Define a body frame $\mathbf{e}_{a}(t), a=1,2,3$ of a rotating rigid body, and show that there exists a vector $\boldsymbol{\omega}=\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ such that

$$\dot{\mathbf{e}}_{a}=\boldsymbol{\omega} \times \mathbf{e}_{a}$$

Let $\mathbf{L}=I_{1} \omega_{1}(t) \mathbf{e}_{1}+I_{2} \omega_{2}(t) \mathbf{e}_{2}+I_{3} \omega_{3}(t) \mathbf{e}_{3}$ be the angular momentum of a free rigid body expressed in the body frame. Derive the Euler equations from the conservation of angular momentum.

Verify that the kinetic energy $E$, and the total angular momentum $L^{2}$ are conserved. Hence show that

$$\dot{\omega}_{3}^{2}=f\left(\omega_{3}\right),$$

where $f\left(\omega_{3}\right)$ is a quartic polynomial which should be explicitly determined in terms of $L^{2}$ and $E$.