,g\circ h{\}}_M=\{f,g{\}}_N$.