Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 8D / Solution

A Lagrange top is an axially symmetric rigid body with one point fixed, centre of mass a distance $l$ along its symmetry axis, principal moments $I_1=I_2,I_3$, mass $M$, and Euler angles $\theta ,\phi ,\psi$ in gravity $g$.

The cyclic coordinates give $p_{\psi }=I_3(\dot{\psi }+\dot{\phi }\cos \theta )$ and $p_{\phi }=I_1\dot{\phi }{\sin }^2\theta +p_{\psi }\cos \theta$; the energy $E$ is the third integral. Eliminating $\dot{\phi },\dot{\psi }$ from $E$ and setting $u=\cos \theta$, with $\dot{u}=-\dot{\theta }\sin \theta$, yields ${\dot{u}}^2=f(u)$ after multiplication by $1-u^2$; all denominators cancel and $f$ is cubic.

Solved by gpt-5.6-sol high.