On the finiteness issue of four-body balanced configurations in the plane

TL;DR

This paper proves that the number of four-body balanced configurations in the plane remains finite when the symmetric matrix involved is close to a specific numerical matrix, addressing a key question in celestial mechanics.

Contribution

It establishes the finiteness of four-body balanced configurations for matrices near a given numerical matrix, advancing understanding of symmetric configurations in celestial mechanics.

Findings

Number of four-body balanced configurations is finite near a specific matrix.

Finiteness holds when the symmetric matrix is close to a numerical matrix.

Addresses a key question in the study of celestial configurations.

Abstract

We show that the number of $\mathbf{S}$-balanced configurations of four bodies in the plane is finite, provided that the symmetric matrix $\mathbf{S}$ is close to a numerical matrix.