In a more general setting, ( v , q ) are local coordinates on the tangent bundle T M of a manifold M . For each q, L ( v , q ) is a convex function of the tangent space V q. The Legendre transform gives the Hamiltonian H ( p , q ) as a function of the coordinates (p, q) of the cotangent bundle T ∗ ∗ M ; the inner product used to define the Legendre transform is inherited from the pertinent canonical symplectic structure. In this abstract setting, the Legendre transformation corresponds to the tautological one-form.