Variational Constructions for Some Satellite Orbits in Periodic Gravitational Force Fields

In a gravitational force field, a bounded orbit of an infinitesimal point mass is called a satellite orbit. The purpose of this paper is to establish a variational theory for the existence of some relative periodic satellite orbits in periodic gravitational force fields. The gravitation field can be generated by a relative equilibrium or a uniformly rotating asteroid. Our approach is to regard the aggregate of primaries together with the small satellite as a restricted full two-body problem, and then look for direct and retrograde relative periodic satellite orbits by direct methods of calculus of variations. Regularity of the action-minimizing satellite orbit is obtained by providing satisfactory lower bound estimates for the distance between the satellite and the mass center. An upper bound estimate for this distance is also provided as a contrast to classical existence proofs by continuation from infinity.