# Tautological one-form

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Canonical differential form

Not to be confused with [Symplectic manifold § Definition](/source/Symplectic_manifold#Definition), or [Symplectic vector space](/source/Symplectic_vector_space).

This article relies largely or entirely on a single source. Please help improve this article by citing more sources. Find sources: "Tautological one-form" – news · newspapers · books · scholar · JSTOR (March 2025)

In [mathematics](/source/Mathematics), the **tautological one-form** is a special [1-form](/source/1-form) defined on the [cotangent bundle](/source/Cotangent_bundle) T ∗ Q {\displaystyle T^{*}Q} of a [manifold](/source/Manifold) Q . {\displaystyle Q.} In [physics](/source/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](/source/Lagrangian_mechanics) and [Hamiltonian mechanics](/source/Hamiltonian_mechanics) (on the manifold Q {\displaystyle Q} ).

The [exterior derivative](/source/Exterior_derivative) of this form defines a [symplectic form](/source/Symplectic_manifold), giving T ∗ Q {\displaystyle T^{*}Q} the structure of a [symplectic manifold](/source/Symplectic_manifold). The tautological one-form plays an important role in relating the formalism of [Hamiltonian mechanics](/source/Hamiltonian_mechanics) and [Lagrangian mechanics](/source/Lagrangian_mechanics). The tautological one-form is sometimes also called the **Liouville one-form**, the **Poincaré one-form**, the **[canonical](/source/Canonical_form) one-form**, or the **symplectic potential**. A similar object is the [canonical vector field](/source/Canonical_vector_field) on the [tangent bundle](/source/Tangent_bundle).

## Definition in coordinates

To define the tautological one-form, select a coordinate chart U {\displaystyle U} on T ∗ Q {\displaystyle T^{*}Q} and a [canonical coordinate](/source/Canonical_coordinate) system on U . {\displaystyle U.} Pick an arbitrary point m ∈ T ∗ Q . {\displaystyle m\in T^{*}Q.} By definition of cotangent bundle, m = ( q , p ) , {\displaystyle m=(q,p),} where q ∈ Q {\displaystyle q\in Q} and p ∈ T q ∗ Q . {\displaystyle p\in T_{q}^{*}Q.} The tautological one-form θ m : T m T ∗ Q → R {\displaystyle \theta _{m}:T_{m}T^{*}Q\to \mathbb {R} } is given by θ m = ∑ i = 1 n p i d q i , {\displaystyle \theta _{m}=\sum _{i=1}^{n}p_{i}\,dq^{i},} with n = dim ⁡ Q {\displaystyle n=\mathop {\text{dim}} Q} and ( p 1 , … , p n ) ∈ U ⊆ R n {\displaystyle (p_{1},\ldots ,p_{n})\in U\subseteq \mathbb {R} ^{n}} being the coordinate representation of p . {\displaystyle p.}

Any coordinates on T ∗ Q {\displaystyle T^{*}Q} that preserve this definition, up to a total differential ([exact form](/source/Exact_form)), may be called canonical coordinates; transformations between different canonical coordinate systems are known as [canonical transformations](/source/Canonical_transformation).

The **canonical symplectic form**, also known as the **Poincaré two-form**, is given by ω = − d θ = ∑ i d q i ∧ d p i {\displaystyle \omega =-d\theta =\sum _{i}dq^{i}\wedge dp_{i}}

The extension of this concept to general [fibre bundles](/source/Fibre_bundle) is known as the [solder form](/source/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](/source/Algebraic_geometry) and [complex geometry](/source/Complex_geometry) the term "canonical" is discouraged, due to confusion with the [canonical class](/source/Canonical_class), and the term "tautological" is preferred, as in [tautological bundle](/source/Tautological_bundle).

## Coordinate-free definition

The tautological 1-form can also be defined rather abstractly as a form on [phase space](/source/Phase_space). Let Q {\displaystyle Q} be a manifold and M = T ∗ Q {\displaystyle M=T^{*}Q} be the [cotangent bundle](/source/Cotangent_bundle) or [phase space](/source/Phase_space). Let π : M → Q {\displaystyle \pi :M\to Q} be the canonical fiber bundle projection, and let d π : T M → T Q {\displaystyle \mathrm {d} \pi :TM\to TQ} be the [induced](/source/Induced_homomorphism) [tangent map](/source/Tangent_map). Let m {\displaystyle m} be a point on M . {\displaystyle M.} Since M {\displaystyle M} is the cotangent bundle, we can understand m {\displaystyle m} to be a map of the tangent space at q = π ( m ) {\displaystyle q=\pi (m)} : m : T q Q → R . {\displaystyle m:T_{q}Q\to \mathbb {R} .}

That is, we have that m {\displaystyle m} is in the fiber of q . {\displaystyle q.} The tautological one-form θ m {\displaystyle \theta _{m}} at point m {\displaystyle m} is then defined to be θ m = m ∘ d m π . {\displaystyle \theta _{m}=m\circ \mathrm {d} _{m}\pi .}

It is a linear map θ m : T m M → R {\displaystyle \theta _{m}:T_{m}M\to \mathbb {R} } and so θ : M → T ∗ M . {\displaystyle \theta :M\to T^{*}M.}

### 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 V ∈ T ω ( T ∗ Q ) {\displaystyle V\in T_{\omega }(T^{*}Q)} , where ω ∈ T q ∗ Q {\displaystyle \omega \in T_{q}^{*}Q} is its base point (a covector), and q ∈ Q {\displaystyle q\in Q} is *its* base point. Then, there are 3 effects of moving infinitesimally from ω {\displaystyle \omega } to ω + V δ t {\displaystyle \omega +V\delta t} : shifting the base point q {\displaystyle q} , rotating the hyperplane of the covector ker ⁡ ω {\displaystyle \ker \omega } , and changing the distance separating between the hyperplane pairs. In particular, the shifting of the base point creates a vector d π ( V ) ∈ T q Q {\displaystyle d\pi (V)\in T_{q}Q} , which can be fed into the covector.

The tautological 1-form computes θ ( V ) {\displaystyle \theta (V)} by feeding to ω {\displaystyle \omega } the vector created by shifting the base point, and ignoring the other two effects, which cannot be fed into the covector, giving θ ( V ) = ω ( d π ( V ) ) {\displaystyle \theta (V)=\omega (d\pi (V))} .

## Symplectic potential

The symplectic potential is generally defined a bit more freely, and also only defined locally: it is any one-form ϕ {\displaystyle \phi } such that ω = − d ϕ {\displaystyle \omega =-d\phi } ; in effect, symplectic potentials differ from the canonical 1-form by a [closed form](/source/Closed_differential_form).

## Properties

The tautological one-form is the unique one-form that "cancels" [pullback](/source/Pullback_(differential_geometry)). That is, let β {\displaystyle \beta } be a 1-form on Q . {\displaystyle Q.} β {\displaystyle \beta } is a [section](/source/Section_(fiber_bundle)) β : Q → T ∗ Q . {\displaystyle \beta :Q\to T^{*}Q.} For an arbitrary 1-form σ {\displaystyle \sigma } on T ∗ Q , {\displaystyle T^{*}Q,} the pullback of σ {\displaystyle \sigma } by β {\displaystyle \beta } is, by definition, β ∗ σ := σ ∘ β ∗ . {\displaystyle \beta ^{*}\sigma :=\sigma \circ \beta _{*}.} Here, β ∗ : T Q → T T ∗ Q {\displaystyle \beta _{*}:TQ\to TT^{*}Q} is the [pushforward](/source/Pushforward_(differential)) of β . {\displaystyle \beta .} Like β , {\displaystyle \beta ,} β ∗ σ {\displaystyle \beta ^{*}\sigma } is a 1-form on Q . {\displaystyle Q.} The tautological one-form θ {\displaystyle \theta } is the only form with the property that β ∗ θ = β , {\displaystyle \beta ^{*}\theta =\beta ,} for every 1-form β {\displaystyle \beta } on Q . {\displaystyle Q.}

Proof. For a chart ( { q i } i = 1 n , U ) {\displaystyle (\{q^{i}\}_{i=1}^{n},U)} on Q {\displaystyle Q} (where U ⊆ R n ) , {\displaystyle U\subseteq \mathbb {R} ^{n}),} let { p i , q i } i = 1 n {\displaystyle \{p_{i},q^{i}\}_{i=1}^{n}} be the coordinates on T ∗ Q , {\displaystyle T^{*}Q,} where the fiber coordinates { p i } i = 1 n {\displaystyle \{p_{i}\}_{i=1}^{n}} are associated with the linear basis { d q i } i = 1 n . {\displaystyle \{dq^{i}\}_{i=1}^{n}.} By assumption, for every q = ( q 1 , … , q n ) ∈ U , {\displaystyle {\mathbf {q} }=(q^{1},\ldots ,q^{n})\in U,} β ( q ) = ∑ i = 1 n β i ( q ) d q i , {\displaystyle \beta ({\mathbf {q} })=\sum _{i=1}^{n}\beta _{i}(\mathbf {q} )\,dq^{i},} or q = ( q 1 , … , q n ) → β ( q 1 , … , q n ⏟ q , β 1 ( q ) , … , β n ( q ⏟ p ) ) . {\displaystyle \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} })).} It follows that β ∗ ( ∂ ∂ q i | q ) = ∂ ∂ q i | β ( q ) + ∑ j = 1 n ∂ β j ∂ q i | q ⋅ ∂ ∂ p j | β ( q ) {\displaystyle \beta _{*}\left({\frac {\partial }{\partial q^{i}}}{\Biggl |}_{\mathbf {q} }\right)={\frac {\partial }{\partial q^{i}}}{\Biggl |}_{\beta (\mathbf {q} )}+\sum _{j=1}^{n}{\frac {\partial \beta _{j}}{\partial q^{i}}}{\Biggl |}_{\mathbf {q} }\cdot {\frac {\partial }{\partial p_{j}}}{\Biggl |}_{\beta (\mathbf {q} )}} which implies that ( β ∗ d q i ) ( ∂ / ∂ q j ) q = d q i [ β ∗ ( ∂ / ∂ q j ) q ] = δ i j . {\displaystyle (\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}.} Step 1. We have ( β ∗ θ ) ( ∂ / ∂ q i ) q = θ ( β ∗ ( ∂ / ∂ q i ) q ) = ( ∑ j = 1 n p j d q j ) ( β ∗ ( ∂ / ∂ q i ) q ) = β i ( q ) = β ( ∂ / ∂ q i ) q . {\displaystyle {\begin{aligned}(\beta ^{*}\theta )\left(\partial /\partial q^{i}\right)_{\mathbf {q} }&=\theta \left(\beta _{*}\left(\partial /\partial q^{i}\right)_{\mathbf {q} }\right)=\left(\sum _{j=1}^{n}p_{j}dq^{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{aligned}}} Step 1'. For completeness, we now give a coordinate-free proof that β ∗ θ = β , {\displaystyle \beta ^{*}\theta =\beta ,} for any 1-form β . {\displaystyle \beta .} Observe that, intuitively speaking, for every q ∈ Q {\displaystyle q\in Q} and p ∈ T q ∗ Q , {\displaystyle p\in T_{q}^{*}Q,} the linear map d π ( q , p ) {\displaystyle d\pi _{(q,p)}} in the definition of θ {\displaystyle \theta } projects the tangent space T ( q , p ) T ∗ Q {\displaystyle T_{(q,p)}T^{*}Q} onto its subspace T q Q . {\displaystyle T_{q}Q.} As a consequence, for every q ∈ Q {\displaystyle q\in Q} and v ∈ T q Q , {\displaystyle v\in T_{q}Q,} d π β ( q ) ( β ∗ q v ) = v , {\displaystyle d\pi _{\beta (q)}(\beta _{*q}v)=v,} where β ∗ q {\displaystyle \beta _{*q}} is the instance of β ∗ {\displaystyle \beta _{*}} at the point q ∈ Q , {\displaystyle q\in Q,} that is, β ∗ q : T q Q → T β ( q ) T ∗ Q . {\displaystyle \beta _{*q}:T_{q}Q\to T_{\beta (q)}T^{*}Q.} Applying the coordinate-free definition of θ {\displaystyle \theta } to θ β ( q ) , {\displaystyle \theta _{\beta (q)},} obtain ( β ∗ θ ) q v = θ β ( q ) ( β ∗ q v ) = β ( q ) ( d π β ( q ) ( β ∗ q v ) ) = β ( q ) v . {\displaystyle (\beta ^{*}\theta )_{q}v=\theta _{\beta (q)}(\beta _{*q}v)=\beta (q)(d\pi _{\beta (q)}(\beta _{*q}v))=\beta (q)v.} Step 2. It is enough to show that α = 0 {\displaystyle \alpha =0} if β ∗ α = 0 , {\displaystyle \beta ^{*}\alpha =0,} for every one-form β . {\displaystyle \beta .} Let α = ∑ i = 1 n α q i ( p , q ) d q i + ∑ i = 1 n α p i ( p , q ) d p i , {\displaystyle \alpha =\sum _{i=1}^{n}\alpha _{q^{i}}(\mathbf {p} ,\mathbf {q} )\,dq^{i}+\sum _{i=1}^{n}\alpha _{p_{i}}(\mathbf {p} ,\mathbf {q} )\,dp_{i},} where α p i , α q i ∈ C ∞ ( R n × U , R ) . {\displaystyle \alpha _{p^{i}},\alpha _{q^{i}}\in C^{\infty }(\mathbb {R} ^{n}\times U,\mathbb {R} ).} Substituting v = ( ∂ / ∂ q i ) q {\displaystyle v=\left(\partial /\partial q_{i}\right)_{\mathbf {q} }} into the identity α ( β ∗ v ) = 0 {\displaystyle \alpha (\beta _{*}v)=0} obtain α ( ∂ / ∂ q i ) β ( q ) + ∑ j = 1 n ( ∂ β j / ∂ q i ) q ⋅ α ( ∂ / ∂ p j ) β ( q ) = 0 , {\displaystyle \alpha (\partial /\partial q^{i})_{\beta (\mathbf {q} )}+\sum _{j=1}^{n}(\partial \beta _{j}/\partial q^{i})_{\mathbf {q} }\cdot \alpha (\partial /\partial p_{j})_{\beta (\mathbf {q} )}=0,} or equivalently, for any choice of n {\displaystyle n} functions p i = β i ( q ) , {\displaystyle p_{i}=\beta _{i}(\mathbf {q} ),} α q i ( p , q ) + ∑ j = 1 n ∂ p j / ∂ q i ⋅ α p j ( p , q ) = 0. {\displaystyle \alpha _{q^{i}}(\mathbf {p} ,\mathbf {q} )+\sum _{j=1}^{n}\partial p_{j}/\partial q^{i}\cdot \alpha _{p_{j}}(\mathbf {p} ,\mathbf {q} )=0.} Let β = ∑ j = 1 n c j d q j , {\displaystyle \beta =\sum _{j=1}^{n}c_{j}dq^{j},} where c j = const . {\displaystyle c_{j}={\text{const}}.} In this case, β j = c j . {\displaystyle \beta _{j}=c_{j}.} For every q ∈ U {\displaystyle \mathbf {q} \in U} and c j ∈ R , {\displaystyle c_{j}\in \mathbb {R} ,} α q i ( p , q ) | j = 1 … n p j = c j = 0. {\displaystyle \alpha _{q^{i}}(\mathbf {p} ,\mathbf {q} ){\bigl |}_{j=1\ldots n}^{p_{j}=c_{j}}=0.} This shows that α q i ( p , q ) = 0 {\displaystyle \alpha _{q^{i}}(\mathbf {p} ,\mathbf {q} )=0} on R n × U , {\displaystyle \mathbb {R} ^{n}\times U,} and the identity ∑ j = 1 n ∂ p j / ∂ q i ⋅ α p j ( p , q ) = 0 {\displaystyle \sum _{j=1}^{n}\partial p_{j}/\partial q^{i}\cdot \alpha _{p_{j}}(\mathbf {p} ,\mathbf {q} )=0} must hold for an arbitrary choice of functions p i = β i ( q ) . {\displaystyle p_{i}=\beta _{i}(\mathbf {q} ).} If β = ∑ j = 1 n c j q j d q j {\displaystyle \beta =\sum _{j=1}^{n}c_{j}q^{j}dq^{j}} (with j {\displaystyle {}^{j}} indicating superscript) then β j = c j q j , {\displaystyle \beta _{j}=c_{j}q^{j},} and the identity becomes α p i ( p , q ) | j = 1 … n p j = c j q j = 0 , {\displaystyle \alpha _{p_{i}}(\mathbf {p} ,\mathbf {q} ){\bigl |}_{j=1\ldots n}^{p_{j}=c_{j}q^{j}}=0,} for every q ∈ U {\displaystyle \mathbf {q} \in U} and c j ∈ R . {\displaystyle c_{j}\in \mathbb {R} .} Since c j = p j / q j , {\displaystyle c_{j}=p^{j}/q^{j},} we see that α p i ( p , q ) = 0 , {\displaystyle \alpha _{p_{i}}(\mathbf {p} ,\mathbf {q} )=0,} as long as q j ≠ 0 {\displaystyle q^{j}\neq 0} for all j . {\displaystyle j.} On the other hand, the function α p i {\displaystyle \alpha _{p_{i}}} is continuous, and hence α p i ( p , q ) = 0 {\displaystyle \alpha _{p_{i}}(\mathbf {p} ,\mathbf {q} )=0} on R n × U . {\displaystyle \mathbb {R} ^{n}\times U.}

So, by the commutation between the pull-back and the exterior derivative, β ∗ ω = − β ∗ d θ = − d ( β ∗ θ ) = − d β . {\displaystyle \beta ^{*}\omega =-\beta ^{*}\,d\theta =-d(\beta ^{*}\theta )=-d\beta .}

## Action functional

If H {\displaystyle H} is a [Hamiltonian](/source/Hamiltonian_mechanics) on the [cotangent bundle](/source/Cotangent_bundle) and X H {\displaystyle X_{H}} is its [Hamiltonian vector field](/source/Hamiltonian_vector_field), then the corresponding [action](/source/Action_(physics)) S {\displaystyle S} is given by S = θ ( X H ) . {\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](/source/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](/source/Action-angle_variables): S ( E ) = ∑ i ∮ p i d q i {\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 E {\displaystyle E} constant: H = E = const . {\displaystyle H=E={\text{const}}.}

## On Riemannian and Pseudo-Riemannian Manifolds

If the manifold Q {\displaystyle Q} has a Riemannian or pseudo-Riemannian [metric](/source/Metric_tensor) g , {\displaystyle g,} then corresponding definitions can be made in terms of [generalized coordinates](/source/Generalized_coordinates). Specifically, if we take the metric to be a map g : T Q → T ∗ Q , {\displaystyle g:TQ\to T^{*}Q,} then define Θ = g ∗ θ {\displaystyle \Theta =g^{*}\theta } and Ω = − d Θ = g ∗ ω {\displaystyle \Omega =-d\Theta =g^{*}\omega }

In generalized coordinates ( q 1 , … , q n , q ˙ 1 , … , q ˙ n ) {\displaystyle (q^{1},\ldots ,q^{n},{\dot {q}}^{1},\ldots ,{\dot {q}}^{n})} on T Q , {\displaystyle TQ,} one has Θ = ∑ i j g i j q ˙ i d q j {\displaystyle \Theta =\sum _{ij}g_{ij}{\dot {q}}^{i}dq^{j}} and Ω = ∑ i j g i j d q i ∧ d q ˙ j + ∑ i j k ∂ g i j ∂ q k q ˙ i d q j ∧ d q k {\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 T ∗ Q . {\displaystyle T^{*}Q.} The canonical one-form restricted to this sphere forms a [contact structure](/source/Contact_structure); the contact structure may be used to generate the [geodesic flow](/source/Geodesic_flow) for this metric.

## References

- [Ralph Abraham](/source/Ralph_Abraham_(mathematician)) and [Jerrold E. Marsden](/source/Jerrold_E._Marsden), *Foundations of Mechanics*, (1978) Benjamin-Cummings, London [ISBN](/source/ISBN_(identifier)) [0-8053-0102-X](https://en.wikipedia.org/wiki/Special:BookSources/0-8053-0102-X) *See section 3.2*.

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Adapted from the Wikipedia article [Tautological one-form](https://en.wikipedia.org/wiki/Tautological_one-form) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Tautological_one-form?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.