On the Sectional Curvature Along Central Configurations

In this paper we characterize planar central configurations in terms of a sectional curvature value of the Jacobi–Maupertuis metric. This characterization works for the $N$-body problem with general masses and any $1/r^\alpha$ potential with $\alpha > 0$. We also obtain dynamical consequences of these curvature values for relative equilibrium solutions. These curvature methods work well for strong forces $(\alpha \geqslant 2)$.