canonical 1-form by a closed form. Properties The tautological one-form is the unique one-form that "cancels" pullback. That is, let be a 1-form on is a section For an arbitrary 1-form on the pullback of by is, by definition, Here, is the pushforward of Like is a 1-form on The tautological one-form is the only form with the property that for every 1-form on So, by the commutation between the pull-back and the exterior derivative, Action If is a Hamiltonian on the cotangent bundle and is its Hamiltonian vector field, then the corresponding action is given by In more prosaic terms, the Hamiltonian flow represents the classical trajectory of a mechanical system obeying the Hamilton-Jacobi equations of motion. The Hamiltonian flow is the integral of the Hamiltonian vector field, and so one writes, using traditional notation for action-angle variables: with the integral understood to be taken over the manifold defined by holding the energy constant: On Riemannian and Pseudo-Riemannian Manifolds If the manifold has a Riemannian or pseudo-Riemannian metric then corresponding definitions can be made in terms of generalized coordinates. Specifically, if we take the metric to be a map then define and In generalized coordinates on one has and The metric allows one to define a unit-radius sphere in The canonical one-form restricted to this sphere forms a contact structure; the contact structure may be used to generate the geodesic flow for this metric. References Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London See section 3.2. Symplectic geometry Hamiltonian mechanics Lagrangian mechanics