Euler Equations & Rigid Lamina Moment of Inertia

[PeteSampras]

"A rigid lamina (i.e. a two dimensional object) has principal moments of inertia about the centre of mass given by $I_1=u^2-1$, $I_2=u^2+1$, $I_3=2u^2$

Choose the initial angular velocity to be $ω = µN \hat{e_1} + N \hat{e_2}$. Define tan α = ω2/ω1,
which is the angle the component of ω in the plane of the lamina makes with e1. Show that it satisfies:
$\ddot{α}+ N^2 \cos α \sin α = 0$"

(the problem does not says what is N). The problem is on http://www.damtp.cam.ac.uk/user/tong/dynamics/mf3.pdf

I tried used the Euler equation, considering that $N_1=N_2=0$

My Euler equation are:

$\dot{\omega_1}+ \omega_2 \omega_3=0$
$\dot{\omega_2}- \omega_1 \omega_3=0$
$2 \mu^2 \dot{\omega_3}+2 \omega_1 \omega_3 =N_3$

I tried of several ways of combining this equations , using that tan α = ω2/ω1, but i do not get to $\ddot{α} + N^2 \cos α \sin α = 0$.

¿some hint please?

Some Hint to do this exercice this?...

 

[Chestermiller]

I don't know whether this helps, but the first two equations imply that

x₂=C sinα

x₁=C cosα

where C is a constant.

Also,

x₃=α'

Chet