Critical Configurations of planar Multiple Penduli

Abstract

We consider the oriented area function A on the moduli space M(P) of mechanical linkage P representing a planar multiple pendulum. For generic lengths of the sides of P, it is proved that A is a Morse function on M(P) and its critical points are given by the cyclic configurations of P satisfying an additional geometric condition. For triple penduli, the main result is complemented by a rather comprehensive analysis of the structure of critical configurations. Moreover, we discuss the critical configurations of another natural function on the moduli space of a planar multiple pendulum and the image of the cross-ratio mapping into the plane. A number of related results and open problems are also presented.