In mathematics, the tautological oneform is a special 1form defined on the cotangent bundle of a manifold 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 ).
The exterior derivative of this form defines a symplectic form giving the structure of a symplectic manifold. The tautological oneform plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics. The tautological oneform is sometimes also called the Liouville oneform, the Poincaré oneform, the canonical oneform, or the symplectic potential. A similar object is the canonical vector field on the tangent bundle.
To define the tautological oneform, select a coordinate chart on and a canonical coordinate system on Pick an arbitrary point By definition of cotangent bundle, where and The tautological oneform is given by
Any coordinates on 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 twoform, 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.
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
That is, the tautological oneform 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 oneanother. 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 oneform can be thought of as a device to convert from Lagrangian mechanics to Hamiltonian mechanics.
The tautological 1form 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
That is, we have that is in the fiber of The tautological oneform at point is then defined to be
It is a linear map
The symplectic potential is generally defined a bit more freely, and also only defined locally: it is any oneform such that ; in effect, symplectic potentials differ from the canonical 1form by a closed form.
The tautological oneform is the unique oneform that "cancels" pullback. That is, let be a 1form on is a section For an arbitrary 1form on the pullback of by is, by definition, Here, is the pushforward of Like is a 1form on The tautological oneform is the only form with the property that for every 1form on
Proof. 
For a chart on (where let be the coordinates on where the fiber coordinates are associated with the linear basis By assumption, for every
or
It follows that
which implies that
Step 1. We have Step 1'. For completeness, we now give a coordinatefree proof that for any 1form Observe that, intuitively speaking, for every and the linear map in the definition of projects the tangent space onto its subspace As a consequence, for every and
where is the instance of at the point that is,
Applying the coordinatefree definition of to obtain
Step 2. It is enough to show that if for every oneform Let
where
Substituting into the identity obtain
or equivalently, for any choice of functions
Let where In this case, For every and
This shows that on and the identity
must hold for an arbitrary choice of functions If (with indicating superscript) then and the identity becomes
for every and Since we see that as long as for all On the other hand, the function is continuous, and hence on

So, by the commutation between the pullback and the exterior derivative,
If is a Hamiltonian on the cotangent bundle and is its Hamiltonian flow, then the corresponding action is given by
In more prosaic terms, the Hamiltonian flow represents the classical trajectory of a mechanical system obeying the HamiltonJacobi equations of motion. The Hamiltonian flow is the integral of the Hamiltonian vector field, and so one writes, using traditional notation for actionangle variables:
If the manifold has a Riemannian or pseudoRiemannian metric 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 one has
The metric allows one to define a unitradius sphere in The canonical oneform restricted to this sphere forms a contact structure; the contact structure may be used to generate the geodesic flow for this metric.