Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 2 / 15C / c

A canonical transformation preserves the symplectic form and therefore the form of Hamilton's equations for arbitrary Hamiltonians. In one degree of freedom,Thus the transformation is canonical exactly when $[Q,P]=1$. Equivalently the chain rule gives $[F,G{]}_{q,p}=[Q,P{]}_{q,p}[F,G{]}_{Q,P}$ for arbitrary transformed functions, so the bracket and Hamiltonian dynamics are preserved precisely with this factor one. The condition also makes the Jacobian nonzero and hence supplies local invertibility; a global canonical change of variables additionally requires the stipulated global bijectivity on its domain.