= Linear stability of principal-axis rotation
For $0<I_1<I_2<I_3$, the transverse <linear stability> rates about steady rotation with signed component $\Omega\ne0$ are
$$
\nu_1=|\Omega|\sqrt{\frac{(I_2-I_1)(I_3-I_1)}{I_2I_3}},\qquad
\nu_3=|\Omega|\sqrt{\frac{(I_3-I_2)(I_3-I_1)}{I_1I_2}},\qquad
\lambda_2=|\Omega|\sqrt{\frac{(I_2-I_1)(I_3-I_2)}{I_1I_3}}.
$$
Axes one and three have transverse <eigenvalues> $\pm i\nu_1$ and $\pm i\nu_3$, producing bounded <oscillations>. Axis two has transverse <eigenvalues> $\pm\lambda_2$, giving growing and decaying <exponential functions>. A neutral longitudinal <perturbation> corresponds to a small change of $\Omega$. Generic intermediate-axis disturbances contain a growing component; a precisely chosen decaying direction is exceptional. For the <normalized intermediate-axis model> with $|\Omega|=1$, the values are $\nu_1=1/\sqrt3$, $\nu_3=1$ and $\lambda_2=1/\sqrt3$.
Proof outline: the <linearization> of the <Euler equations for a torque-free rigid body> about the third <principal axis> gives transverse <perturbations> obeying
$$
\dot\omega_1=\frac{I_2-I_3}{I_1}\Omega\omega_2,\qquad
\dot\omega_2=\frac{I_3-I_1}{I_2}\Omega\omega_1,
$$
and hence
$$
\ddot\omega_1=-\Omega^2\frac{(I_3-I_2)(I_3-I_1)}{I_1I_2}\omega_1.
$$
Cyclic permutation gives the other rates. For nonlinear <Lyapunov stability> of the extreme rotations, use the <torque-free rigid-body invariants>: the conserved <quadratic forms> $L^2-2I_1E$ and $2I_3E-L^2$ have strictly positive transverse coefficients about axes one and three respectively. They bound transverse <angular velocity>, while $E$ bounds its longitudinal change near nonzero rotation. Imaginary transverse <eigenvalues> alone would not establish this nonlinear conclusion.<Classical Dynamics by David Tong/3. The Motion of Rigid Bodies>[Classical Dynamics by David Tong, §3.4.3, equations (3.41)–(3.45), and §3.4.4, invariants (3.46)–(3.47); the invariant-bound argument here is an original short derivation]{ref}
The practical lesson is to choose a stable <principal axis> when using rotation to maintain an instrument's pointing direction. Both extreme <principal moments of inertia> are stable in the ideal torque-free model; if internal <damping> lowers <rotational kinetic energy> while approximately preserving <angular momentum>, the minimum energy $E=L^2/(2I_3)$ instead favors the largest <principal moment of inertia>. A control system must account for <torque>, changing <mass> distributions and flexibility before applying this ideal result.
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