Introduction.

We shall now pass to the study of the general form and disposition of the orbits of dynamical systems. For simplicity we shall in the present chapter chiefly consider the motion of a particle which is free to move in a plane under the action of conservative forces, but many of the results obtained can be readily extended to more general dynamical systems.

It has already been observed (§ 104) that the determination of the motion of a particle with two degrees of freedom under the action of conservative forces is reducible to the problem of finding the geodesies on a surface with a given line-element; an account of the properties of geodesies might therefore be regarded as falling within the scope of the discussion. Many of these properties are however of no importance for our present purpose: and as the theory of geodesies is fully treated in many works on Differential Geometry, we shall only consider those theorems which are of general dynamical interest.

Periodic solutions.

Great interest has attached in recent years to the, investigation of those particular modes of motion of dynamical systems in which the same configuration of the system is repeated at regular intervals of time, so that the motion is purely periodic. Such modes of motion are called periodic solutions.