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 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 is given by

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

Contents

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, however, 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 \beta^* be its pullback. Then

\beta^*\theta = \beta

so that, by the commutation between the pull-back and the exterior derivative:

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

This can be most easily understood in terms of coordinates:

\beta^*\theta = \beta^*\sum_i p_idq^i = \sum_i \beta^*p_idq^i = \sum_i \beta_idq^i = \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 %2B \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.

See also

References