Consider a rigid body with angular velocity $\boldsymbol{\omega}$, angular momentum $\mathbf{L}$ and position vector $\mathbf{r}$, in its body frame.

(a) Use the expression for the kinetic energy of the body,

$$\frac{1}{2} \int d^{3} \mathbf{r} \rho(\mathbf{r}) \dot{\mathbf{r}}^{2},$$

to derive an expression for the tensor of inertia of the body, I. Write down the relationship between $\mathbf{L}, \mathbf{I}$ and $\boldsymbol{\omega}$.

(b) Euler's equations of torque-free motion of a rigid body are

$$\begin{aligned} &I_{1} \dot{\omega}_{1}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3} \\ &I_{2} \dot{\omega}_{2}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1} \\ &I_{3} \dot{\omega}_{3}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2} \end{aligned}$$

Working in the frame of the principal axes of inertia, use Euler's equations to show that the energy $E$ and the squared angular momentum $\mathbf{L}^{2}$ are conserved.

(c) Consider a cuboid with sides $a, b$ and $c$, and with mass $M$ distributed uniformly.

(i) Use the expression for the tensor of inertia derived in (a) to calculate the principal moments of inertia of the body.

(ii) Assume $b=2 a$ and $c=4 a$, and suppose that the initial conditions are such that

$$\mathbf{L}^{2}=2 I_{2} E$$

with the initial angular velocity $\omega$ perpendicular to the intermediate principal axis $\mathbf{e}_{2}$. Derive the first order differential equation for $\omega_{2}$ in terms of $E, M$ and $a$ and hence determine the long-term behaviour of $\boldsymbol{\omega}$.