Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 15A / b / Solution

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.