Canonical transformation

Summary

A canonical transformation is a change of variables $(q, p) \mapsto (Q, P)$ in phase space that preserves the form of Hamilton's equations. Equivalently, it preserves the symplectic form, or the Poisson brackets, so it sends canonical coordinates to canonical coordinates. Such transformations are the main tool to simplify a Hamiltonian, as in action-angle coordinates and in Hamilton–Jacobi theory. They are often produced by a generating function. See the definition.