In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T ∗ Q {\displaystyle T^{*}Q} of a manifold Q . {\displaystyle 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 {\displaystyle Q} ). The exterior derivative of this form defines a symplectic form, giving T ∗ Q {\displaystyle 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.
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 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), 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
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 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 {\displaystyle Q} be a manifold and M = T ∗ Q {\displaystyle M=T^{*}Q} be the cotangent bundle or 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 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)} :
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