= Intermediate axis theorem
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For a <rigid body> with three distinct positive <principal moments of inertia> $0<I_1<I_2<I_3$ and zero external <torque> about its <centre of mass>, steady rotation about the first or third <principal axis> is <Lyapunov stable> in <angular velocity>, while rotation about the intermediate <principal axis> is unstable. Here steady rotation means $\boldsymbol\omega=\Omega\hat e_i$ in the <principal body frame>, with constant $\Omega\ne0$. A small sideways <perturbation> remains small about the extreme <principal moments of inertia>, but generically grows about the intermediate one. Stable does not mean that the wobble decays: without <damping>, there is no <asymptotic stability>. Coincident <principal moments of inertia> and a stationary body are excluded.<Classical Dynamics by David Tong/3. The Motion of Rigid Bodies>[Classical Dynamics by David Tong, §3.4.3, The Asymmetric Top: Stability, printed pp. 57–58, equations (3.40)–(3.45)]{ref}
No external <torque> is needed for a body to tumble. <Conservation of angular momentum> fixes $\mathbf L$ in the <fixed space frame>, but the body can turn relative to $\mathbf L$, so its <angular velocity> components in the <principal body frame> change. The <torque-free rigid-body invariants> constrain this motion. A supported <gyroscope> with gravitational <torque> has different governing assumptions.
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