Transformation Theory of Non-Linear Differential Equations of the Second Order

Abstract

Interest in the field of non-linear differential equations in bygone years centered around the dynamics of conservative systems, the problems arising mainly in celestial mechanics. Current interest is focused mainly on non-conservative systems. Nevertheless many of the advances made in the past are of the greatest relevance to the problems of current interest. Therefore we shall give an account here of certain of these past results relevant to current problems. We shall also adapt certain methods so as to get results for these current problems. So far as the non-linear equations that we shall consider are concerned, the most challenging problem mathematically appears to be that of finding means of excluding the possibility of certain singular situations. We shall describe some of these singular possibilities. As yet there is no method available to indicate under what conditions they cannot arise. Transformation theory as a method in differential equations is due to Poincar6. The type of transformation we shall be interested in here is that of the Euclidean plane into itself. Certain methods and results of Birkhoff will be shown to be of interest. In connection with their work on second order non-linear equations, the non-linear terms of which have a small parameter as a coefficient, Kryloff and Bogoliuboff' make use of transformation theory, particularly the theory that has been developed in the study of curves on a torus by Poincar6 and Denjoy. The second order equation