Scattering of Light in a Medium Adjacent to a Reflecting Surface

Abstract

cifically, no mention was made of the fact that the solution of the fundamental equation, as well as its time derivative, are to be continuously dependent upon the original parameter IX and initial value perturbations. As mentioned before, the entire treatment in this paper which centers around the questions of existence and deter­ mination of periodic solutions of the differential equations may be compared with the Poincare treatment and theory. A discussion of the latter may be found in Fundamentals of Poincare's Theory by K. 0. Friedrichs in Proceedings of the Symposium on Nonlinear Circuit Analysis/' New York, April 23 and 24, 1953, p. 56. The principal difference between Shteinberg's and Poin­ care's theories is the following: The former method, treated in this paper, considers a periodic solution of the simplified system of differential equations with \x = 6, and given initial conditions. The problem is to find a uniquely corresponding periodic solution of the fundamental set of equations whose initial conditions are those of the respective simplified set plus respective perturbations which are continuous functions of M and which vanish when /* = 0. In other words, the periodic solutions of the fundamental set reduce those of the simplified set when \x vanishes. The criterion which estab­ lishes such solutions is the vanishing of a certain Jacobian determinant of the initial solution and its time derivative with respect to the perturbations of initial conditions when they are zero. This eventually leads to the equivalent cri­ terion that these perturbations are continuous functions of H and vanish when At = 0, and, thereby, reduce to periodic solutions of the simplified equations. Poincare's treatment first considers the two sets of equa­ tions in the Shteinberg paper. However, the problem here is to consider a variational equation associated with the original equation, nonzero parameter, //. Now the essence of this theory is that a periodic solution of the aforementioned simplified system may be continued to a periodic solution of the fundamental system if the above variational equation has no periodic solution. The above continued periodic solution in question depends continuously upon /J. and reduces to the periodic solution of the simplified system when n — 0 Of course, the additional feature of the present paper of Shteinberg is the occurrence of discontinuous terms of the fundamental and simplified sets of equations. Although no particular mathematical difficulties arise because of these terms, their presence serves part of the intended purpose of Shteinberg's paper. One important topic that was lacking in this paper was the discussion of stability of periodic solutions. However, the existence and stability of limit cycles were discussed in an example that was presented at the end of the paper. Apart from occasional inconsistencies in the use of sub­ script symbols and some minor algebraic errors all of which were easily rectified, the paper is not at all difficult to follow. From the point of view of the study and investigations in orbital mechanics and similar studies, the present paper appears to give a satisfactorily useful method for determining periodic solutions of second-order differential equations of special types. An example of a counterpart, nonperturbation method which has been used successfully for equations similar in type to those of this paper is van der Pol's method. As another example of a nonperturbative method for periodic solutions, the phase plane method is quite appropriate for the types of equations as Shteinberg's since they are autonomous systems. There are no indications in the paper which point to de­ cisively new contributions to the overall theory of periodic solutions of second-order differential equations. It is felt, though, that the material in this paper could be extended to systems somewhat more general than those treated here. As a final comment, it might be stated that in comparing the method presented in Shteinberg's paper with Poincare's method, the latter method does not make as stringent re­ quirements on continuity and differentiability as the former does.