Tautological one-form

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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 canonical one-form, or the symplectic potential.

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

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

1 Coordinate-free definition
2 Properties
3 Action
4 On metric spaces
5 See also
6 References

[edit] 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 space or phase space. Let

$\pi:M\to Q$

be the canonical fiber bundle projection, and let

$T_\pi:TM \to TQ$

be the induced pushforward. 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 $q = π(m)$ :

$m:T_qQ \to \mathbb{R}$ .

That is, we have that m is in the fiber of q. The tautological one-form $θ m$ 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

$\theta:TM \to \mathbb{R}$ .

[edit] Properties

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

$\beta:Q\to T^*Q$

be any 1-form on Q, and $β *$ be its pullback. Then

β * θ = β

and

β * ω = - d β

This can be most easily understood in terms of coordinates:

β ^* θ = β ^*	∑	p_idqⁱ =	∑	β ^* p_idqⁱ =	∑	β_idqⁱ = β
	i		i		i

[edit] 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 = θ(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: $H = E = c o n s t .$ .

[edit] 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

Θ = g * θ

and

Ω = - d Θ = g * ω

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.