Hilbert's "monkey saddle" and other curiosities in the problem of the Riesz force equilibrium of three point particles on a circle

Michael K. -H. Kiessling, Renna Yi

Published 2017-12-14Version 1

This article determines all possible Riesz $s$-force equilibrium arrangements (proper as well as pseudo) 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 $V_s(r)= s^{-1}\left(r^{-s}-1 \right)$ when the real parameter $s \neq 0$, and by $V_0(r) := \lim_{s\to0}V_s(r) = -\ln r$; 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.