Canonical symplectic form

In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T*Q of a 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 one-form, or the symplectic potential. A similar object is the canonical vector field on the tangent bundle. 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.

In canonical coordinates, the tautological one-form is given by

Equivalently, any coordinates on phase space which preserve this structure for the canonical one-form, 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

The extension of this concept to general fibre bundles is known as the solder form.

...
Wikipedia