Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S7 / 3 / c / Solution

The radial and azimuthal Hamilton equations giveAlso $\dot{\theta }=p_{\theta }/(m{r}^2)$ and ${\dot{p}}_{\theta }=p_{\varphi }^2\cos \theta /(m{r}^2{\sin }^3\theta )$. Differentiating the angular-momentum square,Therefore both $p_{\varphi }$ and $\boxed{J^2=p_{\theta }^2+p_{\varphi }^2/{\sin }^2\theta }$ are constants. Rotational symmetry conserves the full angular-momentum vector; these are its axial component and squared magnitude in spherical canonical variables.