Relative equilibria, linear stability and electromagnetic curvature

In this paper we study the linear stability of relative equilibria in the Newtonian $n$-body problem from the viewpoint of electromagnetic systems. We first examine the effect of the ambient dimension on stability, starting from the Lagrange equilateral triangle solutions of the three-body problem in $\mathbb R^4$. We then initiate a new approach to stability based on electromagnetic curvature. In a two-dimensional model, we relate linear stability to both the Ma\~n\'e critical value and to the behavior of the zero set of the electromagnetic curvature, highlighting a change in its topology at the stability threshold. This criterion is then applied to the planar $n$-body problem: in the three-body case, we recover Routh's classical criterion, and, more generally, we obtain an instability criterion for relative equilibria whose reduced linearized dynamics splits along invariant symplectic planes. These results suggest a new geometric perspective on linear stability and on questions related to Moeckel's conjecture.