Past exam of the theoretical physics I course of the University of Cambridge / 2015 / Paper 1 / 3 / a

For canonical coordinates $(q_i,p_i)$, the Poisson bracket isIt is bilinear and antisymmetric, obeys the Leibniz rule in either argument, and satisfies the Jacobi identity. These properties encode the symplectic structure of phase space. The two printed inline applications are treated separately below.

An invertible change $(q,p)\mapsto (Q,P)$ is canonical when it preserves the Poisson bracket. Its fundamental brackets must beThe chain rule then makes the bracket of any two observables identical in the old and new coordinates. Consequently Hamilton's equations retain their canonical form. A time-dependent canonical transformation also requires the corresponding extra term in the transformed Hamiltonian; bracket preservation by itself concerns each fixed time.

Along Hamiltonian motion,Substituting ${\dot{q}}_i=\partial H/\partial p_i$ and ${\dot{p}}_i=-\partial H/\partial q_i$ givesFor the printed $O(q,p)$ there is no explicit time term. In particular, an autonomous observable is conserved when its Poisson bracket with $H$ vanishes. Taking $O=q_i$ or $O=p_i$ recovers Hamilton's equations.