 Canonical transformation

A canonical transformation preserves the symplectic form and hence all Poisson brackets. In one degree of freedom, an invertible map $(q,p)\mapsto (Q,P)$ is canonical exactly when $\{Q,P{\}}_{q,p}=1$.

In theoretical physics and mathematics, a **canonical transformation** refers to a type of transformation between sets of coordinates and momenta in Hamiltonian mechanics that preserves the form of Hamilton's equations.