(i) An axially symmetric top rotates freely about a fixed point $O$ on its axis. The principal moments of inertia are $A, A, C$ and the centre of gravity $G$ is a distance $h$ from $O .$

Define the three Euler angles $\theta, \phi$ and $\psi$, specifying the orientation of the top. Use Lagrange's equations to show that there are three conserved quantities in the motion. Interpret them physically.

(ii) Initially the top is spinning with angular speed $n$ about $O G$, with $O G$ vertical, before it is slightly disturbed.

Show that, in the subsequent motion, $\theta$ stays close to zero if $C^{2} n^{2}>4 m g h A$, but if this condition fails then $\theta$ attains a maximum value given approximately by

$$\cos \theta \approx \frac{C^{2} n^{2}}{2 m g h A}-1$$

Why is this only an approximation?