In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle
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 and Hamiltonian mechanics (on the manifold
the structure of a symplectic manifold.
The tautological one-form plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics.
A similar object is the canonical vector field on the tangent bundle.
To define the tautological one-form, select a coordinate chart
and a canonical coordinate system on
Pick an arbitrary point
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.
The tautological 1-form can also be defined rather abstractly as a form on phase space.
be the cotangent bundle or phase space.
be the canonical fiber bundle projection, and let
to be a map of the tangent space at
The symplectic potential is generally defined a bit more freely, and also only defined locally: it is any one-form
; in effect, symplectic potentials differ from the canonical 1-form by a closed form.
For completeness, we now give a coordinate-free proof that
must hold for an arbitrary choice of functions
So, by the commutation between the pull-back and the exterior derivative,
is its Hamiltonian vector field, then the corresponding action
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
has a Riemannian or pseudo-Riemannian metric
then corresponding definitions can be made in terms of generalized coordinates.
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.