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
$$ \omega(X_f,-)=df(-) $$How it is applied to another function $g$? It is satisfied that
$$ X_f(g)=dg(X_f)=\omega(X_g,X_f)=-\omega(X_f,X_g) $$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, the the parameter is $x$, the position.
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 $M$.
________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: