Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 2 / 9C / Solution

The components in Euler equations for a torque-free rigid body are evaluated in the body-fixed principal-axis frame, with $L_i=I_i{\omega }_i$. In an inertial space frame the angular momentum vector is constant; its components in the rotating body frame need not be.

Assume $\Omega \ne 0$ and distinct positive principal moments of inertia. Write $\omega =(\Omega +{\delta }_1,{\delta }_2,{\delta }_3)$. To first order,soThe coefficient is positive when $I_1$ lies between the other moments, giving an exponentially growing mode and instability. When $I_1$ is an extreme moment, it is negative and the transverse perturbations oscillate.

To establish actual nonlinear Lyapunov stability in the latter case, use the conserved kinetic energy $E=\frac{1}{2}\sum I_j{\omega }_j^2$ and squared angular momentum $L^2=\sum I_j^2{\omega }_j^2$. Their combination isFor an extreme $I_1$ its two coefficients have the same sign. Conservation controls both transverse components by their initially small values, while conservation of $E$ controls ${\omega }_1^2$ near ${\Omega }^2$. Continuity preserves its initial sign for sufficiently small perturbations. Thus nonzero steady rotation is stable exactly about the largest or smallest principal moment, the intermediate axis theorem. It is not asymptotically stable, since energy is conserved. If $\Omega =0$, energy makes the zero angular-velocity state stable for every ordering; the nonzero-rotation assumption is essential to the stated equivalence.