Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 14B (Classical Dynamics)

In the Euler angles for a symmetric top convention, $\theta$ is the inclination of the body symmetry axis ${\mathbf{e}}_3$ from the fixed vertical, $\varphi$ is the azimuth of the line of nodes in the fixed horizontal plane, and $\psi$ is rotation about ${\mathbf{e}}_3$ measured from that line:

The coordinates $\varphi$ and $\psi$ are cyclic, so their canonical momenta are conserved:Because the Lagrangian has no explicit time dependence, the conserved energy isThese are the three integrals of motion requested.

Solving the two momentum equations givesSubstitution into the conserved energy reduces the heavy symmetric top to one-dimensional effective potential motion:whereThis is the heavy symmetric top reduction.

Put $L_z=L_3(1-{ϵ}^2/2)$ and seek $\theta =O(ϵ)$. Using $\cos \theta =1-{\theta }^2/2+O({\theta }^4)$ and ${\sin }^2\theta ={\theta }^2+O({\theta }^4)$, the $\theta$-dependent part of the effective potential isIt has a local minimum precisely whenwhich is the gyroscopic stabilization of an inverted symmetric top condition. Differentiating the displayed approximation givesand henceTaking $L_3>0$ without loss of orientation, the steady precession rate isso