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.

Figure 1. A phase-space grid and blob (left) and their image under a canonical (or non-canonical) transformation morphed continuously from the identity (right), with the Poisson bracket and the area ratio. Method: analytic Jacobian Poisson bracket; shoelace phase-area; symplectic M^T J M check; one-parameter morph from identity.

map

parameter1.70

morph t1.00

energy E1.00

WHAT TO TRY

Switch the map: harmonic scaling squashes a circle into an ellipse, a phase rotation spins it, and an area-preserving squeeze stretches one axis while shrinking the other. The grid deforms with the blob.
Watch the two diagnostic curves: the solid line (full map) stays pinned at 1, while the dashed line (the same q-stretch with p left untouched) drifts away. The gap is exactly what the p-transformation cancels, so the flat line is the symplectic condition {Q,P} = 1 doing its work, not an empty plot.
Pick the p-doubling map, which is NOT canonical: the area ratio jumps away from 1, and the blob inflates. A transformation that changes phase-space area cannot come from any Hamiltonian.