Finiteness of classes of relative equilibria for the $n$-body problem

TL;DR

This paper proves that for fixed masses in the celestial $n$-body problem, the number of distinct classes of relative equilibria solutions is finite, providing a key insight into the solution structure.

Contribution

It establishes the finiteness of classes of relative equilibria for given masses in the $n$-body problem, a significant theoretical advancement.

Findings

Number of classes of relative equilibria is finite for fixed masses.

Provides a foundational result for understanding solution diversity.

Supports future classification and stability analysis of solutions.

Abstract

We prove that if the masses of the $n$-body problem of celestial mechanics are given, the number of classes of relative equilibria that solve the $n$-body problem is finite.