Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 14B / b / Solution

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.