Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 15E (Classical Dynamics)

The kinetic energy and squared angular-momentum magnitude areDifferentiate and substitute Euler's equations. In $\dot{T}$, the coefficient of ${\omega }_1{\omega }_2{\omega }_3$ isThe analogous weighted sum in $d{L}^2/dt$ also cancels, so both quantities are conserved.

The initial relation implies $L^2=2I_{2}T$, and conservation preserves it. With $I_3=I_1+I_2$, this identity reduces toThe energy then givesThe first Euler equation is ${\dot{\omega }}_1=-{\omega }_2{\omega }_3$. Squaring and substituting the preceding identities yields

SetThen ${\omega }_1=A\mathrm{sech}(\Omega t)$ satisfies the squared equation and the stated initial condition after choosing the origin of time at the maximum of ${\omega }_1$.

With compatible signs,Thus as $t\to \infty$, ${\omega }_1$ and ${\omega }_3$ vanish while ${\omega }_2\to A$. The body approaches steady rotation about the intermediate principal axis; the full separatrix connects the two opposite intermediate-axis rotations.