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 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

:$heta\; =\; sum\_i\; p\_i\; dq^i.,$

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

:$omega\; =\; -d\; heta\; =\; sum\_i\; dq^i\; wedge\; dp\_i.$

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

:$pi:M\; o\; Q$

be the canonical fiber bundle projection, and let

:$T\_pi:TM\; o\; TQ$

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 $q=pi(m)$:

:$m:T\_qQ\; o\; mathbb\{R\}$.

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

:$heta\_m\; =\; m\; circ\; T\_pi$

It is a linear map

:$heta\_m:T\_mM\; o\; mathbb\{R\}$

and so

:$heta:TM\; o\; mathbb\{R\}$.

Properties

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

:

be any 1-form on "Q", and be its pullback. Then

:

and

:

This can be most easily understood in terms of coordinates:

:

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

:$S=\; heta\; (X\_H)$.

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

:$g:TQ\; o\; T^*Q$,

then define

:$Theta\; =\; g^*\; heta$

and

:$Omega\; =\; -dTheta\; =\; g^*omega$

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

:$Omega=\; sum\_\{ij\}\; g\_\{ij\}\; ;\; dq^i\; wedge\; ddot\; q^j\; +sum\_\{ijk\}\; frac\{partial\; g\_\{ij\{partial\; q^k\}\; ;\; dot\; q^i,\; dq^j\; wedge\; dq^k$

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.

ee also

* fundamental class

References

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