Editing Canonical coordinates (section)

==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 <math>\left(q^i, p_j\right)</math> or <math>\left(x^i, p_j\right)</math> 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-form]]s 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
:<math>\sum_i p_i\,\mathrm{d}q^i</math>

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.