The pendulum as a dynamical billiard

Abstract

We consider a simple example of a dynamical billiard consisting of a mass point moving in a circle under the influence of a homogeneous gravitational field. The point reflects by the mirror elastic law when it encounters the circular boundary. The problem is integrable between one collision and another, and also when the particle moves on the bounding circle. This makes it possible to build the conditions of existence and stability (in a linear and, at times, in a nonlinear sense, too) of the families of basic periodic trajectories determining the phase space topology for a fixed energy level. The numerical implementation of the Poincaré mapping offers a means of describing the phase pictures with regular and chaotic regions in more detail as well as their evolution as the energy changes. In a weak gravitational field, numerical experiments reveal only periodic trajectories that are symmetric about the vertical diameter of the circle. An analytic proof is given that the imposition of a weak gravitational field causes the disappearance of nonsymmetric two-, three-, four-, and six-link trajectories. The phenomenon arises from the superposition of two factors: the gravitation and the perfect symmetry of the circular billiard. We also consider motion evolution in the special case of the perfectly inelastic reflection law.