Hamiltonian group action

1-parameter case

Consider a 1-dimensional symplectic group action of $G$ on $(M,\omega )$. An infinitesimal generator $\xi \in \mathfrak{g}$ gives rise to the fundamental vector field ${\xi }_M$ on $M$, which could be a Hamiltonian vector field, i.e., there exists a smooth function $A$ such that ${\xi }_M=\{-,A\}$; or it might not be. In the affirmative case it is called a 1-parameter Hamiltonian group action.

General case

A Hamiltonian group action is a group action that is Hamiltonian in the sense that it is generated by Hamiltonian vector fields.
In other words, a symplectic group action of $G$ on a symplectic manifold $(M,\omega )$ is Hamiltonian if there exists a momentum mapping, i.e., a map

where ${\mathfrak{g}}^{\ast }$ is the dual of the Lie algebra of $G$, such that for all $\xi \in \mathfrak{g}$,

where ${\xi }_M$ is the vector field generated by the action of $\xi$.
If we consider a 1-parameter subgroup generated by $\xi$, then the infinitesimal generator is $X_{{\xi }_M}$. Thus, if the group action is Hamiltonian, we have

and so ${\xi }_M=X_{⟨J,\xi ⟩}$ so the infinitesimal generators or the group action are Hamiltonian vector fields

In other words, the momentum mapping provides a method for constructing the function $A$ associated to the infinitesimal generator ${\xi }_M=X_A$ of a Hamiltonian group action.