The components of the angular velocity $\omega$ of a rigid body and of the position vector $\mathbf{r}$ are given in a body frame.

(a) The kinetic energy of the rigid body is defined as

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

Given that the centre of mass is at rest, show that $T$ can be written in the form

$$T=\frac{1}{2} I_{a b} \omega_{a} \omega_{b},$$

where the explicit form of the tensor $I_{a b}$ should be determined.

(b) Explain what is meant by the principal moments of inertia.

(c) Consider a rigid body with principal moments of inertia $I_{1}, I_{2}$ and $I_{3}$, which are all unequal. Derive Euler's equations of torque-free motion

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

(d) The body rotates about the principal axis with moment of inertia $I_{1}$. Derive the condition for stable rotation.