# 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.

## Keywords

N-body problem Relative equilibria Linear stability## Mathematics Subject Classification

70F10 37N05 70Fxx 37Cxx## References

- 1.Albouy, A.: On a paper of Moeckel on central configurations. Regul. Chaotic Dyn.
**8**(2), 133–142 (2002)MathSciNetCrossRefMATHGoogle Scholar - 2.Albouy, A., Cabral, H.E., Santos, A.A.: Some problems on the classical \(n\)-body problem. Celest. Mech. Dyn. Astron.
**113**, 369–375 (2012)MathSciNetCrossRefGoogle Scholar - 3.Albouy, A., Fu, Y., Sun, S.: Symmetry of planar four-body convex central configurations. Proc. R. Soc. A
**464**, 1355–1365 (2008)MathSciNetCrossRefMATHGoogle Scholar - 4.Brumberg, V.A.: Permanent configurations in the problem of four bodies and their stability. Sov. Astron.
**1**, 57–79 (1957)MathSciNetGoogle Scholar - 5.Leandro, E.S.G.: Factorization of the Stability Polynomials of Ring Systems. arXiv: 1705.02701v1 [math.DS] (2017)
- 6.Longley, W.R.: Some particular solutions in the problem of \(n\) bodies. Bull. Am. Math. Soc.
**13**(7), 324–335 (1907)MathSciNetCrossRefMATHGoogle Scholar - 7.MacMillan, W.D., Bartky, W.: Permanent configurations in the problem of four bodies. Trans. Am. Math. Soc.
**34**, 838–875 (1932)MathSciNetCrossRefMATHGoogle Scholar - 8.Moeckel, R.: Linear stability analysis of some symmetrical classes of relative equilibria. In: Dumas, H.S., Meyer, K.R., Schmidt, D.S. (eds.) Hamiltonian Dynamical Systems: History, Theory and Applications, IMA, vol. 63, pp. 291–317. Springer, New York (1995)CrossRefGoogle Scholar
- 9.Roberts, G.: Linear stability of the \(1+n\)-gon relative equilibrium. In: Delgado, J., Lacomba, E.A., Pérez-Chavela, E., Llibre, J. (eds.) Hamiltonian Systems and Celestial Mechanics (HAMSYS-98), Proceedings of the III International Symposium, World Scientific Monograph Series in Mathematics, vol. 6, pp. 303–330. World Scientific Publishing Co., Singapore (2000)Google Scholar
- 10.Roberts, G.: Stability of relative equilibria in the planar n-vortex problem. SIAM J. Appl. Dyn. Syst.
**12**(2), 1114–1134 (2013)MathSciNetCrossRefMATHGoogle Scholar - 11.Schmidt, D.S.: Central configurations and relative equilibria for the \(N\)-body problem. In: Cabral, H., Diacu, F. (eds.) Classical and Celestial Mechanics, The Recife Lectures, pp. 1–34. Princeton University Press, Princeton (2002)Google Scholar
- 12.Serre, J.-P.: Linear Representations of Finite Groups. Springer, New York (1997)Google Scholar