Canonical Transformations

A change of phase-space coordinates $(q,p)\to(Q,P)$ is canonical when it preserves the symplectic structure: the Poisson bracket $\{Q,P\}=\partial_qQ\,\partial_pP-\partial_pQ\,\partial_pP=1$, equivalently the Jacobian is unimodular, equivalently it preserves phase-space area (Liouville). The left panel is a blob of phase points; the right is its image under the chosen map. The harmonic scaling turns the energy ellipse into a circle; a rotation spins it; a squeeze stretches it thin but keeps the area; a deliberately non-canonical $p$-doubling balloons it to twice the area, with $\{Q,P\}=2$. A canonical map need not be a symmetry: the squeeze is canonical yet changes the Hamiltonian.