Behaviour of the pendulum with a singular configuration space

Abstract

The flat double mathematical pendulum is considered, the loose end of which moves along an ellipse. In the general case, the configuration space of this pendulum is two disjoint curves. It is possible to choose parametres so that these curves intersect transversally. The observed trajectory of motion in this case forms an angle. Moreover, there are special parameters in which the curves have a first-order tangency. In this case, there is a geometric uncertainty how the pendulum have to move after passing a singular point. It is shown that for the transversal case the inverse dynamic problem is not solvable, and the Lagrange multipliers tend to infinity as they move to a singular point in the configuration space. The observed motion is dynamically determined. The pendulum always moves from one branch of movement to another during the passage of a singular point. A qualitative explanation of this effect is proposed. Refs 11. Figs 3.