A numerical analysis of planar central and balanced configurations in the (n+1)-body problem with a small mass

TL;DR

This paper introduces two numerical algorithms for analyzing planar central and balanced configurations in the (n+1)-body problem with a small mass, combining stochastic optimization and analytic-continuation methods.

Contribution

It presents novel numerical algorithms that efficiently compute configurations in the (n+1)-body problem with a small mass, integrating stochastic and analytic techniques.

Findings

Demonstrated configurations for n=4,5,6 cases.

Validated effectiveness of the algorithms through numerical examples.

Provided insights into the structure of solutions in small-mass (n+1)-body problems.

Abstract

Two numerical algorithms for analyzing planar central and balanced configurations in the $(n+1)$-body problem with a small mass are presented. The first one relies on a direct solution method of the $(n+1)$-body problem by using a stochastic optimization approach, while the second one relies on an analytic-continuation method, which involves the solutions of the $n$-body and the restricted $(n+1)$-body problem, and the application of a local search procedure to compute the final $(n+1)$-body configuration in the neighborhood of the configuration obtained at the first two steps. Some exemplary central and balanced configurations in the cases $n=4,5,6$ are shown.