= Solution
The <action-angle variables> $(\boldsymbol\theta,\mathbf J)$ are <canonical coordinates>, with $[\theta_a,J_b]=\delta_{ab}$. In the convention $\dot g=\partial_tg+[g,H]$, their <Poisson bracket> is
$$
\boxed{[g,h]=\sum_{a=1}^3\left(\frac{\partial g}{\partial\theta_a}\frac{\partial h}{\partial J_a}-\frac{\partial g}{\partial J_a}\frac{\partial h}{\partial\theta_a}\right)
=\partial_{\boldsymbol\theta}g\cdot\partial_{\mathbf J}h-\partial_{\mathbf J}g\cdot\partial_{\boldsymbol\theta}h.}
$$
In particular, <Hamilton's equations> read $\dot{\boldsymbol\theta}=\partial_{\mathbf J}H$ and $\dot{\mathbf J}=-\partial_{\boldsymbol\theta}H$, fixing the sign needed below.
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