Circular non-collision orbits for a large class of n-body problems

TL;DR

This paper characterizes circular orbits in various n-body problems, showing they must be regular polygons or rotopulsators, and provides explicit formulas and uniqueness results for these configurations.

Contribution

It establishes the conditions under which circular orbits are regular polygons or rotopulsators across multiple n-body problem variants, including curved spaces.

Findings

Circular orbits are either regular polygons or rotopulsators.

Explicit formulas for mass in terms of orbit radius are derived.

Uniqueness of polygonal homographic orbits for each mass order is proven.

Abstract

We prove for a large class of n-body problems including a subclass of quasihomogeneous n-body problems, the classical n-body problem, the n-body problem in spaces of negative constant Gaussian curvature and a restricted case of the n-body problem in spaces of positive constant curvature for the case that all masses are equal and not necessarily constant that any solution for which the point masses move on a circle of not necessarily constant size has to be either a regular polygonal homographic orbit in flat space, or a regular polygonal rotopulsator in curved space, under the constraint that the minimal distance between point masses attains its minimum in finite time. Additionally, we prove that the same holds true if we add an extra mass at the center of that circle and find an explicit formula for the mass of each point particle in terms of the radius of the circle. Finally, we prove that for each order of the masses there is at most one polygonal homographic orbit for the case that the masses need not be constant.