Regularity and chaos in classical mechanics, illustrated by three deformations of a circular 'billiard'

Abstract

The theory of motion of particles bouncing inside a curve B is employed to illustrate different types of orbit in classical mechanics. In the 'phase space' whose two coordinates are s, the position around B, and p, the direction of impact, orbital dynamics is a discrete area-preserving mapping between successive bounces; and orbit may be zero dimensional (i.e. closed, and so returning repeatedly to a finite number of points), one dimensional (eventually filling an 'invariant curve') or two dimensional (eventually filling an area chaotically). When B is a circle, s, p space is covered with invariant curves and no closed orbits are isolated. Different deformations of a circle generate very different orbits: stadia give ergodic motion (almost all orbits explore almost all s, p values) with extreme unpredictability (chaos), ellipses give motion entirely confined to invariant curves whose topology is organised by two isolated closed orbits, a family of ovals gives (generic) motion in which phase space is intricately divided into chaotic areas and areas covered with invariant curves. The nature of the motion is determined by whether the closed orbits are stable, unstable or neutrally stable.