Fixed points of symplectic tranformations

Let (M,ω) be a closed symplectic manifold. Given a function H : M → R the Hamiltonian vector field XH determined by the Hamiltonian H is defined by the formula XH ω = −dH. Then LXHω = 0, and hence the flow generated by XH preserves the symplectic form ω. If one has a family of functions Ht : M → R, t ∈ [0, 1], one gets a family of Hamiltonian vector fields XHt which generate an isotopy ft : M → M which starts at f0 = Id and defined by the differential equation