Hilbert's "monkey saddle" and other curiosities in the equilibrium problem of three point particles on a circle for repulsive power law forces

This article determines all possible equilibrium arrangements (proper as well as pseudo) under repulsive power law forces of three point particles on the unit circle. These are the critical points of the sum over the three (standardized) Riesz pair interaction terms, each given by $V_s®= s^{-1}\left(r^{-s}-1 \right)$whentherealparameter$s \neq 0$,andby$V_0® := \lim_{s\to0}V_s® = -\ln r$;here,$r$ is the chordal distance between the particles in the pair. The bifurcation diagram which exhibits all these equilibrium arrangements together as functions of $s$ features three obvious “universal” equilibria, which do not depend on $s$, and two not-so-obvious continuous families of $s$-dependent non-universal isosceles triangular equilibria. The two continuous families of non-universal equilibria are disconnected, yet they bifurcate off of a common universal limiting equilibrium (the equilateral triangular configuration), at $s=-4$, where the graph of the total Riesz energy of the 3-particle configurations has the shape of a “monkey saddle.” In addition, one of the families of non-universal equilibria also bifurcates off of another universal equilibrium (the antipodal arrangement), at $s=-2$. While the bifurcation at $s=-4$ is analytical, the one at $s=-2$ is not. The bifurcation analysis presented here is intended to serve as template for the treatment of similar N-point equilibrium problems on the d-dimensional sphere for small N.