Structure and Stability of the Rhombus Family of Relative Equilibria under General Homogeneous Forces

Abstract

Let $$N>2$$ and $$n>1$$ . Among the classes of symmetric relative equilibria of the N-body problem whose symmetry group is one of the dihedral groups $$D_n$$ , the rhombus family stands out as the only noncollinear family for which the symmetry implies that the stability polynomial is fully factorizable. The present paper discusses basic properties and linear stability of the rhombus family assuming the forces of interaction depend on the mutual distances raised to an arbitrary real exponent $$2a+1$$ . In a suitable parameter plane, the family of rhombus relative equilibria forms a pincel of graphs which foliates the union of an open unit square and an open rectangle obtained from the unit square by a reflection and an inversion. We show that all rhombus relative equilibria are linearly stable if $$a>-1$$ , that they are all unstable for a in the interval bound by $$-4-2\sqrt{2}\approx -6.82$$ and $$4(\sqrt{3}-2)\approx -1.07$$ , and that stability and instability depend on mass values for the remaining values of a. These results impose limitations on the validity of Moeckel’s dominant mass stability conjecture in the context of generalized N-body problems.