We consider the equilibrium points of the electrostatic potential of three mutually repelling point charges with Coulomb interaction placed at the vertices of a given triangle T. It is proven that for each point P inside the triangle T, there exists a unique collection of positive point charges, called stationary charges for P in T, such that P is a critical point of the electrostatic potential of these point charges placed at vertices of T in a fixed order. Explicit formulas for stationary charges are given, which are used to investigate the existence and geometry of stable equilibria arising in this setting. In particular, symbolic computations and computer experiments reveal that for an isosceles triangle T, the set S(T) of points P that are stable equilibria of their stationary charges is a non-empty open set containing the incenter of a triangle T. For a regular triangle, using symbolic computations, it appears possible to verify that the formulas for stationary charges define a stable mapping in the sense of Whitney having a deltoid caustic with three ordinary cusps. An interpretation of our results in terms of electrostatic ion traps is also given, and several plausible conjectures are presented.

中文翻译：

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三角形和静电离子阱

我们考虑放置在给定三角形T的顶点处的具有库仑相互作用的三个相互排斥的点电荷的静电势的平衡点。事实证明，对每个点P的三角形内Ť，存在的正的点电荷的唯一集合，称为静止收费P在Ť，使得P是放置在顶点这些点电荷的静电电位的临界点吨以固定的顺序。给出了固定电荷的显式公式，用于研究在这种情况下出现的稳定平衡的存在性和几何形状。特别是，符号计算和计算机实验表明，对于等腰三角形T，其固定电荷的稳定平衡点P的集合S ( T )是包含三角形T 中心的非空开集. 对于正三角形，使用符号计算，似乎可以验证固定电荷的公式在惠特尼具有三个普通尖点的三角焦散的意义上定义了稳定映射。还给出了关于我们的静电离子阱结果的解释，并提出了一些合理的猜想。