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
Proof. |
For a chart on (where let be the coordinates on where the fiber coordinates are associated with the linear basis By assumption, for every or It follows that which implies that Step 1. We have Step 1'. For completeness, we now give a coordinate-free proof that for any 1-form Observe that, intuitively speaking, for every and the linear map in the definition of projects the tangent space onto its subspace As a consequence, for every and where is the instance of at the point i.e. Applying the coordinate-free definition of to obtain Step 2. It is enough to show that if for every one-form Let where Substituting into the identity obtain or equivalently, for any choice of functions Let where In this case, For every and This shows that on and the identity must hold for an arbitrary choice of functions If (with indicating superscript) then and the identity becomes for every and Since we see that as long as for all On the other hand, the function is continuous, and hence 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.