Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 9E (Classical Dynamics)

Assume the velocity-to-momentum Legendre map is locally invertible. Differentiate the Hamiltonian while regarding $q,p,t$ as independent:The velocity variations cancel because $p_i=\partial L/\partial {\dot{q}}_i$. Combining the remaining coefficients with the Euler-Lagrange equations gives Hamilton's equationsAn ignorable coordinate does not occur explicitly in the Lagrangian, so its conjugate momentum is conserved. For the spherical motion, $\varphi$ is ignorable and the Hamiltonian has no explicit time dependence. The two constants are $p_{\varphi }$ and $H$.

The polar Hamilton equations areA constant polar angle therefore requires $p_{\theta }=0$ and vanishing of the second right-hand side. For $0<\theta <\pi$ with $\cos \theta \ne 0$, this isThe expression must be nonnegative to represent a real momentum. Once it holds, $\varphi$ advances uniformly at $p_{\varphi }/(m{a}^2{\sin }^2\theta )$, so it is also sufficient with those initial data. At the equator the undivided equation instead requires $V^{\prime }(\pi /2)=0$, with any $p_{\varphi }$; dividing by cosine would wrongly discard this special case. The polar coordinate chart itself excludes the poles.