Past exam of the kinetic theory course of the University of Oxford / 2023 / 3 / a / Solution

The action-angle variables $(\mathbf{\theta },\mathbf{J})$ are canonical coordinates, with $[{\theta }_a,J_b]={\delta }_{ab}$. In the convention $\dot{g}={\partial }_{t}g+[g,H]$, their Poisson bracket isIn particular, Hamilton's equations read $\dot{\mathbf{\theta }}={\partial }_{\mathbf{J}}H$ and $\dot{\mathbf{J}}=-{\partial }_{\mathbf{\theta }}H$, fixing the sign needed below.