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 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 interpretationEdit

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 definitionEdit

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 potentialEdit

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 an closed form.

PropertiesEdit

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,

 .

ActionEdit

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 spacesEdit

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.

ReferencesEdit

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