This article determines all possible (proper as well as pseudo) equilibrium arrangements under a repulsive power law force 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 \(s \ne 0\), and by \(V_0(r) := \lim _{s\rightarrow 0}V_s(r) = -\ln r\); 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. The bifurcation analysis presented here is intended to serve as template for the treatment of similar N-point equilibrium problems on \({\mathbb {S}}^d\) for small N.

中文翻译：

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希尔伯特的“猴子鞍”和其他好奇心在排斥力法力圆周上的三点质点平衡问题中

本文在单位圆上的三点粒子的排斥力定律力的作用下，确定了所有可能的（正确的和伪的）平衡布置。这些是三个（标准化）Riesz对交互项上总和的临界点，每个项由\（V_s（r）= s ^ {-1} \ left（r ^ {-s} -1 \ right）\ ）当实数参数\（s \ ne 0 \）并由\（V_0（r）：= \ lim _ {s \ rightarrow 0} V_s（r）=-\ ln r \）时；此处，r是该对粒子之间的弦距。分叉图将所有这些平衡布置展示为s的函数，具有三个明显的“普遍”均衡，它们不依赖于s，以及两个不太明显的s依赖非等腰等腰三角形平衡的连续族。非通用平衡的两个连续族断开，但它们在\（s = -4 \）处的共同通用极限平衡（等边三角形构型）处分叉，其中3的总Riesz能量图-粒子配置具有“猴子鞍”的形状。另外，非普遍平衡的一个族也从\（s = -2 \）的另一个普遍平衡（对映排列）分叉。\（s = -4 \）处的分叉是解析的，而\（s = -2 \）处的分叉是解析的不是。这里提出的分岔分析的目的是作为模板的相似治疗Ñ -点均衡问题上\（{\ mathbb {S}} ^ d \）为小 Ñ。