Consider a heavy symmetric top of mass $M$, pinned at point $P$, which is a distance $l$ from the centre of mass.

(a) Working in the body frame $\left(\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right)$ (where $\mathbf{e}_{3}$ is the symmetry axis of the top) define the Euler angles $(\psi, \theta, \phi)$ and show that the components of the angular velocity can be expressed in terms of the Euler angles as

$$\boldsymbol{\omega}=(\dot{\phi} \sin \theta \sin \psi+\dot{\theta} \cos \psi, \dot{\phi} \sin \theta \cos \psi-\dot{\theta} \sin \psi, \dot{\psi}+\dot{\phi} \cos \theta)$$

(b) Write down the Lagrangian of the top in terms of the Euler angles and the principal moments of inertia $I_{1}, I_{3}$.

(c) Find the three constants of motion.