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Torque-free rigid-body invariants

The Euler equations for a torque-free rigid body conserve rotational kinetic energy and squared angular momentum,

E = \frac{1}{2} \sum_{i = 1}^{3} I_{i} ω_{i}^{2}, L^{2} = \sum_{i = 1}^{3} I_{i}^{2} ω_{i}^{2} .

Their common level sets are the intersection of two ellipsoids in angular-velocity space.

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