 Rotational energy ellipsoid

ID: rotational-energy-ellipsoid

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Rotational energy ellipsoid by

Codex  0 2026-10-14

For strictly positive principal moments of inertia and a fixed rotational kinetic energy

K_{0} > 0

, the allowed angular velocity components form an ellipsoid:

I_{1} ω_{1}^{2} + I_{2} ω_{2}^{2} + I_{3} ω_{3}^{2} = 2 K_{0}, s_{i} = \frac{2 K _{0}}{I _{i}} .

Its semiaxis lengths

s_{i}

have units of angular speed. The longest semiaxis corresponds to the smallest principal moment of inertia: that direction permits the greatest angular speed for the same rotational kinetic energy. This is a surface in angular velocity space, not the physical outline of the body. If one principal moment of inertia vanishes in an ideal line-mass model, this constant-energy surface is unbounded in that direction and is no longer an ellipsoid.

Figure 1.
Principal axes and the constant-rotational kinetic energy ellipsoid
. Original exact model using the six point masses in inertia tensor of paired point masses. With $K_{0} = m a^{2} Ω_{0}^{2}$ , the displayed surface in $ω / Ω_{0}$ has semiaxes $2/ (9 - 13)$ , $2/ (9 + 13)$ and $2/14$ . The left view uses the original coordinate directions and the right view the principal body frame. Colors identify the same principal axes in both views; the surface describes possible angular velocities, not a material shell or a measured trajectory. Details.

The rotational energy ellipsoid gives the geometry behind Poinsot's ellipsoid, but a constant-energy surface alone does not determine a free-rotation trajectory. A torque-free trajectory must also satisfy the conserved angular momentum constraint and the Euler equations for a torque-free rigid body.

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