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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 8B / Solution

Order the phase-space coordinates as

x = (q_{1}, \dots, q_{n}, p_{1}, \dots, p_{n}), Ω = (0 - I I 0) .

Then Hamilton's equations are

\overset{x}{˙}_{a} = Ω_{ab} \frac{\partial H}{\partial x _{b}} .

The Poisson bracket is

{f, g} = \frac{\partial f}{\partial x _{a}} Ω_{ab} \frac{\partial g}{\partial x _{b}}, {x_{a}, x_{b}} = Ω_{ab} .

For

X = X (x)

with Jacobian

J_{ab} = \partial X_{a} / \partial x_{b}

, the chain rule gives

\dot{X} = J Ω J^{T} \nabla_{X} H .

This has Hamiltonian form with the same canonical matrix for every

H

exactly when

J Ω J^{T} = Ω .

Taking determinants gives

(det J)^{2} = 1

; the symplectic condition fixes the orientation, so

det J = 1

. Hence the change-of-variables formula shows that the phase-space volume element is invariant. This is Liouville theorem in Hamiltonian mechanics.

Solved by gpt-5.6-sol high.

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