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 m0 ≫ 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 form U ∼ 1 r α , where α ∈ [1, 2) 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 Γ-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 Γ-converges to the action functional of a Kepler problem, defined on a suitable set of loops. Minimizers of the Γ-limit problem can be easily found, and they are useful to understand the motion of the satellites for large values of m0. We discuss some examples, where the symmetry is defined by an action of the groups Z4 , Z2 × Z2 and the rotation groups of Platonic polyhedra on the set of loops.