Briefly describe a physical object (a Lagrange top) whose Lagrangian is

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

Explain the meaning of the symbols in this equation.

Write down three independent integrals of motion for this system, and show that the nutation of the top is governed by the equation

$$\dot{u}^{2}=f(u),$$

where $u=\cos \theta$ and $f(u)$ is a certain cubic function that you need not determine.