Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 14B (Classical Dynamics)

Scale the unit ball by $(u,v,w)\mapsto (au,bv,cw)$. Symmetry makes all products of inertia vanish, while the unit-ball averages satisfy
$⟨u^2⟩=⟨v^2⟩=⟨w^2⟩=1/5$. The inertia tensor about the centre is therefore

For torque-free motion, the inertial angular momentum $\mathbf{L}=I\mathbf{\omega }$ and energy
$T=\frac{1}{2}\mathbf{\omega }\cdot I\mathbf{\omega }$ are constant. Consider the energy ellipsoidwhich is fixed in the body. At its point $\mathbf{x}=\mathbf{\omega }$, its normal is
$I\mathbf{\omega }=\mathbf{L}$, and its tangent plane isThis plane has fixed normal and fixed distance $2T/|\mathbf{L}|$ from the origin, so it is the invariable plane. The material velocity of the contact vector is
$\mathbf{\omega }\times \mathbf{\omega }=0$. Thus the contact point is instantaneously at rest and the ellipsoid rolls without slipping on the plane. This is the Poinsot construction.

When $a=b<c$, the ellipsoid is an axisymmetric rigid body withThe angle between its symmetry axis and the fixed angular momentum is constant. In an inertial frame the symmetry axis precesses uniformly around $\mathbf{L}$, while the body simultaneously spins about that axis. The angular-velocity vector also traces a circular cone about $\mathbf{L}$. The limiting cases are pure rotation about the symmetry axis and pure rotation about a perpendicular principal axis.