The classification of orbits

Abstract

The present paper is concerned with a simple type of dynamical system, a system having only two freedoms and of “separable” type. The essential simplification introduced by this restriction is that the motions in the two coordinates can be discussed to some extent independently of each other. In Part I a method of classification of the possible orbits is explained. Four constants are needed to define the motion completely, but the general nature of the orbit depends on two constants suitably chosen, and the classification is effected by reference to a plane in which the two constants are taken as Cartesian coordinates. It is shewn that the plane of reference is divided into regions by critical curves, and that the orbits represented by points in the same region (each point representing a set of orbits, not a single orbit) are of the same general type. The various possible types of orbits, and their stability, in a special sense, are discussed. In Part II the theory is applied by way of illustration to a somewhat trivial example, the orbits for a particle under a central attraction proportional to the inverse (n + 1)th power of the distance. The most striking thing here is that the possible types of orbits are essentially the same for all values of n greater than 2. In Part III the theory is applied to the classical problem of a particle subject to an attraction of Newtonian type to two fixed centres, a problem of some interest in relation to the general problem of three bodies, and to some current questions in atomic dynamics. The problem is of course not a new one, but it seems that a satisfactory classification of all the possible orbits has not previously been given. It must be confessed that the complete classification for this problem is very laborious.