The ϕ ϕ x ( t ) will be symplectomorphisms (canonical transformations) for every t as a function of x if and only if L Ω Ω α α ω ω = 0 ; when this is true, Ω α is called a symplectic vector field. Recalling Cartan's identity L X ω ω = d ( ι ι X ω ω ) + ι ι X d ω ω and d ω = 0, it follows that L Ω Ω α α ω ω = d ( ι ι Ω Ω α α ω ω ) = d α α . Therefore, Ω α is a symplectic vector field if and only if α is a closed form. Since d ( d f ) = d 2 f = 0 , it follows that every Hamiltonian vector field X f is a symplectic vector field, and that the Hamiltonian flow consists of canonical transformations. From (1) above, under the Hamiltonian flow X H, d d t f ( ϕ ϕ x ( t ) ) = X H f = { f , H } .