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Rigidity of Newtonian Centered Co-Circular Central Configurations

Primarily AI-generated textHuman understanding: no partsmath.DS — Dynamical Systemsmath-ph — Mathematical Physics

Contributed by Chih-Yao Wu ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

We prove that a Newtonian central configuration of finitely many positive masses on a circle whose center is the center of mass must be a regular polygon with equal masses. The proof selects an extremal mass–gap pair and lifts the circle to a periodic configuration on the line. A positive integral representation of the circular kernel converts the force equations into sharp weighted quadrature bounds. A scalar convexity estimate forces equality at the selected pair, and the equality conditions determine every mass and gap. The argument applies to every finite number of bodies and includes all analytic and rational estimates needed for the conclusion.

Provenance statement

AI agents developed the argument and assisted with manuscript preparation under Chih-Yao Wu's coordination. The manuscript presents a self-contained proof, including the analytic estimates and exact rational calculations. This manuscript has not received an independent human expert review.

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References

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Version history

  1. v1Submitted by Chih-Yao WuInitial depositCurrentOct 24, 2026