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Wikipedia / Tautological one-form

In mathematics, the tautological one-form is a special 1-form 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 and Hamiltonian mechanics).

The exterior derivative of this form defines a symplectic form 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.

Definition in coordinates

and a canonical coordinate

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

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.

Coordinate-free definition

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 induced tangent map

Intuition

Visually, the tautological 1-form can be described as follows. Like how a vector can be pictured as an ordered pair of points, a 1-form can be pictured as an ordered pair of hyperplanes.

is its, which can be fed into the covector.

.

Symplectic potential

; in effect, symplectic potentials differ from the canonical 1-form by a closed form.

Properties

The tautological one-form is the unique one-form that "cancels" pullback is a section is the pushforward

Action functional

is a Hamiltonian on the cotangent bundle 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

On Riemannian and Pseudo-Riemannian Manifolds

has a Riemannian or pseudo-Riemannian metric then corresponding definitions can be made in terms of generalized coordinates

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.

References