On the planar central configurations of rhomboidal and triangular four- and five-body problems

TL;DR

This paper analyzes symmetric four- and five-body gravitational configurations, deriving regions of possible central configurations analytically and numerically, and investigates their dynamical behavior including chaos and periodicity.

Contribution

It introduces analytical and numerical methods to identify central configurations in symmetric four- and five-body problems with specific mass arrangements.

Findings

Regions of possible central configurations are explicitly derived.

The phase space exhibits both chaotic and periodic orbits.

Regularization techniques facilitate the dynamical analysis.

Abstract

We consider a symmetric five-body problem with three unequal collinear masses on the axis of symmetry. The remaining two masses are symmetrically placed on both sides of the axis of symmetry. Regions of possible central configurations are derived for the four- and five-body problems. These regions are determined analytically and explored numerically. The equations of motion are regularized using Levi-Civita type transformations and then the phase space is investigated for chaotic and periodic orbits by means of Poincar\'e surface of sections.