A symmetric top of mass $M$ rotates about a fixed point that is a distance $l$ from the centre of mass along the axis of symmetry; its principal moments of inertia about the fixed point are $I_{1}=I_{2}$ and $I_{3}$. The Lagrangian of the top is

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

(i) Draw a diagram explaining the meaning of the Euler angles $\theta, \phi$ and $\psi$.

(ii) Derive expressions for the three integrals of motion $E, L_{3}$ and $L_{z}$.

(iii) Show that the nutational motion is governed by the equation

$$\frac{1}{2} I_{1} \dot{\theta}^{2}+V_{\text {eff }}(\theta)=E^{\prime}$$

and derive expressions for the effective potential $V_{\mathrm{eff}}(\theta)$ and the modified energy $E^{\prime}$ in terms of $E, L_{3}$ and $L_{z}$.

(iv) Suppose that

$$L_{z}=L_{3}\left(1-\frac{\epsilon^{2}}{2}\right)$$

where $\epsilon$ is a small positive number. By expanding $V_{\text {eff }}$ to second order in $\epsilon$ and $\theta$, show that there is a stable equilibrium solution with $\theta=O(\epsilon)$, provided that $L_{3}^{2}>4 M g l I_{1}$. Determine the equilibrium value of $\theta$ and the precession rate $\dot{\phi}$, to the same level of approximation.