In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T*Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system to it's momentum, thus providing a bridge between Lagrangian mechanics with Hamiltonian mechanics (on the 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 variables are meant to be understood as generalized coordinates, so that a point is a point in configuration space. The tangent space corresponds to velocities, so that if is moving along a path , the instantaneous velocity at corresponds a point
on the tangent manifold , for the given location of the system at point . Velocities are appropriate for the Lagrangian formulation of classical mechanics, but in the Hamiltonian formulation, one works with momenta, and not velocities; the tautological one-form is a device that converts velocities into momenta.
That is, the tautological one-form assigns a numerical value to the momentum for each velocity , and more: it does so such that they point "in the same direction", and linearly, such that the magnitudes grow in proportion. It is called "tautological" precisely because, "of course", velocity and momenta are necessarily proportional to one-another. It is a kind of solder form, because it "glues" or "solders" each velocity to a corresponding momentum. The tautological one-form can be thought of as a device to convert from Lagrangian mechanics to Hamiltonian mechanics.
be the canonical fiber bundle projection, and let
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 , 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,
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.