A stochastic optimization algorithm for analyzing planar central and balanced configurations in the $n$-body problem

TL;DR

This paper introduces a stochastic optimization method to analyze planar central and balanced configurations in the $n$-body problem, successfully identifying configurations up to $n=12$ and exploring symmetric and asymmetric cases.

Contribution

The paper presents a novel stochastic optimization algorithm for analyzing $n$-body configurations, providing comprehensive lists and examples for various $n$ values.

Findings

Identified all equal mass central configurations satisfying Morse equality up to $n=12

Discovered asymmetric balanced configurations for specific $n$ values

Provided examples of balanced configurations without symmetry for $n=4$ and $n=10$

Abstract

A stochastic optimization algorithm for analyzing planar central and balanced configurations in the $n$-body problem is presented. We find a comprehensive list of equal mass central configurations satisfying the Morse equality up to $n=12$. We show some exemplary balanced configurations in the case $n=5$, as well as some balanced configurations without any axis of symmetry in the cases $n=4$ and $n=10$.