Consider a heavy symmetric top of mass $M$ with principal moments of inertia $I_{1}$, $I_{2}$ and $I_{3}$, where $I_{1}=I_{2} \neq I_{3}$. The top is pinned at point $P$, which is at a distance $l$ from the centre of mass, $C$, as shown in the figure.

Its angular velocity in a body frame $\left(\mathbf{e}_{\mathbf{1}}, \mathbf{e}_{\mathbf{2}}, \mathbf{e}_{\mathbf{3}}\right)$ is given by

$$\boldsymbol{\omega}=[\dot{\phi} \sin \theta \sin \psi+\dot{\theta} \cos \psi] \mathbf{e}_{1}+[\dot{\phi} \sin \theta \cos \psi-\dot{\theta} \sin \psi] \mathbf{e}_{2}+[\dot{\psi}+\dot{\phi} \cos \theta] \mathbf{e}_{3}$$

where $\phi, \theta$ and $\psi$ are the Euler angles.

(a) Assuming that $\left\{\mathbf{e}_{a}\right\}, a=1,2,3$, are chosen to be the principal axes, write down the Lagrangian of the top in terms of $\omega_{a}$ and the principal moments of inertia. Hence find the Lagrangian in terms of the Euler angles.

(b) Find all conserved quantities. Show that $\omega_{3}$, the spin of the top, is constant.

(c) By eliminating $\dot{\phi}$ and $\dot{\psi}$, derive a second-order differential equation for $\theta$.