If ι ι v ω ω is the interior product or contraction operation defined by ( ι ι v ω ω ) ( u ) = ω ω ( v , u ) , then non-degeneracy is equivalent to saying that for every one-form α α there is a unique vector field Ω Ω α α such that ι ι Ω Ω α α ω ω = α α . Alternatively, Ω Ω d H = ω ω − − 1 ( d H ) . Then if H is a smooth function on M , the Hamiltonian vector field X H can be defined to be Ω Ω d H . It is easy to see that X p i = ∂ ∂ ∂ ∂ q i X q i = − − ∂ ∂ ∂ ∂ p i .