# A subtlety about Lyapunov stability of stationary rotations of rigid body

On Page 145 of Arnold's mechanics book there is the intermediate axis theorem:

"The stationary solutions of the Euler equations corresponding to the largest and smallest principal axes [of the inertia ellipsoid for a rigid body] are stable while the solution corresponding to the middle axis is unstable."

Right afterwards there is a problem:

"Problem: Are stationary rotations of the body around the largest and smallest principal axes Lyapunov stable? Answer: No."

How is this not a contradiction?

I suppose one could describe a body using a reference frame whose origin was away from the body's center of mass. Then obviously rotation of such a body about the inertia ellipsoid's axes might not be a stable rotation. But such a reference frame seems out of context for this discussion.

• Oh, it's the angular momentum trajectories that are stable or unstable. The rotations of the body themselves over time are not stable. – enthdegree Feb 10 at 1:57

1. The major axis $$\vec{\Omega}\approx (\Omega_1,0,0)$$ is stable because $$\ddot{\Omega}_i~\approx~-\omega^2_1 \Omega_i, \qquad i\in{2,3}, \tag{1a}$$ where $$\omega_1~:=~\Omega_1\sqrt{\frac{(I_1-I_2)(I_1-I_3)}{I_2I_3}}. \tag{1b}$$
2. The intermediate axis $$\vec{\Omega}\approx (0,\Omega_2,0)$$ is $$\color{red}{\text{unstable}}$$ because $$\ddot{\Omega}_i~\approx~\color{red}{+}\omega^2_2 \Omega_i, \qquad i\in{1,3}, \tag{2a}$$ where $$\omega_2~:=~\Omega_2\sqrt{\frac{(I_1-I_2)(I_2-I_3)}{I_1I_3}}. \tag{2b}$$
3. The minor axis $$\vec{\Omega}\approx (0,0,\Omega_3)$$ is stable because $$\ddot{\Omega}_i~\approx~-\omega^2_1 \Omega_i, \qquad i\in{1,2}, \tag{3a}$$ where $$\omega_3~:=~\Omega_3\sqrt{\frac{(I_1-I_3)(I_2-I_3)}{I_1I_2}}. \tag{3b}$$
For each of the stable axes, we see that 2 small perpendicular angular velocity components $$\Omega_i$$ are 2 independent harmonic oscillators. If they at some instant happens to vanish, there is no guarantee that they will stay that way in the future. In particular, they are not Lyapunov stable.