Jacques Philippe Marie Binet showed that the angular momentum vector lies on the intersection of a sphere and an ellipsoid. The angular velocity along the principal axis of a freely rotating rigid body is described by Euler's equations from the point of view of an observer rotating with the body, a motion known as nutation (as in the case of a top). The angular velocity vector is a constant of motion viewed from the body system.

Euler's equations take the form of the autonomous system

,

,

.

References

[1] R. Kent Nagle, E. B. Saff, A. D. Snider, Fundamentals of Differential Equations and Boundary Value Problems, 6th ed., Boston: Pearson Addison-Wesley, 2012 pp. 275–276.