 Solution

ID: past-exam-of-the-theoretical-physics-i-course-of-the-university-of-cambridge/2015/paper-1/3/a/i/solution

(q, p) \mapsto (Q, P)

is canonical when it preserves the Poisson bracket. Its fundamental brackets must be

{Q_{i}, Q_{j}} = 0, {P_{i}, P_{j}} = 0, {Q_{i}, P_{j}} = δ_{ij} .

The 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.

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