Intermediate axis theorem

For $I_1<I_2<I_3$, steady torque-free rotation about the first or third principal axis is stable, while rotation about the intermediate principal axis is unstable.

Linearize the Euler equations for a torque-free rigid body about rotation with angular speed $\Omega$ along the third principal axis. The transverse perturbations obeyand henceCyclic permutation gives the corresponding equations about the other two axes. For $I_1<I_2<I_3$, the squared frequency is positive about axes one and three, whereas the intermediate-axis equation has a positive exponential-growth coefficient.

On the invariant level $L^2=2E{I}_2$, the torque-free rigid-body orbit is a separatrix joining the two intermediate-axis rotations. Withone orientation has ${\omega }_2(t)=\mu \tanh (\lambda (t-t_0))$; the other components are constant multiples of $\mathrm{sech}(\lambda (t-t_0))$.