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, also known as the Poincaré two-form, is given by
The extension of this concept to general fibre bundles is known as the solder form.
The tautological 1-form can also be defined rather abstractly as a form on phase space. Let be a manifold and be the cotangent bundle or phase space. Let
be the canonical fiber bundle projection, and let
be the induced tangent map. Let m be a point on M. 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
The tautological one-form is the unique horizontal one-form that "cancels" a pullback. That is, let
be any 1-form on , and (considering it as a map from to ) let denote the operation of pulling back by . Then
which can be most easily understood in terms of coordinates:
So, by the commutation between the pull-back and the exterior derivative,
If H is a Hamiltonian on the cotangent bundle and is its Hamiltonian flow, then the corresponding action S 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 metric spaces
If the manifold Q has a Riemannian or pseudo-Riemannian metric g, then corresponding definitions can be made in terms of generalized coordinates. Specifically, if we take the metric to be a map
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.