Hamiltonian flows, cotangent lifts, and momentum maps

Let (M,ω) and (N, η) be symplectic manifolds. A symplectomorphism F : M → N is a diffeomorphism such that ω = F ∗η. Recall that for x ∈ M and v1, v2 ∈ TxM , (F η)x(v1, v2) = ηF (x)((TxF )v1, (TxF )v2); TxF : TxM → TF (x)N . (A tangent vector at x ∈ M is pushed forward to a tangent vector at F (x) ∈ N , while a differential 2-form on N is pulled back to a differential 2-form on M .) In these notes the only symplectomorphisms in which we are interested are those from a symplectic manifold to itself.