Tautological one-form

{{Short description|Canonical differential form}} {{distinguish|Symplectic manifold#Definition|Symplectic vector space}}

{{One source|date=March 2025}} In [[mathematics]], the '''tautological one-form''' is a special [[1-form]] defined on the [[cotangent bundle]] <math>T^{*}Q</math> of a [[manifold]] <math>Q.</math> 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 <math>Q</math>).

The [[exterior derivative]] of this form defines a [[Symplectic manifold|symplectic form]], giving <math>T^{*}Q</math> 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 form|canonical]] one-form''', or the '''symplectic potential'''. A similar object is the [[canonical vector field]] on the [[tangent bundle]].

== Definition in coordinates == To define the tautological one-form, select a coordinate chart <math> U </math> on <math>T^*Q </math> and a [[canonical coordinate]] system on <math> U. </math> Pick an arbitrary point <math>m \in T^*Q.</math> By definition of cotangent bundle, <math>m = (q,p),</math> where <math>q \in Q</math> and <math>p \in T_q^*Q.</math> The tautological one-form <math>\theta_m : T_mT^*Q \to \R</math> is given by <math display="block">\theta_m = \sum^n_{i=1} p_i \, dq^i,</math> with <math>n = \mathop{\text{dim}}Q</math> and <math>(p_1, \ldots, p_n) \in U \subseteq \R^n</math> being the coordinate representation of <math>p. </math>

Any coordinates on <math>T^*Q</math> 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 transformation]]s.

The '''canonical symplectic form''', also known as the '''Poincaré two-form''', is given by <math display="block">\omega = -d\theta = \sum_i dq^i \wedge dp_i</math>

The extension of this concept to general [[fibre bundle]]s 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 <math>Q</math> be a manifold and <math>M=T^*Q</math> be the [[cotangent bundle]] or [[phase space]]. Let <math display=block>\pi : M \to Q</math> be the canonical fiber bundle projection, and let <math display=block>\mathrm{d} \pi : TM \to TQ </math> be the [[Induced homomorphism|induced]] [[tangent map]]. Let <math>m</math> be a point on <math>M.</math> Since <math>M</math> is the cotangent bundle, we can understand <math>m</math> to be a map of the tangent space at <math>q=\pi(m)</math>: <math display=block>m : T_qQ \to \R.</math>

That is, we have that <math>m</math> is in the fiber of <math>q.</math> The tautological one-form <math>\theta_m</math> at point <math>m</math> is then defined to be <math display=block>\theta_m = m \circ \mathrm{d}_m \pi.</math>

It is a linear map <math display=block>\theta_m : T_mM \to \R</math> and so <math display=block>\theta : M \to T^*M.</math>

=== Intuition === Visually, the tautological 1-form can be described as follows. Like how a vector can be pictured as an ordered pair of points, a 1-form can be pictured as an ordered pair of hyperplanes.

Consider any vector in the cotangent bundle <math>V \in T_\omega(T^* Q)</math>, where <math>\omega \in T^*_q Q </math> is its base point (a covector), and <math>q \in Q </math> is ''its'' base point. Then, there are 3 effects of moving infinitesimally from <math>\omega </math> to <math>\omega + V \delta t </math>: shifting the base point <math>q </math>, rotating the hyperplane of the covector <math>\ker\omega </math>, and changing the distance separating between the hyperplane pairs. In particular, the shifting of the base point creates a vector <math>d\pi(V) \in T_q Q </math>, which can be fed into the covector.

The tautological 1-form computes <math>\theta(V) </math> by feeding to <math>\omega </math> the vector created by shifting the base point, and ignoring the other two effects, which cannot be fed into the covector, giving <math>\theta(V) = \omega(d\pi(V)) </math>.

== Symplectic potential== The symplectic potential is generally defined a bit more freely, and also only defined locally: it is any one-form <math>\phi</math> such that <math>\omega=-d\phi</math>; in effect, symplectic potentials differ from the canonical 1-form by a [[Closed differential form|closed form]].

==Properties== The tautological one-form is the unique one-form that "cancels" [[pullback_(differential geometry)|pullback]]. That is, let <math>\beta</math> be a 1-form on <math>Q.</math> <math>\beta</math> is a [[Section (fiber_bundle)|section]] <math>\beta: Q \to T^*Q.</math> For an arbitrary 1-form <math>\sigma</math> on <math>T^*Q,</math> the pullback of <math>\sigma</math> by <math>\beta</math> is, by definition, <math>\beta^*\sigma := \sigma \circ \beta_*.</math> Here, <math>\beta_* : TQ\to TT^*Q</math> is the [[Pushforward (differential)|pushforward]] of <math>\beta.</math> Like <math>\beta,</math> <math>\beta^*\sigma</math> is a 1-form on <math>Q.</math> The tautological one-form <math>\theta</math> is the only form with the property that <math>\beta^*\theta = \beta,</math> for every 1-form <math>\beta</math> on <math>Q.</math>

{| role="presentation" class="wikitable mw-collapsible mw-collapsed" | '''Proof.''' |- | For a chart <math>(\{q^i\}^n_{i=1},U)</math> on <math>Q</math> (where <math>U \subseteq \R^n),</math> let <math>\{p_i,q^i\}^n_{i=1}</math> be the coordinates on <math>T^*Q,</math> where the fiber coordinates <math>\{p_i\}^n_{i=1}</math> are associated with the linear basis <math>\{dq^i\}^n_{i=1}.</math> By assumption, for every <math>{\mathbf q}=(q^1,\ldots,q^n) \in U,</math> <math display=block>\beta({\mathbf q}) = \sum^n_{i=1} \beta_i(\mathbf{q})\,dq^i,</math> or <math display=block>\mathbf{q}=(q^1,\ldots,q^n)\ \stackrel{\beta}{\to}\ (\underbrace{q^1,\ldots,q^n}_{\mathbf{q}},\underbrace{\beta_1(\mathbf{q}),\ldots,\beta_n(\mathbf{q}}_{\mathbf{p}})).</math> It follows that <math display=block>\beta_*\left(\frac{\partial}{\partial q^i}\Biggl|_\mathbf{q}\right) = \frac{\partial}{\partial q^i} \Biggl|_{\beta(\mathbf{q})} + \sum^n_{j=1}\frac{\partial \beta_j}{\partial q^i}\Biggl|_{\mathbf{q}} \cdot \frac{\partial}{\partial p_j}\Biggl|_{\beta(\mathbf{q})}</math> which implies that <math display=block>(\beta^*\,dq^i)\left({\partial/\partial q^j}\right)_\mathbf{q}=dq^i\left[\beta_*\left({\partial/\partial q^j}\right)_\mathbf{q}\right] = \delta_{ij}.</math>

'''Step 1.''' We have <math display=block>\begin{align} (\beta^*\theta)\left(\partial / \partial q^i\right)_\mathbf{q} &= \theta\left( \beta_*\left(\partial/\partial q^i\right)_\mathbf{q}\right) = \left(\sum^{n}_{j=1}p_jdq^j\right)\left(\beta_*\left(\partial/\partial q^i\right)_\mathbf{q}\right) \\ &= \beta_i(\mathbf{q}) = \beta\left(\partial/\partial q^i\right)_\mathbf{q}. \end{align}</math>

'''Step 1'.''' For completeness, we now give a coordinate-free proof that <math>\beta^*\theta = \beta,</math> for any 1-form <math>\beta.</math>

Observe that, intuitively speaking, for every <math>q \in Q</math> and <math>p \in T^*_qQ,</math> the linear map <math>d\pi_{(q,p)}</math> in the definition of <math> \theta </math> projects the tangent space <math>T_{(q,p)}T^*Q</math> onto its subspace <math>T_qQ.</math> As a consequence, for every <math>q \in Q</math> and <math>v \in T_qQ,</math> <math display=block>d\pi_{\beta(q)}(\beta_{*q} v) = v,</math> where <math>\beta_{*q}</math> is the instance of <math>\beta_*</math> at the point <math>q \in Q,</math> that is, <math display=block>\beta_{*q} : T_qQ \to T_{\beta(q)}T^*Q.</math> Applying the coordinate-free definition of <math>\theta</math> to <math>\theta_{\beta(q)},</math> obtain <math display=block>(\beta^*\theta)_qv=\theta_{\beta(q)}(\beta_{*q}v) = \beta(q)(d\pi_{\beta(q)}(\beta_{*q} v)) = \beta(q) v.</math>

'''Step 2.''' It is enough to show that <math>\alpha=0</math> if <math>\beta^*\alpha = 0,</math> for every one-form <math>\beta.</math> Let <math display=block>\alpha = \sum^n_{i=1} \alpha_{q^i}(\mathbf{p},\mathbf{q})\,dq^i + \sum^n_{i=1} \alpha_{p_i}(\mathbf{p},\mathbf{q})\,dp_i,</math> where <math>\alpha_{p^i},\alpha_{q^i} \in C^\infty(\R^n \times U,\R).</math>

Substituting <math>v = \left(\partial / \partial q_i\right)_{\mathbf q}</math> into the identity <math>\alpha(\beta_*v) = 0</math> obtain <math display="block">\alpha(\partial / \partial q^i)_{\beta(\mathbf q)} + \sum^n_{j=1}(\partial \beta_j / \partial q^i)_{\mathbf{q}}\cdot \alpha(\partial / \partial p_j)_{\beta(\mathbf{q})} = 0,</math> or equivalently, for any choice of <math>n</math> functions <math>p_i = \beta_i(\mathbf{q}),</math> <math display="block">\alpha_{q^i}(\mathbf{p},\mathbf{q}) + \sum^n_{j=1} \partial p_j / \partial q^i \cdot \alpha_{p_j}(\mathbf{p},\mathbf{q}) = 0.</math> Let <math>\beta = \sum^n_{j=1} c_jdq^j,</math> where <math>c_j=\text{const}.</math> In this case, <math>\beta_j = c_j.</math> For every <math>\mathbf{q} \in U</math> and <math>c_j \in \R,</math> <math display="block">\alpha_{q^i}(\mathbf{p},\mathbf{q})\bigl|_{j=1\ldots n}^{p_j=c_j} = 0.</math> This shows that <math>\alpha_{q^i}(\mathbf{p},\mathbf{q}) = 0</math> on <math>\R^n \times U,</math> and the identity <math display="block">\sum^n_{j=1} \partial p_j / \partial q^i \cdot \alpha_{p_j}(\mathbf{p},\mathbf{q}) = 0 </math> must hold for an arbitrary choice of functions <math>p_i=\beta_i(\mathbf{q}).</math> If <math>\beta = \sum^n_{j=1}c_jq^jdq^j</math> (with <math>{}^j</math> indicating superscript) then <math>\beta_j = c_jq^j,</math> and the identity becomes <math display="block">\alpha_{p_i}(\mathbf{p},\mathbf{q})\bigl|_{j=1\ldots n}^{p_j=c_jq^j} = 0,</math> for every <math>\mathbf{q} \in U</math> and <math>c_j \in \R.</math> Since <math>c_j = p^j / q^j,</math> we see that <math>\alpha_{p_i}(\mathbf{p},\mathbf{q}) = 0,</math> as long as <math>q^j \neq 0</math> for all <math>j.</math> On the other hand, the function <math>\alpha_{p_i}</math> is continuous, and hence <math>\alpha_{p_i}(\mathbf{p},\mathbf{q}) = 0</math> on <math>\R^n \times U.</math> |}

So, by the commutation between the pull-back and the exterior derivative, <math display=block>\beta^*\omega = -\beta^* \, d\theta = -d (\beta^*\theta) = -d\beta.</math>

==Action functional== If <math>H</math> is a [[Hamiltonian mechanics|Hamiltonian]] on the [[cotangent bundle]] and <math>X_H</math> is its [[Hamiltonian vector field]], then the corresponding [[action (physics)|action]] <math>S</math> is given by <math display=block>S = \theta(X_H).</math>

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]]: <math display=block>S(E) = \sum_i \oint p_i\,dq^i</math> with the integral understood to be taken over the manifold defined by holding the energy <math>E</math> constant: <math>H=E=\text{const}.</math>

==On Riemannian and Pseudo-Riemannian Manifolds== If the manifold <math>Q</math> has a Riemannian or pseudo-Riemannian [[Metric tensor|metric]] <math>g,</math> then corresponding definitions can be made in terms of [[generalized coordinates]]. Specifically, if we take the metric to be a map <math display=block>g : TQ \to T^*Q,</math> then define <math display=block>\Theta = g^*\theta</math> and <math display=block>\Omega = -d\Theta = g^*\omega</math>

In generalized coordinates <math>(q^1,\ldots,q^n,\dot q^1,\ldots,\dot q^n)</math> on <math>TQ,</math> one has <math display=block>\Theta = \sum_{ij} g_{ij} \dot q^i dq^j</math> and <math display=block>\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</math>

The metric allows one to define a unit-radius sphere in <math>T^*Q.</math> 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 (mathematician)|Ralph Abraham]] and [[Jerrold E. Marsden]], ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London {{isbn|0-8053-0102-X}} ''See section 3.2''.

[[Category:Symplectic geometry]] [[Category:Hamiltonian mechanics]] [[Category:Lagrangian mechanics]]