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 (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.

Definition in coordinates

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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.

Coordinate-free definition

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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

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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

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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

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If   is a Hamiltonian on the cotangent bundle and   is its Hamiltonian vector field, 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 Riemannian and Pseudo-Riemannian Manifolds

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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

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