Symmetric Constellations of Satellites Moving Around a Central Body of Large Mass

Abstract

We consider a (1 + N )-body problem in which one particle has mass m 0 1 and the remaining N have unitary mass. We can assume that the body with larger mass (central body) is at rest at the origin, coinciding with the center of mass of the N bodies with smaller masses (satellites). The interaction force between two particles is defined through a potential of the form2) and r is the distance between the particles. Imposing symmetry and topological constraints, we search for periodic orbits of this system by variational methods. Moreover, we use -convergence theory to study the asymptotic behaviour of these orbits, as the mass of the central body increases. It turns out that the Lagrangian action functional -converges to the action functional of a Kepler problem, defined on a suitable set of loops. In some cases, minimizers of the -limit problem can be easily found, and they are useful to understand the motion of the satellites for large values of m 0 . We discuss some examples, where the symmetry is defined by an action of the groups Z 4 , Z 2 × Z 2 and the rotation groups of Platonic polyhedra on the set of loops.