Torque-free rigid-body invariants

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.