Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 15A (Classical Dynamics)

Euler's equations are $I_1{\dot{\omega }}_1=(I_2-I_3){\omega }_2{\omega }_3$ and cyclic permutations. Setting two components to zero gives uniform rotation about any principal axis.

Multiplying Euler's equations by ${\omega }_i$ and by $I_i{\omega }_i$ respectively shows $T=\frac{1}{2}\sum I_i{\omega }_i^2$ and $L^2=\sum I_i^2{\omega }_i^2$ are constant. Their ellipsoids are tangent when their normals are parallel, which occurs on the principal axes.

Eliminating ${\omega }_2^2$ between $2T$ and $L^2$ givesThe intersections are closed near axes 1 and 3 but hyperbolic near axis 2. Thus rotations about the smallest and largest inertia axes are stable, while the intermediate-axis rotation is unstable.

Linearization about $(0,\Omega ,0)$ gives $\dot{u}=-Av$, $\dot{v}=-Bu$, where $A=(I_3-I_2)\Omega /I_1$, $B=(I_2-I_1)\Omega /I_3$. With $\kappa =\sqrt{AB}$,The exponential growth confirms intermediate-axis instability.