Tautological one-form

In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T*Q of a manifold Q. The exterior derivative of this form defines a symplectic form giving T*Q 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. 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.

In canonical coordinates, the tautological one-form is given by

\theta = \sum_i p_i dq^i

Equivalently, any coordinates on phase space which preserve this structure for the canonical one-form, 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

\omega = -d\theta = \sum_i dq^i \wedge dp_i

The extension of this concept to general fibre bundles is known as the solder form.

Coordinate-free definition

The tautological 1-form can also be defined rather abstractly as a form on phase space. Let $Q$ be a manifold and $M=T^*Q$ be the cotangent bundle or phase space. Let

\pi:M\to Q

be the canonical fiber bundle projection, and let

T_\pi:TM \to TQ

be the induced tangent map. Let m be a point on M. Since M is the cotangent bundle, we can understand m to be a map of the tangent space at $q=\pi(m)$ :

m:T_qQ \to \mathbb{R}

That is, we have that m is in the fiber of q. The tautological one-form $\theta_m$ at point m is then defined to be

\theta_m = m \circ T_\pi

It is a linear map

\theta_m:T_mM \to \mathbb{R}

and so

\theta:M \to T^*M

Properties

The tautological one-form is the unique horizontal one-form that "cancels" a pullback. That is, let

\beta:Q\to T^*Q

be any 1-form on Q, and (considering it as a map from Q to T*Q ) let $\beta^*$ be its pullback. Then

\beta^*\theta = \beta

which can be most easily understood in terms of coordinates:

\beta^*\theta = \beta^*(\sum_i p_i\, dq^i) = \sum_i \beta^*p_i\, dq^i = \sum_i \beta_i\, dq^i = \beta.

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

\beta^*\omega = -\beta^*d\theta = -d (\beta^*\theta) = -d\beta

Action

If H is a Hamiltonian on the cotangent bundle and $X_H$ is its Hamiltonian flow, then the corresponding action S is given by

S=\theta (X_H)

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:

S(E) = \sum_i \oint p_i\,dq^i

with the integral understood to be taken over the manifold defined by holding the energy $E$ constant: $H=E=const.$ .

On metric spaces

If the manifold Q has a Riemannian or pseudo-Riemannian metric g, then corresponding definitions can be made in terms of generalized coordinates. Specifically, if we take the metric to be a map

g:TQ\to T^*Q

then define

\Theta = g^*\theta

and

\Omega = -d\Theta = g^*\omega

In generalized coordinates $(q^1,\ldots,q^n,\dot q^1,\ldots,\dot q^n)$ on TQ, one has

\Theta=\sum_{ij} g_{ij} \dot q^i dq^j

and

\Omega= \sum_{ij} g_{ij} \; dq^i \wedge d\dot q^j + \sum_{ijk} \frac{\partial g_{ij}}{\partial q^k} \; \dot q^i\, dq^j \wedge dq^k

The metric allows one to define a unit-radius sphere in $T^*Q$ . 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.

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