(In algebraic terms, one would say that the C ∞ ∞ ( M , R ) -modules Vect ( M ) and Ω Ω 1 ( M ) are isomorphic). If ⁠ H ∈ ∈ C ∞ ∞ ( M × × R t , R ) ⁠, then, for every fixed ⁠ t ∈ ∈ R t ⁠, ⁠ d H ∈ ∈ Ω Ω 1 ( M ) ⁠, and ⁠ J ( d H ) ∈ ∈ Vect ( M ) ⁠. J ( d H ) is known as a Hamiltonian vector field. The respective differential equation on M x ˙ ˙ = J ( d H ) ( x ) is called Hamilton's equation. Here x = x ( t ) and J ( d H ) ( x ) ∈ ∈ T x M is the (time-dependent) value of the vector field J ( d H ) at ⁠ x ∈ ∈ M ⁠.