Canonical Coordinates - Definition, On Cotangent Bundles

Definition, On Cotangent Bundles

Canonical coordinates are defined as a special set of coordinates on the cotangent bundle of a manifold. They are usually written as a set of or with the x 's or q 's denoting the coordinates on the underlying manifold and the p 's denoting the conjugate momentum, which are 1-forms in the cotangent bundle at point q in the manifold.

A common definition of canonical coordinates is any set of coordinates on the cotangent bundle that allow the canonical one form to be written in the form

up to a total differential. A change of coordinates that preserves this form is a canonical transformation; these are a special case of a symplectomorphism, which are essentially a change of coordinates on a symplectic manifold.

In the following exposition, we assume that the manifolds are real manifolds, so that cotangent vectors acting on tangent vectors produce real numbers.