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

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

(i) Starting with the fixed space frame $\left(\tilde{\mathbf{e}}_{\mathbf{1}}, \tilde{\mathbf{e}}_{2}, \tilde{\mathbf{e}}_{3}\right)$ and choosing $O$ at its origin, sketch the top with embedded body frame axis $\mathbf{e}_{3}$ being the symmetry axis. Clearly identify the Euler angles $(\theta, \phi, \psi)$.

(ii) Obtain the momenta $p_{\theta}, p_{\phi}$ and $p_{\psi}$ and the Hamiltonian $H\left(\theta, \phi, \psi, p_{\theta}, p_{\phi}, p_{\psi}\right)$. Derive Hamilton's equations. Identify the three conserved quantities.