Reaction Forces of a Singular Pendulum

In this paper, various types of behavior of reaction forces and Lagrange multipliers for the motion of mechanical systems with a configuration space singularity are studied. The motion of a one-dimensional double pendulum (or a singular pendulum) with a transversal singular point or a first-order tangency singular point is considered. Depending on the properties of the curve along which the free vertex of the double pendulum moves, the configuration space of the mechanical system is two smooth curves on the torus without common points, two transversely intersecting smooth curves, or two curves with first-order tangency. The reaction forces on a two-dimensional torus are found to study the pendulum motion. The expressions for the reaction forces in angular coordinates are obtained analytically. In the case of a transverse intersection, it is proven that the reaction forces must be zero at the singular point. In the case of a first-order tangency singularity, the reaction forces are nonzero at the singular point. The Lagrange multiplier, which depends on the motion along the ellipse, becomes unlimited near the singular point. Two mechanisms with a different type of singular points in the configuration space are described: a nonsmooth singular pendulum and a broken singular pendulum. There are no smooth regular curves passing through a singular point in the configuration spaces of these mechanical systems. For a nonsmooth singular pendulum, the Lagrange multiplier, which depends on the motion along the ellipse, becomes undefined while passing the singular point. For a broken singular pendulum, the Lagrange multiplier makes a jump from a finite value to an infinite one.