Behavior of a Pendulum with a Singular Configuration Space

A 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 nonintersecting curves. It is possible to choose its parameters 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: along which branch will the pendulum move after passing the singular point? It is shown that for the transversal case the inverse dynamics 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 passage of the singular point. A qualitative explanation of this effect is proposed.