Coulomb Control of Polygonal Linkages

Abstract

Equilibria of polygonal linkage with respect to Coulomb potential of point charges placed at the vertices of linkage are considered. It is proved that any convex configuration of a quadrilateral linkage is the point of global minimum of Coulomb potential for appropriate values of charges of vertices. Similar problems are treated for the equilateral pentagonal linkage. Some corollaries and applications in the spirit of control theory are also presented.

Point Charges and Polygonal Linkages

“…As a visual illustration of the paradigm we briefly recall the main results of [10]. For a quadrilateral linkage, we put charges equal to 1 at three vertices, whereas the fourth vertex is charged by t. Then we have…”

“…The "picture of what happens" which is behind the proofs of [10] is such. The configuration space of a quadrilateral is a circle, and therefore the relation between the diagonals is important since both diagonals appear in E. We remind that four points A 1 , ..., A 4 are coplanar whenever their Cayley-Menger determinant vanishes.…”

“…The rest of the proof from [10] uses the idea of convexity. Namely, we can consider E as a convex function defined on the (x, y)-plane.…”

“…The ultimate goal is to establish the possibility of controlling the shape of a pentagonal linkage by varying the values of two charges at the vertices. Following a paradigm developed in our previous paper [10] we consider Coulomb potential as a meromorphic function on the planar moduli space of linkage and investigate its critical points. The basic assumption and motivation for accepting such an approach is that a vertex-charged linkage subject only to Coulomb interaction of its charged vertices should take the shape with the minimal Coulomb potential.…”

Equilibria of three constrained point charges

“…We deal with equilibrium configurations of point charges with Coulomb interaction satisfying certain geometric constraints. Our approach to this topic is similar to the paradigms used in [4], [5]. Namely, we consider the Coulomb energy as a function on a certain configuration space naturally associated with the constraints in question, and investigate its critical points.…”