Why the intermediate axis is unstable

Order three distinct positive principal moments as I₁ < I₂ < I₃, all about the center of mass of a freely moving rigid body. Exact rotation with body angular velocity (0, Ω, 0) solves the torque-free Euler equations. Introduce small body components u along axis one and v along axis three. The change in the second component is second order because its Euler equation contains the product uv. At linear order, hold that component equal to Ω while solving for the two transverse perturbations.

The resulting first-order equations couple u and v with coefficients of the same sign when Ω is positive. Differentiate once to eliminate v. The coefficient multiplying u in its second derivative is positive, so the solutions include an exponentially increasing term. A negative coefficient would instead give an oscillation. Notice that this is a statement about body components near a particular steady solution; it does not say that the magnitude of the full angular momentum grows, or that the rigid body gains kinetic energy.

The positive growth rate depends on the gaps between the middle moment and the two extreme moments. Doubling Ω halves the local growth time. Making either gap small weakens this particular linear instability. If two principal moments coincide, this distinct-axis calculation reaches a degenerate limit and cannot be used to claim exponential growth with a finite rate. Near the smallest or largest principal axis, repeating the same calculation gives a negative second-derivative coefficient and bounded linear oscillations of the transverse components.

Linear stability has a local meaning. The approximation predicts an exponential departure only while u and v remain small compared with Ω. Once the body tumbles substantially, neglected terms matter and the full Euler equations must be used. The finite body cannot sustain unlimited exponential growth of its angular speed with fixed energy. In the ideal conservative problem, nearby extreme-axis motions remain close in the appropriate stability sense, but they do not relax back to the exact axis because there is no damping.

Express the rotational energy in terms of body angular-momentum components and restrict attention to the sphere of fixed angular-momentum magnitude. Rotation about the largest moment minimizes that energy on the sphere, rotation about the smallest moment maximizes it, and rotation about the middle moment is a saddle. Nearby intersections of the energy ellipsoid with the momentum sphere therefore have different geometry near the three axes. Conservation restricts the trajectory but does not require every allowed steady motion to be stable.

Use the quadratic-function lab as a local-curvature illustration. Predict that changing coefficient a from positive to negative reverses the sign of the tangent slope at the same positive position x, while the stationary point stays at zero. Check that prediction using the displayed slope 2ax. This single parabola illustrates a curvature sign, not both directions of a saddle and not the Euler equations. No control in this lab specifies principal moments or predicts a three-dimensional flip; the growth rate must come from the mechanical calculation above.