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

For each canonical coordinate and momentum, Hamilton's equations areThey use canonical rather than arbitrary mechanical momenta, a distinction important in a magnetic field. The bracket convention printed in the source is consistent with $\dot{F}=[F,H]$ for a function with no explicit time dependence.

Apply the chain rule to the phase-space function:Thus $\boxed{[F,H]=0⟹F\text{ is conserved}}$. If an explicitly time-dependent function is allowed, the correct equation is $dF/dt={\partial }_{t}F+[F,H]$, so a vanishing bracket alone would not suffice. The source specifies $F(\mathbf{q},\mathbf{p})$, for which that extra term is absent.

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.

In the equatorial plane,Differentiating the given combination produces a cosine termand a sine term $p_r{p}_{\varphi }\dot{\varphi }\sin \varphi /m-p_{\varphi }^2\dot{r}\sin \varphi /(m{r}^2)=0$. Thus $\boxed{\dot{G}=0}$. The expression is a Cartesian component of the Laplace-Runge-Lenz vector with a conventional overall normalization, reflecting the additional symmetry of the inverse-square-force problem.

With the apsidal origin chosen as specified, $G=0$. For $p_{\varphi }\ne 0$, rearrangement givesLet $u=1/r$. Since $du/d\varphi =-p_r/p_{\varphi }$, the equation is $u^{\prime }=-(u-mk/p_{\varphi }^2)\tan \varphi$. Its solution, continued through points where the intermediate division fails, isAs printed, $e$ has units of inverse length and is not the usual dimensionless eccentricity. Writing $\ell =p_{\varphi }^2/(mk)$ gives the standard form $r=\ell /(1+\epsilon \cos \varphi )$ with $\epsilon =e\ell$. The orbit is an ellipse for $|\epsilon |<1$, parabolic for unit magnitude and hyperbolic for magnitude greater than one on its allowed angular interval. The purely radial case $p_{\varphi }=0$ must be handled separately and is excluded by these divisions.