Triangles and electrostatic ion traps

Abstract

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.

Hooke and Coulomn Energy of Tripod Spiders

“…Given a triangle with vertices A, B, C, referred to as a reference triangle (or base triangle), and a point X in the same plane, the triple of normalized stationary charges Q(X) = {q i (X; )} is defined as a triple of non-zero real numbers q i such that ∇E(X, Q) = 0. As was proven in [GK1] these charges are uniquely determined by the ratio's…”

“…From the viewpoint of control theory, it is also interesting and practically important to consider the Coulomb potential of point charges which are placed at the fixed foot points of a spider and can be varied in order to change the position of its body endowed with a fixed charge. A similar scenario was studied in big detail in the papers [GK1], [GK2] and our exposition in this section relies on the constructions and results given in those two papers.…”

“…This suggests that Coulomb potential is a reasonable candidate for developing Morse theory for a spider linkage along the same lines as in the case of Hooke's energy considered above. The aim of this section is to present a number of results in this direction and present an application in the spirit of robust Coulomb control discussed in [KPS], [GK1], [GK2]. Another relevant topic in our context is the so-called Maxwell's conjecture on point charges stating that the number of isolated equilibrium points of n point charges in 2-dimensional Euclidean space does not exceed (n − 1) 2 (see, e.g., a recent review in [GNS]).…”

“…In section 3 we study the Coulomb potential of point charges placed at the foots of a 3leg spider linkage. We deal with the so-called stationary charges of the spider's center (the common point of legs) and the so-called trapping domain of stationary charges determined in [GK1], [GK2]. This enables us to determine the domain in the workspace of spider, where the position of spider's center can be robustly controlled by the values of stationary charges using the Coulomb control scenario developed in [KPS], [GK1].…”

“…We deal with the so-called stationary charges of the spider's center (the common point of legs) and the so-called trapping domain of stationary charges determined in [GK1], [GK2]. This enables us to determine the domain in the workspace of spider, where the position of spider's center can be robustly controlled by the values of stationary charges using the Coulomb control scenario developed in [KPS], [GK1]. For a symmetric spider based on a regular triangle and having a contractible workspace, we show that the domain of robust Coulomb control is a non-void open subset of the workspace containing the center of the reference triangle.…”