Consider a system of coordinates rotating with angular velocity $\boldsymbol{\omega}$ relative to an inertial coordinate system.

Show that if a vector $\mathbf{v}$ is changing at a rate $d \mathbf{v} / d t$ in the inertial system, then it is changing at a rate

$$\left.\frac{d \mathbf{v}}{d t}\right|_{\text {rot }}=\frac{d \mathbf{v}}{d t}-\boldsymbol{\omega} \wedge \mathbf{v}$$

with respect to the rotating system.

A solid body rotates with angular velocity $\omega$ in the absence of external torque. Consider the rotating coordinate system aligned with the principal axes of the body.

(a) Show that in this system the motion is described by the Euler equations

$\left.I_{1} \frac{d \omega_{1}}{d t}\right|_{\text {rot }}=\omega_{2} \omega_{3}\left(I_{2}-I_{3}\right) \quad,\left.\quad I_{2} \frac{d \omega_{2}}{d t}\right|_{\text {rot }}=\omega_{3} \omega_{1}\left(I_{3}-I_{1}\right) \quad,\left.\quad I_{3} \frac{d \omega_{3}}{d t}\right|_{\text {rot }}=\omega_{1} \omega_{2}\left(I_{1}-I_{2}\right)$, where $\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ are the components of the angular velocity in the rotating system and $I_{1,2,3}$ are the principal moments of inertia.

(b) Consider a body with three unequal moments of inertia, $I_{3}<I_{2}<I_{1}$. Show that rotation about the 1 and 3 axes is stable to small perturbations, but rotation about the 2 axis is unstable.

(c) Use the Euler equations to show that the kinetic energy, $T$, and the magnitude of the angular momentum, $L$, are constants of the motion. Show further that

$$2 T I_{3} \leq L^{2} \leq 2 T I_{1} \text {. }$$