Tautological one-form explained

T ^{*} Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus providing a bridge between Lagrangian mechanics and Hamiltonian mechanics (on the 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.

To define the tautological one-form, select a coordinate chart

U on T ^{*Q} and a canonical coordinate system on U. Pick an arbitrary point m \in T ^{*Q.} By definition of cotangent bundle, m = (q,p), where q \in Q and p \in T ^{*Q.} _{q} The tautological one-form \theta _{m} : T ^{*Q} _{mT} \to \R is given by$$\backslash theta\_m\; =\; \backslash sum^n\_\; p\_i\; \backslash ,\; dq^i,$$with n = dim Q and (p _{1,} \ldots, p _{n)} \in U \subseteq \R ^{n} being the coordinate representation of p.

Any coordinates on

T ^{*Q} that preserve this definition, 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$$\backslash omega\; =\; -d\backslash theta\; =\; \backslash sum\_i\; dq^i\; \backslash wedge\; dp\_i$$

The extension of this concept to general fibre bundles is known as the solder form. By convention, one uses the phrase "canonical form" whenever the form has a unique, canonical definition, and one uses the term "solder form", whenever an arbitrary choice has to be made. 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.

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 $$\backslash pi\; :\; M\; \backslash to\; Q$$be the canonical fiber bundle projection, and let $$\backslash mathrm\; \backslash pi\; :\; TM\; \backslash to\; TQ$$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) :$$m\; :\; T\_qQ\; \backslash to\; \backslash R.$$

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$$\backslash theta\_m\; =\; m\; \backslash circ\; \backslash mathrm\_m\; \backslash pi.$$

It is a linear map$$\backslash theta\_m\; :\; T\_mM\; \backslash to\; \backslash R$$and so $$\backslash theta\; :\; M\; \backslash to\; T^*M.$$

Symplectic potential

The symplectic potential is generally defined a bit more freely, and also only defined locally: it is any one-form

\phi such that \omega =- d\phi ; in effect, symplectic potentials differ from the canonical 1-form by a closed form.

Properties

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

\beta be a 1-form on Q. \beta is a section \beta: Q \to T ^{*Q.} For an arbitrary 1-form \sigma on T ^{*Q,} the pullback of \sigma by \beta is, by definition, \beta ^{*\sigma} : = \sigma \circ \beta _{*.} Here, \beta _{*} : TQ\to TT ^{*Q} is the pushforward of \beta. Like \beta, \beta ^{*\sigma} is a 1-form on Q. The tautological one-form \theta is the only form with the property that \beta ^{*\theta} = \beta, for every 1-form \beta on Q.

Proof.
For a chart (\{q ^{i\}} ^{n} ^{i =1} ,U) on Q (where U \subseteq \R ^{n),} let \{p ^{i\}} _{i,q} ^{n} ^{i =1} be the coordinates on T ^{*Q,} where the fiber coordinates \{p ^{n} _{i =1} are associated with the linear basis \{dq ^{i\}} ^{n} ^{i =1} . By assumption, for every { q}=(q ^{1,\ldots,q} ^{n)} \in U, $$\backslash beta\; =\; \backslash sum^n\_\; \backslash beta\_i(\backslash mathbf)\backslash ,dq^i,$$or$$\backslash mathbf=(q^1,\backslash ldots,q^n)\backslash \; \backslash stackrel\backslash \; (\backslash underbrace\_,\backslash underbrace\_)).$$It follows that$$\backslash beta\_*\backslash left(\backslash frac\backslash Biggl$$	_\mathbf\right) =\frac \Biggl	_+ \sum^n_\frac\Biggl	_ \cdot \frac\Biggl	_which implies that$$(\backslash beta^*\backslash ,dq^i)\backslash left(\backslash right)\_\backslash mathbf=dq^i\backslash left[\backslash beta\_*\backslash left(\{\backslash partial/\backslash partial\; q^j\}\backslash right)\_\backslash mathbf\{q\}\backslash right]\; =\backslash delta\_.$$ Step 1. We have Step 1'. For completeness, we now give a coordinate-free proof that \beta ^{*\theta} = \beta, for any 1-form \beta. Observe that, intuitively speaking, for every q \in Q and p \in T ^{*} _{qQ,} the linear map d\pi _{(q,p)} in the definition of \theta projects the tangent space T _{(q,p)} T ^{*Q} onto its subspace T _{qQ.} As a consequence, for every q \in Q and v \in T _{qQ,} $$d\backslash pi\_(\backslash beta\_\; v)\; =\; v,$$where \beta _{*q} is the instance of \beta _{*} at the point q \in Q, that is,$$\backslash beta\_\; :\; T\_qQ\; \backslash to\; T\_T^*Q.$$Applying the coordinate-free definition of \theta to \theta _{\beta(q)} , obtain$$(\backslash beta^*\backslash theta)\_qv=\backslash theta\_(\backslash beta\_v)\; =\; \backslash beta(q)(d\backslash pi\_(\backslash beta\_\; v))\; =\; \backslash beta(q)\; v.$$ Step 2. It is enough to show that \alpha =0 if \beta ^{*\alpha} = 0 , for every one-form \beta. Let$$\backslash alpha\; =\; \backslash sum^n\_\; \backslash alpha\_(\backslash mathbf,\backslash mathbf)\backslash ,dq^i\; +\; \backslash sum^n\_\; \backslash alpha\_(\backslash mathbf,\backslash mathbf)\backslash ,dp\_i,$$where \alpha _{p ^{i}} ,\alpha _{q ^{i}} \in C ^{infty(\R} ^{n} x U,\R). Substituting v = \left(\partial / \partial q _{i\right)} _{q} into the identity \alpha(\beta _{*v)} = 0 obtain$$\backslash alpha(\backslash partial\; /\; \backslash partial\; q^i)\_\; +\; \backslash sum^n\_(\backslash partial\; \backslash beta\_j\; /\; \backslash partial\; q^i)\_\backslash cdot\; \backslash alpha(\backslash partial\; /\; \backslash partial\; p\_j)\_\; =\; 0,$$or equivalently, for any choice of n functions p _{i} = \beta _{i(q),} $$\backslash alpha\_(\backslash mathbf,\backslash mathbf)\; +\; \backslash sum^n\_\; \backslash partial\; p\_j\; /\; \backslash partial\; q^i\; \backslash cdot\; \backslash alpha\_(\backslash mathbf,\backslash mathbf)\; =\; 0.$$Let \beta = \sum ^{n} _{j =1} c ^{j,} _{jdq} where c _{j=const.} In this case, \beta _{j} = c _{j.} For every q \in U and c _{j} \in \R, $$\backslash alpha\_(\backslash mathbf,\backslash mathbf)\backslash bigl$$	_^ = 0.This shows that \alpha _{q ^{i}} ( p , q ) = 0 on \R ^{n} x U, and the identity$$\backslash sum^n\_\; \backslash partial\; p\_j\; /\; \backslash partial\; q^i\; \backslash cdot\; \backslash alpha\_(\backslash mathbf,\backslash mathbf)\; =\; 0$$must hold for an arbitrary choice of functions p _{i=\beta} _{i(q).} If \beta = \sum ^{n} _{j =1} c ^{jdq} _{jq} ^{j} (with {} ^{j} indicating superscript) then \beta _{j} = c ^{j,} _{jq} and the identity becomes$$\backslash alpha\_(\backslash mathbf,\backslash mathbf)\backslash bigl$$	_^ = 0,for every q \in U and c _{j} \in \R. Since c _{j} = p ^{j} / q ^{j,} we see that \alpha _{p _{i}} ( p , q ) = 0 , as long as q ^{j} ≠ 0 for all j. On the other hand, the function \alpha _{p _{i}} is continuous, and hence \alpha _{p _{i}} ( p , q ) = 0 on \R ^{n} x U.

So, by the commutation between the pull-back and the exterior derivative,$$\backslash beta^*\backslash omega\; =\; -\backslash beta^*\; \backslash ,\; d\backslash theta\; =\; -d\; (\backslash beta^*\backslash theta)\; =\; -d\backslash beta.$$

Action

If

H is a Hamiltonian on the cotangent bundle and X _{H} is its Hamiltonian vector field, then the corresponding action S is given by$$S\; =\; \backslash theta(X\_H).$$

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:$$S(E)\; =\; \backslash sum\_i\; \backslash oint\; p\_i\backslash ,dq^i$$with the integral understood to be taken over the manifold defined by holding the energy

E constant: H = E = const .

On Riemannian and Pseudo-Riemannian Manifolds

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$$g\; :\; TQ\; \backslash to\; T^*Q,$$then define$$\backslash Theta\; =\; g^*\backslash theta$$and $$\backslash Omega\; =\; -d\backslash Theta\; =\; g^*\backslash omega$$

In generalized coordinates

(q ^{1,\ldots,q} ^{• n, q} ^{• 1 ,\ldots, q} ^{n)} on TQ, one has$$\backslash Theta\; =\; \backslash sum\_\; g\_\; \backslash dot\; q^i\; dq^j$$and $$\backslash Omega\; =\; \backslash sum\_\; g\_\; \backslash ;\; dq^i\; \backslash wedge\; d\backslash dot\; q^j\; +\backslash sum\_\; \backslash frac\; \backslash ;\; \backslash dot\; q^i\backslash ,\; dq^j\; \backslash wedge\; dq^k$$

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 See section 3.2.