Use this tag for questions related to canonical transformations, which are changes of canonical coordinates that preserve the form of Hamilton's equations.
In Hamiltonian mechanics, a canonical transformation, sometimes called form invariance, is a change of canonical coordinates (q, p, t) → (Q, P, t) that preserves the form of hamilton-equations. The transformation need not preserve the form of the Hamiltonian itself. Canonical transformations are useful in their own right and also form the basis for the hamilton-jacobi-equation (used for calculating conserved quantities) and Liouville's theorem (the basis for classical statistical mechanics).
Because Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates q → Q do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if momentum is simultaneously changed by a Legendre transformation into $$P_i = \frac {\partial L}{\partial \dot {Q}_i}.$$
Coordinate transformations (also called point transformations) are therefore a type of canonical transformation. However, the class of canonical transformations is much broader because the old generalized coordinates, momenta and even time, may be combined to form the new generalized coordinates and momenta. Canonical transformations that do not include time explicitly are called restricted canonical transformations.
