it was shown that there is a bijective correspondence between smooth actions and complete vector fields. Similarly, there is a bijective correspondence between symplectic actions (the induced diffeomorphisms are all symplectomorphisms) and complete symplectic vector fields. A closely related idea is that of Hamiltonian vector fields. Given a symplectic manifold , we say that is a Hamiltonian vector field if there exists a smooth function satisfying where the map is the interior product. This motivatives the definition of a Hamiltonian group action as follows: If is a Lie group with Lie algebra and is a group action of on a smooth manifold , then we say that is a Hamiltonian group action if there exists a moment map such that for each , where and is the fundamental vector field of References Lie groups Symplectic geometry Hamiltonian mechanics Smooth manifolds