arXiv subjects

G. Khimshiashvili

Publications and source records attributed to G. Khimshiashvili.

2 recordsLinked to original sources

arXiv2016

Equilibria of three constrained point charges

We study the critical points of Coulomb energy considered as a function on configuration spaces associated with certain geometric constraints. Two settings of such kind are discussed in some detail. The first setting arises by considering polygons of fixed perimeter with freely sliding positively charged vertices. The second one is concerned with triples of positive charges constrained to three concentric circles. In each of these cases the Coulomb energy is generically a Morse function. We describe the minima and other stationary points of Coulomb energy and show that, for three charges, a pitchfork bifurcation takes place accompanied by an effect of the Euler's Buckling Beam type.

math.MG↗

arXiv2012

Critical configurations of planar robot arms

It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, $P$ can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive a formula for the Morse index of a critical configuration.

math.DG↗