On Centered Co-circular Central Configurations of the n-Body Problem

Abstract

We study the co-circular central configurations of the nbody problem for which the center of mass and the center of the common circle coincide. In particular, we prove that there are no central configurations of this type with all the masses equal except one. This provides more evidences for the veracity of the conjecture that the regular n-gon with equal masses is the unique co-circular central configuration of the n-body problem whose center of mass is the center of the circle. Our result remains valid if we consider power-law potentials.

Exploring Talents Training Model in Applied Universities—Taking the School of Economics and Management of Chengdu Technological University as an Example

“…In [6], Hampton showed that there are no centered co-circular central configurations formed by n equal masses plus one infinitesimal mass in the case of α = 1. This result was subsequently expanded upon by Corbera and Valls in [3] to general powerlaw potentials for masses formed by n equal masses plus one arbitrary mass. In [10], we proved the nonexistence of such configurations in several cases: when there are equal masses and exactly two unequal masses; when there are two groups of equal masses; and when there are a mix of equal masses and some heavier (or lighter ) ones, if ordered in some specific way.…”

“…Another interesting approach to Chenciner's question was initiated by Hampton in [8], where he proved that there are no centered cocircular central configuration formed by n equal masses plus one infinitesimal mass in the case of α = 1, or, we may say that he proved the nonexistence of such configurations for masses in the form of n + ϵ. This result was subsequently expanded upon by Corbera and Valls in [5] to general power-law potentials and masses in the form of n + 1, i.e., n equal masses plus one arbitrary mass.…”

Stacked central configurations with a homogeneous potential in $\mathbb{R}^3$

“…Hampton [18] gave the uniqueness of the case n = 4. Corbera and Valls [8] gave some characterization of the general case for n-body problems, and it is still open for n ≥ 5. For the regular n-gon central configuration itself, Cors et al [11], Wang [38] (generalizing the work of [35] by Perko and Walter) and Hampton [21] studied the homogeneous potential case, and gave some characterization from different aspects.…”