(a) Explain what is meant by a Lagrange top. You may assume that such a top has the Lagrangian

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

in terms of the Euler angles $(\theta, \phi, \psi)$. State the meaning of the quantities $I_{1}, I_{3}, M$ and $l$ appearing in this expression.

Explain why the quantity

$$p_{\psi}=\frac{\partial L}{\partial \dot{\psi}}$$

is conserved, and give two other independent integrals of motion.

Show that steady precession, with a constant value of $\theta \in\left(0, \frac{\pi}{2}\right)$, is possible if

$$p_{\psi}^{2} \geqslant 4 M g l I_{1} \cos \theta .$$

(b) A rigid body of mass $M$ is of uniform density and its surface is defined by

$$x_{1}^{2}+x_{2}^{2}=x_{3}^{2}-\frac{x_{3}^{3}}{h}$$

where $h$ is a positive constant and $\left(x_{1}, x_{2}, x_{3}\right)$ are Cartesian coordinates in the body frame.

Calculate the values of $I_{1}, I_{3}$ and $l$ for this symmetric top, when it rotates about the sharp point at the origin of this coordinate system.