The Lagrangian for a heavy symmetric top of mass $M$, pinned at a point that is a distance $l$ from the centre of mass, is

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

(a) Find all conserved quantities. In particular, show that $\omega_{3}$, the spin of the top, is constant.

(b) Show that $\theta$ obeys the equation of motion

$$I_{1} \ddot{\theta}=-\frac{d V_{\text {eff }}}{d \theta},$$

where the explicit form of $V_{\text {eff }}$ should be determined.

(c) Determine the condition for uniform precession with no nutation, that is $\dot{\theta}=0$ and $\dot{\phi}=$ const. For what values of $\omega_{3}$ does such uniform precession occur?