Polhode

Remarks

Let $T = \tfrac12 \omega^T I \omega$ and $\abs{L}^2 = \abs{I\omega}^2$, both constant along the motion. The polhode lies on the intersection of Poinsot's ellipsoid $\set{x^T I x = 2T}$ with the ellipsoid $\set{\abs{Ix}^2 = \abs{L}^2}$. In principal axes with $I_1 < I_2 < I_3$, these are the curves $\sum_i I_i x_i^2 = 2T$, $\sum_i I_i^2 x_i^2 = \abs{L}^2$. They are closed curves around the $e_1$-axis or the $e_3$-axis, separated by the curves through the points $\pm\sqrt{2T/I_2}\,e_2$ of the middle axis. The herpolhode lies in the invariable plane, perpendicular to the angular momentum, on which Poinsot's ellipsoid rolls.

Example: for a symmetric body with $I_1 = I_2$, the polhodes are circles $\omega_1^2 + \omega_2^2 = \text{const}$, $\omega_3 = \text{const}$ around the symmetry axis.

Non-example: if a torque acts, $T$ or $\abs{L}$ changes in general. Then the curve of $\omega$ need not lie on a fixed ellipsoid, and it is not a polhode in the sense above.