Angular Kinetic Energy Constraint
In the absence of applied torques, the angular kinetic energy is conserved so .
The angular kinetic energy may be expressed in terms of the moment of inertia tensor and the angular velocity vector
where are the components of the angular velocity vector along the principal axes, and the are the principal moments of inertia. Thus, the conservation of kinetic energy imposes a constraint on the three-dimensional angular velocity vector ; in the principal axis frame, it must lie on an ellipsoid, called inertia ellipsoid.
The ellipsoid axes values are the half of the principal moments of inertia. The path traced out on this ellipsoid by the angular velocity vector is called the polhode (coined by Poinsot from Greek roots for "pole path") and is generally circular or taco-shaped.
Read more about this topic: Poinsot's Ellipsoid
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