Editing Canonical transformation (section)

==Modern mathematical description==
In mathematical terms, [[canonical coordinates]] are any coordinates on the phase space ([[cotangent bundle]]) of the system that allow the [[canonical one-form]] to be written as
<math display="block">\sum_i p_i\,dq^i</math>
up to a total differential ([[exact form]]). The change of variable between one set of canonical coordinates and another is a '''canonical transformation'''. The index of the [[generalized coordinate]]s {{math|'''q'''}} is written here as a ''superscript'' (<math>q^{i}</math>), not as a ''subscript'' as done above (<math>q_{i}</math>). The superscript conveys the [[Covariance and contravariance of vectors|contravariant transformation properties]] of the generalized coordinates, and does ''not'' mean that the coordinate is being raised to a power. Further details may be found at the [[symplectomorphism]] article.