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Coulomb control of polygonal linkages

Giorgi Khimshiashvili, Gaiane Panina, Dirk Siersma

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.

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.

Paper Structure

This paper contains 4 sections, 7 theorems, 26 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Coulomb control problem of convex quadrilaterals
  3. Coulomb control of convex equilateral pentagons
  4. Concluding remarks

Key Result

Lemma 1

For a given convex quadrilateral $P \in M(L)$, there exists a unique $t$ such that $P$ is a critical point of $E$ on $M(L)$. In this case, $t$ is positive.

Figures (7)

  • Figure 1: Diagonal relation in $x,y$ (left) and in $x^2,y^2$ (right)
  • Figure 2: The configuration space $M(L)$ is divided into four parts
  • Figure 3: Critical configurations from Example \ref{['ExManyCrit']}
  • Figure 4:
  • Figure 5:
  • ...and 2 more figures

Theorems & Definitions (13)

  • Remark 1
  • Lemma 1
  • Proposition 1
  • Theorem 1
  • Lemma 2
  • Example 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 3
  • ...and 3 more