Hamiltonian vector field

Given a smooth function $f$ defined on a symplectic manifold $(M,\omega )$ we can define a vector field $X_f$, corresponding to the 1-form $df$ by means of the duality provided by the symplectic form. It is also called the symplectic gradient of $f$. It is defined as the vector field which satisfies

How it is applied to another function $g$? It is satisfied that

An this is precisely the definition of the Poisson bracket, i.e., $X_f(g)=\{g,f\}$. So Hamiltonian vector fields have the form $\{-,f\}$. Therefore they can also be defined in Poisson manifolds.

A Hamiltonian vector field is a particular case of a symplectic vector field, so is the generator of a 1-dimensional symplectic group action and, in particular, a Hamiltonian group action.

A Hamiltonian vector field gives rise to a 1-parameter local group of transformations, and this parameter has usually a physical interpretation. For example, for $f=H$, the energy of a system, the parameter is $t$, the time. If $f=p$, the momentum, then the parameter is $x$, the position.

Symplectic foliation of a Poisson manifold

In the context of a Poisson manifold $M$, the relation of the Poisson bracket with the Lie bracket is telling to us that the set of Hamiltonian vector fields constitute an involutive distribution on $M$. The resulting foliation is called the symplectic foliation of $M$.

Characterization

Consider the matrix

which is the Jacobian of the function $F=(X_1,X_2,\dots ,X_{2n})$

Then $X$ is a Hamiltonian vector field if and only if

for every pair of vectors $u,v\in {\mathbb{R}}^{2n}$