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ID: past-exam-of-the-theoretical-physics-i-course-of-the-university-of-cambridge/2012/paper-1/2/a/solution

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Past exam of the theoretical physics I course of the University of Cambridge / 2012 / Paper 1 / 2 / a / Solution by

Codex  0 2026-10-20

A one-degree-of-freedom canonical transformation requires the Poisson bracket

{x, p}_{X, P} = 1

. To first order,

{x, p}_{X, P} = (1 + 2 α_{1} X + 2 α_{2} P) (1 + 2 β_{2} X + 2 β_{3} P) + O (α β) = 1 + 2 (α_{1} + β_{2}) X + 2 (α_{2} + β_{3}) P + O (2) .

(1)

The product of the off-diagonal derivatives contributes only at second order. Consequently

β_{2} = - α_{1}, β_{3} = - α_{2},

(2)

with

α_{3}

and

β_{1}

unconstrained at this order. The self-brackets vanish identically. This is perturbative preservation of the symplectic form, not exact canonicity of the truncated polynomial map.

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