Analytically solvable dynamical systems which are not integrable

Abstract

The central, two-fold aim of this paper is to expose a class of nonintegrable systems whose solutions can be analytically expressed in terms of computationally tractable algorithms and to explicitly construct a demonstrably meaningful general solution for one member of this class. Toward this end, information theoretic results are reviewed which show that dynamical systems of null metric entropy have general solutions of null algorithmic complexity. Hence, all systems exhibiting null metric entropy, including ergodic (only) and mixing (only) systems, have orbits which can be specified by algorithmically meaningful, analytic expressions whose information content is very much less than that of the orbits they describe. In consequence, we may quite legitimately define the class of systems having null metric entropy ot be algorithmically integrable, called A-integrable herein. After establishing the existence of A-integrability, this paper turns to the problem of exposing a specific nonintegrable but nonetheless A-integrable system whose algorithmically meaningful solution can be explicitly derived. The billiard moving within a plane polygon having the shape of a 60°–120° rhombus is shown to be a simple example. Indeed, the whole class of billiards moving in polygons whose angles are all rational multiples of π have null metric entropy and are therefore A-integrable systems; moreover, all rational billiards appear to have explicitly derivable solutions as is discussed herein and in a recent preprint by Bjorn Birnir. Because rational billiards are dense in the set of all plane billiards, including chaotic ones, the present paper suggests that A-integrable billiards may provide a long sought analytic route to chaos. Finally, it is pointed out that null entropied, A-integrable systems are perhaps the broadest category of systems which can reasonably be termed integrable, for systems having positive metric entropy are truly nonintegrable due to the presence of orbits which are their own simplest description and their own fastest computer.