Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 8B / c / Solution

Fix the two conserved momenta $p_{\psi }=\lambda$ and $p_{\varphi }=\mu$. The remaining canonical variables $(\theta ,p_{\theta })$ form a two-dimensional phase space with Hamiltonianwhere

An upright spinning solution has $\theta =0$, $p_{\theta }=0$, and necessarily $\mu =\lambda$, since the vertical and body-axis angular momenta then coincide. On that momentum level,Near zero,Thus the upright configuration is a strict local minimum of the reduced Hamiltonian, and hence stable, whenHere $\lambda$ is the angular momentum about the symmetry axis. This proves stability for sufficiently large axial angular momentum.