Euler equations for a torque-free rigid body

In principal-axis body coordinates, torque-free angular velocity satisfies

The Euler equations for a torque-free rigid body conserve rotational kinetic energy and squared angular momentum,Their common level sets are the intersection of two ellipsoids in angular-velocity space.

Eliminating ${\omega }_1^2$ and ${\omega }_2^2$ between the conserved energy and squared angular momentum givesThe right-hand side is a quartic polynomial in ${\omega }_3$; its real turning points delimit the nutation of the body-frame angular velocity.

For principal moments $I_1=I_2=I$ and symmetry-axis unit vector $e_3$, angular momentum and angular velocity obeyThus $L$, $\omega$, and the symmetry axis are coplanar at every instant.

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 $\mathrm{sech}(\lambda (t-t_0))$.