Variational Aspects of the Two-Center Problem

Advances on variational approaches for the n-body problem in the past two decades have inspired recent investigations for the n-center problem. Substantial progress about periodic solutions within various homotopy classes has been made for n = 3. One technical tool in such investigations, the local deformation argument, plays an important role. A major challenge of this approach is to construct periodic orbits which have some lobes enclosing only one or two centers. Closer scrutiny for the two-center problem from variational perspectives is therefore essential for further developments in this direction. In this paper we consider the two-center problem and orbits of lemniscate type and planetary type. A lemniscate type orbit consists of two lobes with one center inside each lobe, and a planetary type orbit encloses both centers. We show the existence and minimizing property of planetary type periodic solutions for any given masses of centers at fixed positions, and the period can be arbitrary as long as it is above a mass-dependent threshold value. In contrast, numerical evidence suggests that lemniscate type orbits have higher action values than some homoclinic ejection-collision solutions.