In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T*Q of a manifold Q. The exterior derivative of this form defines a symplectic form giving T*Q the structure of a symplectic manifold. The tautological one-form plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics. The tautological one-form is sometimes also called the Liouville one-form, the Poincaré one-form, the canonical one-form, or the symplectic potential. A similar object is the canonical vector field on the tangent bundle. In algebraic geometry and complex geometry the term "canonical" is discouraged, due to confusion with the canonical class, and the term "tautological" is preferred, as in tautological bundle.
In canonical coordinates, the tautological one-form is given by
Equivalently, any coordinates on phase space which preserve this structure for the canonical one-form, up to a total differential (exact form), may be called canonical coordinates; transformations between different canonical coordinate systems are known as canonical transformations.
The canonical symplectic form is given by
be the canonical fiber bundle projection, and let
be the induced tangent map. Let m be a point on M, however, since M is the cotangent bundle, we can understand m to be a map of the tangent space at :
That is, we have that m is in the fiber of q. The tautological one-form at point m is then defined to be
It is a linear map
be any 1-form on Q, and (considering it as a map from Q to T*Q ) let be its pullback. Then
which can be most easily understood in terms of coordinates:
So, by the commutation between the pull-back and the exterior derivative,
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 metric spaces
In generalized coordinates on TQ, one has
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.