Reaction forces of singular pendulum

Abstract

Various behavior types of reaction forces and Lagrange multipliers for the case of mechanical systems with configuration space singularities are studied in this paper. 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. Properties of the configuration space of singular pendulum depends on the constraint line which the free vertex of the double pendulum moves along. Configuration space of singular pendulum could be represented by two smooth curves on a torus without common points, two transversely intersecting smooth curves or two curves with first-order tangency. In order to study the pendulum motion, the reaction forces on a two-dimensional torus are found. The expressions for the reaction forces are obtained analytically in angular coordinates. In the case of a transverse intersection it is proved 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 to 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. In the case of nonsmooth singular pendulum, the Lagrange multiplier which depends on the motion along the ellipse becomes undefined when the singular point is passed. In the case of broken singular pendulum, the Lagrange multiplier makes a jump from a finite value to an infinite one.