Tautological one-form

Summary

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 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 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 on and a canonical coordinate system on Pick an arbitrary point By definition of cotangent bundle, where and The tautological one-form is given by

with and being the coordinate representation of

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 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 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 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 be a manifold and be the cotangent bundle or phase space. Let

be the canonical fiber bundle projection, and let
be the induced tangent map. Let be a point on Since is the cotangent bundle, we can understand to be a map of the tangent space at :

That is, we have that is in the fiber of The tautological one-form at point 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 such that ; 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. That is, let be a 1-form on is a section For an arbitrary 1-form on the pullback of by is, by definition, Here, is the pushforward of Like is a 1-form on The tautological one-form is the only form with the property that for every 1-form on

So, by the commutation between the pull-back and the exterior derivative,

Action

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 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

If the manifold has a Riemannian or pseudo-Riemannian metric 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 on one has

and

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.

References

  • Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 3.2.