Past exam of the theoretical physics I course of the University of Cambridge / 2022 / 1 / d / Solution

The angular Euler-Lagrange equations areA static equilibrium point has both angular velocities zero and $\sin (\theta -\phi )=0$. Hence the complete equilibrium families, with angles taken modulo $2\pi$, areThe first family is aligned in azimuth and minimizes the spring length; the second is antipodal and maximizes it. Their potentials are $k[(a-A{)}^2+h^2]/2$ and $k[(a+A{)}^2+h^2]/2$, respectively. Common rotations generate a whole family rather than isolated equilibria.