Hint 8

Taking ${\bf e}_3$ to be the principal axis normal to the disc, the principal moments can be expressed (using the perpendicular axes theorem) as

$$I_1=I_2=I,\qquad I_3=2I.$$

In this case Euler's equations simplify to

$$I\dot\omega_1+I\omega_2\omega_3 = -k\omega_1$$

$$I\dot\omega_2-I\omega_3\omega_1 = -k\omega_2,$$

$$\dot\omega_3 = 0$$

These equations are very similar to the ones for the freely rotating symmetric top. They can be solved using the same trick as in the case of free rotation. The complex ``coordinate" $\eta$ defined as in the previous hint satisfies the first-order differential equation

$$I\dot\eta-{\rm i}I\omega_3\eta+k\eta=0.$$

This has solutions of the form

$$\eta(t)=A\exp[-kt/I+{\rm i}(\omega_3 t+\phi)].$$