The following equations model the motion of a rigid body about its centre of mass, where $\vec{\Omega}$ is the angular velocity vector in the body and $I_1,I_2,I_3$ are the principal moments of inertia:
$$
\begin{align*}
I_1\dot{\Omega}_1&=(I_2-I_3)\Omega_2\Omega_3\\
I_2\dot{\Omega}_2&=(I_3-I_1)\Omega_3\Omega_1\\
I_3\dot{\Omega}_3&=(I_1-I_2)\Omega_1\Omega_2.
\end{align*}
$$
Can this be rescaled to the system
$$
\begin{align*}
\dot{u}_1&=u_2u_3\\
\dot{u}_2&=u_3u_1\\
\dot{u}_3&=u_1u_2?
\end{align*}
$$
Is this straightforward (and am I just too stupid)?
My first idea was to rescale time by $t=\lambda\tau$ but this seems not to work.
Best Answer
If you set $Ω_1(t)=c_2c_3u_1(t)$, etc., then you get $$ I_1=(I_2-I_3)c_1^2\\ I_2=(I_3-I_1)c_2^2\\ I_3=(I_1-I_2)c_3^2\\ $$ Now the best one can do is to set $$ c_1=\sqrt{\frac{I_1}{|I_2-I_3|}} $$ etc., so that the reduced equations retain the sign of the difference factor on the right.
This obviously becomes singular when the principal moments are close together or identical.