 Solution

ID: past-exam-of-the-kinetic-theory-course-of-the-university-of-oxford/2023/3/a/solution

(θ, J)

[θ_{a}, J_{b}] = δ_{ab}

. In the convention

\overset{g}{˙} = \partial_{t} g + [g, H]

[g, h] = a = 1 \sum 3 (\frac{\partial g}{\partial θ _{a}} \frac{\partial h}{\partial J _{a}} - \frac{\partial g}{\partial J _{a}} \frac{\partial h}{\partial θ _{a}}) = \partial_{θ} g \cdot \partial_{J} h - \partial_{J} g \cdot \partial_{θ} h .

In particular, Hamilton's equations read

\dot{θ} = \partial_{J} H

and

\dot{J} = - \partial_{θ} H

, fixing the sign needed below.

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