Tautological one-form

In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle formula_1 of a manifold formula_2. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus providing a bridge between Lagrangian mechanics with Hamiltonian mechanics (on the manifold formula_2). 
The exterior derivative of this form defines a symplectic form giving formula_1 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.
To define the tautological one-form, select a coordinate chart formula_5 on formula_6 and a canonical coordinate system on formula_7 Pick an arbitrary point formula_8 By definition of cotangent bundle, formula_9 where formula_10 and formula_11 The tautological one-form formula_12 is given by
with formula_14 and formula_15 being the coordinate representation of formula_16
Any coordinates on formula_17 that preserve this definition, 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. By convention, one uses the phrase "canonical form" whenever the form has a unique, canonical definition, and one uses the term "solder form", whenever an arbitrary choice has to be made. 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.
Physical interpretation.
The variables formula_19 are meant to be understood as generalized coordinates, so that a point formula_20 is a point in configuration space. The tangent space formula_21 corresponds to velocities, so that if formula_22 is moving along a path formula_23, the instantaneous velocity at formula_24 corresponds a point 
on the tangent manifold formula_21, for the given location of the system at point formula_20. 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 formula_28 for each velocity formula_29, 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 choice of gluing is unique; each momentum vector corresponds to only one velocity vector, by definition. The tautological one-form can be thought of as a device to convert from Lagrangian mechanics to Hamiltonian mechanics.
Coordinate-free definition.
The tautological 1-form can also be defined rather abstractly as a form on phase space. Let formula_2 be a manifold and formula_31 be the cotangent bundle or phase space. Let 
be the canonical fiber bundle projection, and let 
be the induced tangent map. Let formula_34 be a point on formula_35. Since formula_35 is the cotangent bundle, we can understand formula_34 to be a map of the tangent space at formula_38:
That is, we have that formula_34 is in the fiber of formula_22. The tautological one-form formula_42 at point formula_34 is then defined to be
It is a linear map
and so 
Symplectic potential.
The symplectic potential is generally defined a bit more freely, and also only defined locally: it is any one-form formula_47 such that formula_48; in effect, symplectic potentials differ from the canonical 1-form by an closed form.
Properties.
The tautological one-form is the unique one-form that "cancels" pullback. That is, let formula_49 
be a 1-form on formula_50 formula_49 is a section formula_52 For an arbitrary 1-form formula_53 on formula_54 the pullback of formula_53 by formula_49 is, by definition, formula_57 Here, formula_58 is the pushforward of formula_59 Like formula_60 formula_61 is a 1-form on formula_50 The tautological one-form formula_63 is the only form with the property that formula_64 for every 1-form formula_49 on formula_50
So, by the commutation between the pull-back and the exterior derivative,
Action.
If formula_68 is a Hamiltonian on the cotangent bundle and formula_69 is its Hamiltonian flow, then the corresponding action formula_70 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 formula_73 constant: formula_74.
On metric spaces.
If the manifold formula_2 has a Riemannian or pseudo-Riemannian metric formula_76, 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 formula_80 on formula_21, one has
and 
The metric allows one to define a unit-radius sphere in formula_17. 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.