A symmetric top of unit mass moves under the action of gravity. The Lagrangian is given by

$$L=\frac{1}{2} I_{1}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+\frac{1}{2} I_{3}(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-g l \cos \theta$$

where the generalized coordinates are the Euler angles $(\theta, \phi, \psi)$, the principal moments of inertia are $I_{1}$ and $I_{3}$ and the distance from the centre of gravity of the top to the origin is $l$.

Show that $\omega_{3}=\dot{\psi}+\dot{\phi} \cos \theta$ and $p_{\phi}=I_{1} \dot{\phi} \sin ^{2} \theta+I_{3} \omega_{3} \cos \theta$ are constants of the motion. Show further that, when $p_{\phi}=I_{3} \omega_{3}$, with $\omega_{3}>0$, the equation of motion for $\theta$ is

$$\frac{d^{2} \theta}{d t^{2}}=\frac{g l \sin \theta}{I_{1}}\left(1-\frac{I_{3}^{2} \omega_{3}^{2}}{4 I_{1} g l \cos ^{4}(\theta / 2)}\right)$$

Find the possible equilibrium values of $\theta$ in the two cases:

(i) $I_{3}^{2} \omega_{3}^{2}>4 I_{1} g l$,

(ii) $I_{3}^{2} \omega_{3}^{2}<4 I_{1} g l$.

By considering linear perturbations in the neighbourhoods of the equilibria in each case, find which are unstable and give expressions for the periods of small oscillations about the stable equilibria.