On the Role of First Integrals in the Perturbation of Periodic Solutions

Abstract

In two previous papers methods were given for appraising the range of the parameter in the theory of perturbation of periodic solutions, originated by Poincare. In the first of these [3 of the bibliography], only the simplest case was considered, in which the variational equations are assumed to have no periodic solutions not identically zero. In the second paper [4] the methods of the first paper were modified so as to be applicable to what may be considered the general case in which the variational equations have one or more linearly independent periodic solutions. The results of neither of these papers are, however, applicable to systems admitting first integrals, unless the first integrals are first used to lower the order of the original system. This can lead to formal complications, which it might be desirable to by-pass. The purpose of this paper is to show how this by-pass can be accomplished in those cases where the number of first integrals is equal to the number of linearly independent periodic solutions of the variational equations. In its reliance upon the methods of integral equations, successive approximations, generalized Green's matrices, and the discussion of the so called bifurcation equations (Verzweigungsgleichungen), this paper stands in intimate relationship with a magnificent paper by E. H6lder [2]. This relationship is considerably closer than in the preceding papers of the present writer; and hence it would be well to point out the two principal contrasts between Holder's work and our own. First, the main purpose of the present paper is to propose a method for appraising the interval for the parameter. Holder makes no mention of the feasibility of doing this, nor does his paper contain those inequalities which might enable the reader easily to form his own estimates. Secondly, whereas Holder focuses attention on the equations of certain problems in celestial mechanics, we wish in this paper to present the theory in a more general setting. Of particular significance is the fact that Holder bases his discussion of the bifurcation equations on the invariance of Hamilton's action integral with respect to a certain group of transformations (rotations in space and translations in time). In this paper a like discussion is based on first integrals. As is generally well known [6] and as H6lder himself explicitly recognized [p. 231, loc. cit.], the invariance of the action integral under a continuous group leads (under suitable hypotheses of regularity) to certain first integrals, namely those * This research was supported by the United States Air Force through the Office of