Simple Choreographic Motions of N Bodies: A Preliminary Study

Abstract

A "simple choreography" for an N-body problem is a periodic solution in which all N masses trace the same curve without colliding. We shall require all masses to be equal and the phase shift between consecutive bodies to be constant. The first 3-body choreography for the Newtonian potential, after Lagrange's equilateral solution, was proved to exist by Chenciner and Montgomery in December 1999 (Chenciner and Montgomery [2000]). In this paper we prove the existence of planar N-body simple choreographies with arbitrary complexity and/or symmetry, and any number N of masses, provided the potential is of strong force type (behaving like 1/r a , a ≥ 2 as r → 0). The existence of simple choreographies for the Newtonian potential is harder to prove, and we fall short of this goal. Instead, we present the results of a numerical study of the simple Newtonian choreographies, and of the evolution with respect to a of some simple choreographies generated by the potentials 1/r a , focusing on the fate of some simple choreographies guaranteed to exist for a ≥ 2 which disappear as a tends to 1.