Abstract
We study the neighborhood of the equal mass regular polygon relative equilibria in the N-body probem, and show that this relative equilibirum is isolated among the co-circular configurations (in which each point lies on a common circle) for which the center of mass is located at the center of the common circle. It is also isolated in the sense that a sufficiently small mass cannot be added to the common circle to form a \(N+1\)-body relative equilibrium. These results provide strong evidence for a conjecture that the equal mass regular polygon is the only co-circular relative equilibrium with its center of mass located at the center of the common circle.
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Hampton, M. Splendid isolation: local uniqueness of the centered co-circular relative equilibria in the N-body problem. Celest Mech Dyn Astr 124, 145–153 (2016). https://doi.org/10.1007/s10569-015-9656-x
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DOI: https://doi.org/10.1007/s10569-015-9656-x