Symmetry and Asymmetry in the 1+N Coorbital Problem

TL;DR

This paper investigates the symmetry properties of coorbital relative equilibria in the planar Newtonian N-body problem, revealing the existence of symmetric configurations with asymmetric masses and establishing conditions for symmetry based on mass equalities.

Contribution

It demonstrates the existence of symmetric relative equilibria with asymmetric masses in certain coorbital problems and links mass equalities to symmetry of configurations.

Findings

Existence of symmetric equilibria with asymmetric masses for N=4,6,8.

Mass equalities imply symmetry in convex relative equilibria.

At most one convex symmetric configuration in the 1+5 problem.

Abstract

The relative equilibria of planar Newtonian $N$-body problem become coorbital around a central mass in the limit when all but one of the masses becomes zero. We prove a variety of results about the coorbital relative equilibria, with an emphasis on the relation between symmetries of the configurations and symmetries in the masses, or lack thereof. We prove that in the $N=4$, $N=6$, and $N=8$ Newtonian coorbital problems there exist symmetric relative equilibria with asymmetric positive masses. This result can be generalized to other homogeneous potentials, and we conjecture similar results hold for larger even numbers of infinitesimal masses. We prove that some equalities of the masses in the $1+4$ and $1+5$ coorbital problems imply symmetry of a class of convex relative equilibria. We also prove there is at most one convex central configuration of the symmetric $1+5$ problem.