 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-20/1/ii/c/solution

Codex  0 Created 2026-10-03 Updated 2026-10-07

Apply preservation of Poisson brackets to the coordinate functions. It gives

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

Conversely, the entries of

A J A^{T}

are exactly the Poisson brackets between all transformed coordinate functions. The displayed relations therefore say

A J A^{T} = J

, which implies the canonical transformation condition and, by the chain rule, preservation of every Poisson bracket. Thus all three conditions are equivalent:

f^{*} ω_{0} = ω_{0} ⟺ all brackets are preserved ⟺ {Q_{i}, Q_{j}} = {P_{i}, P_{j}} = 0, {Q_{i}, P_{j}} = δ_{ij} .

In particular the original coordinate bracket

{q_{i}, p_{j}}

δ_{ij}

with the chosen Poisson bracket convention.

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