On three point charges on a flexible arc

We consider equilibrium configurations of three mutually repelling point charges with the Coulomb interaction confined to a simple arc of a constant length with fixed positions of ends. For given values of the charges, the length of the arc, and the distance between its ends, we calculate all possible equilibrium configurations. We also study the behavior of equilibrium configurations for variable values of charges and show that a unique possible bifurcation is the pitchfork bifurcation. Similar results are presented for elastic loop obeying Hooke's law and for charges interacting via a Riesz potential.