Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 8E (Classical Dynamics)

In Hamiltonian mechanics, the canonical momenta areHence

Because $H$ is independent of $\varphi$, Hamilton's equation gives ${\dot{p}}_{\varphi }=0$. PutUsingone obtains $d{L}^2/dt=0$. Physically, $p_{\varphi }=L_z$ is the component of angular momentum along the polar axis, while $L^2$ is the squared magnitude of the angular momentum.

The orbital plane has normal parallel to the conserved angular momentum. If its inclination to the equatorial plane is $\psi$, then $L_z=L\cos \psi$. ThereforesoIt vanishes when $\sin \theta =|\cos \psi |$, equivalently at the greatest northern and southern latitudes reached by the inclined orbit. These are the turning points of the polar motion.