An axially-symmetric top of mass $m$ is free to rotate about a fixed point $O$ on its axis. The principal moments of inertia about $O$ are $A, A, C$, and the centre of gravity $G$ is at a distance $\ell$ from $O$. Define Euler angles $\theta, \phi$ and $\psi$ which specify the orientation of the top, where $\theta$ is the inclination of $O G$ to the upward vertical. Show that there are three conserved quantities for the motion, and give their physical meaning.

Initially, the top is spinning with angular velocity $n$ about $O G$, with $G$ vertically above $O$, before being disturbed slightly. Show that, in the subsequent motion, $\theta$ will remain close to zero provided $C^{2} n^{2}>4 m g \ell A$, but that if $C^{2} n^{2}<4 m g \ell A$, then $\theta$ will attain a maximum value given by

$$\cos \theta \simeq\left(C^{2} n^{2} / 2 m g \ell A\right)-1$$