Poinsot's Ellipsoid - Derivation of The Polhodes in The Body Frame

Derivation of The Polhodes in The Body Frame

In the principal axis frame (which is rotating in absolute space), the angular momentum vector is not conserved even in the absence of applied torques, but varies as described by Euler's equations. However, in the absence of applied torques, the magnitude of the angular momentum and the kinetic energy are both conserved

where the are the components of the angular momentum vector along the principal axes, and the are the principal moments of inertia.

These conservation laws are equivalent to two constraints to the three-dimensional angular momentum vector . The kinetic energy constrains to lie on an ellipsoid, whereas the angular momentum constraint constrains to lie on a sphere. These two surfaces intersect in taco-shaped curves that define the possible solutions for .

This construction differs from Poinsot's construction because it considers the angular momentum vector rather than the angular velocity vector . It appears to have been developed by Jacques Philippe Marie Binet.