Poisson manifold

It is a manifold $M$ together with an operation ${-, -}$ in $C^{\infty} (M)$ satisfying the properties:

Bilinearity
Antisymmetry: ${f, g} = - {g, f}$ .
Leibniz rule: ${f, g h} = {f, g} h + g {f, h}$ .
Jacobi identity: ${f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0$ .

The operation is called Poisson bracket. Every symplectic manifold is a Poisson manifold but the converse is not true (@olver86 page 391). The algebra of functions on a Poisson manifold constitutes a Poisson algebra.

Bivector form of the bracket. The bracket is encoded by a skew-symmetric bivector (a skew-symmetric matrix of functions),

π = \frac{1}{2} \sum_{i, j} π^{i j} (x) \partial_{x^{i}} \land \partial_{x^{j}}, {f, g} = π (d f, d g) = \sum_{i, j} π^{i j} \partial_{i} f \partial_{j} g .

The sharp map $π^{♯} : T^{*} M \to T M$ sends a covector to a vector, $(π^{♯} α)^{i} = \sum_{j} π^{i j} α_{j}$ , and the Hamiltonian vector field of $f$ is $X_{f} = π^{♯} (d f)$ . Jacobi for ${-, -}$ is equivalent to $[π, π] = 0$

Example
An arbitrary $R^{N} \equiv (p_{i}, q_{i}, z_{j})$ , with $i = 1, \dots, m$ and $j = 1, \dots, l$ ( $N = 2 m + l$ ), can be seen as a Poisson manifold, we can define a Poisson bracket on the space of smooth functions on $R^{N}$ as follows:

For any two smooth functions $f, g : R^{N} \to R$ , we define their Poisson bracket $f, g$ as:

{f, g} = \sum_{i = 1}^{m} (\frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}})

The Poisson bracket of functions which only depends on $z_{j}$ with any other function is zero. These are called distinguished functions or Casimir functions, and can be thought as "observables" which do not provide "evolution parameter", since its associated Hamiltonian vector field is null.
Moreover, they remain constant along the evolution parameter of any Hamiltonian vector fields. They are a kind of "generalized constants". This reminds me the centralizer of an algebra, in case that the bracket is given by the commutator of a product...
$◼$

For every $f$ we have that ${-, f}$ is a differential operator, so it is a vector field. We call it the Hamiltonian vector field associated with $f$ . It coincides with the notion of Hamiltonian vector field for symplectic manifolds. Also, see Poisson bracket#Relation with Lie bracket.

The Poisson bracket can be seen in local coordinates as a skew-symmetric matrix. If we require that the rank of this matrix is everywhere the same as the dimension of the manifold, we recover the notion of symplectic manifold.

The "morphisms" between Poisson manifolds are called Poisson maps.

Poisson systems

Equations $\dot{x} = π^{♯} (d H)$ are called Poisson systems (also Hamiltonian systems on a Poisson manifold, or generalized Hamiltonian systems).

The characteristic distribution /symplectic foliation

Consider $D_{x} = Im π_{x}^{♯} = span {X_{f} (x)}$ , the set of all Hamiltonian vectors at $x$ . It is a (maybe singular) distribution (non constant rank). It is integrable: $[X_{f}, X_{g}] = - X_{{f, g}}$ .
The integral manifolds have a symplectic structure (symplectic leaves); the whole decomposition of $M$ is the symplectic foliation.

Annihilator and Frobenius ideal. $Ann (D)_{x} = \ker π_{x}^{♯}$ , i.e. $\sum_{j} π^{i j} ω_{j} = 0$ for all $i$ . It is localy generated by $r = n - 2 k$ one-forms $θ^{1}, \dots, θ^{r}$ ( $r =$ corank $=$ codimension of the leaves), defined only up to an invertible change of basis.

If a Casimir function $C$ exists, then $d C \in Ann (D)$ . Therefore, Casimir functions cut out the leaves of the symplectic foliation. They are the first integrals of the characteristic distribution.

Generalization: Jacobi manifold