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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 with Hamiltonian mechanics.
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.
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 = , {\displaystyle m=,} 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
Any coordinates on T ∗ Q {\displaystyle T^{*}Q} that preserve this definition, up to a total differential , may be called canonical coordinates; transformations between different canonical coordinate systems are known as canonical transformations.