Existence of Really Perverse Central Configurations in the Spatial $N$-Body Problem

We construct explicit examples of really perverse central configurations in the spatial Newtonian $N$-body problem. A central configuration is called really perverse if it satisfies the central configuration equations for two distinct mass distributions having the same total mass. While such configurations were previously known only in the planar case for large $N$, we prove the existence of spatial really perverse central configurations for $N=27,\dots,55$.

💡 Research Summary

The paper addresses a long‑standing open problem in celestial mechanics: the existence of “really perverse” central configurations in the three‑dimensional Newtonian N‑body problem. A central configuration is a set of positions ({c_k}) together with positive masses ({M_k}) such that the acceleration of each body is proportional to its position vector, i.e. (\lambda M_k c_k = -\sum_{j\neq k} M_k M_j (c_k-c_j)/|c_k-c_j|^3) for some scalar (\lambda). A configuration is called perverse if it satisfies the central‑configuration equations for two distinct mass distributions, and really perverse if those two distributions have the same total mass and the same centre of mass. In the planar case, such configurations were first found for astronomically large numbers of bodies (n ≈ 1369) and later reduced to n ≈ 474, but no spatial example was known.

The authors construct an explicit family of spatial configurations that are really perverse for a whole range of body numbers. The construction is highly symmetric: a central mass (m_0) sits at the origin, (n) equal masses (m_1) are placed at the vertices of a regular (n)-gon of radius (r(t)) in the (xy)-plane, and two additional masses (m_2) are placed on the (z)-axis at ((0,0,\pm \alpha r)). The total mass is (M=m_0+nm_1+2m_2) and the centre of mass is fixed at the origin by design.

Substituting this ansatz into the Newtonian equations of motion and imposing the central‑configuration condition reduces the problem to two scalar equations (labelled (4) and (5) in the paper) involving the parameters (r), (\alpha), and the masses, together with the total‑mass relation (6). The key geometric constant \