Tautological one-form

The variables ${\displaystyle q_{i}}$ are meant to be understood as generalized coordinates, so that a point ${\displaystyle q\in Q}$ is a point in configuration space. The tangent space ${\displaystyle TQ}$ corresponds to velocities, so that if ${\displaystyle q}$ is moving along a path ${\displaystyle q(t)}$, the instantaneous velocity at ${\displaystyle t=0}$ corresponds a point

${\displaystyle \left.{\frac {dq(t)}{dt}}\right|_{t=0}={\dot {q}}\in TQ}$

on the tangent manifold ${\displaystyle TQ}$, for the given location of the system at point ${\displaystyle q\in Q}$. 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 ${\displaystyle p}$ for each velocity ${\displaystyle {\dot {q}}}$, 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.

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

${\displaystyle \pi :M\to Q}$

be the canonical fiber bundle projection, and let

${\displaystyle \mathrm {d} \pi :TM\to TQ}$

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

${\displaystyle m:T_{q}Q\to \mathbb {R} }$.

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

${\displaystyle \theta _{m}=m\circ \mathrm {d} \pi _{m}}$.

It is a linear map

${\displaystyle \theta _{m}:T_{m}M\to \mathbb {R} }$

and so

${\displaystyle \theta :M\to T^{*}M}$.

The symplectic potential is generally defined a bit more freely, and also only defined locally: it is any one-form ${\displaystyle \phi }$ such that ${\displaystyle \omega =-d\phi }$; in effect, symplectic potentials differ from the canonical 1-form by a closed form.

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

${\displaystyle \beta :Q\to T^{*}Q}$

be any 1-form on ${\displaystyle Q}$, and (considering it as a map from ${\displaystyle Q}$ to ${\displaystyle T^{*}Q}$) let ${\displaystyle \beta ^{*}}$ denote the operation of pulling back by ${\displaystyle \beta }$. Then

${\displaystyle \beta ^{*}\theta =\beta }$,

which can be most easily understood in terms of coordinates:

${\displaystyle \beta ^{*}\theta =\beta ^{*}(\sum _{i}p_{i}\,dq^{i})=\sum _{i}\beta ^{*}p_{i}\,dq^{i}=\sum _{i}\beta _{i}\,dq^{i}=\beta .}$

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

${\displaystyle \beta ^{*}\omega =-\beta ^{*}d\theta =-d(\beta ^{*}\theta )=-d\beta }$.

If ${\displaystyle H}$ is a Hamiltonian on the cotangent bundle and ${\displaystyle X_{H}}$ is its Hamiltonian flow, then the corresponding action ${\displaystyle S}$ is given by

${\displaystyle S=\theta (X_{H})}$.

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:

${\displaystyle S(E)=\sum _{i}\oint p_{i}\,dq^{i}}$

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

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

${\displaystyle g:TQ\to T^{*}Q}$,

then define

${\displaystyle \Theta =g^{*}\theta }$

and

${\displaystyle \Omega =-d\Theta =g^{*}\omega }$

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

${\displaystyle \Theta =\sum _{ij}g_{ij}{\dot {q}}^{i}dq^{j}}$

and

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

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

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