# Inverse Problem of Central Configurations in the Collinear 5-Body Problem

## Document Type

Article

## Publication Date

5-1-2018

## Department

Mathematics

## School

Mathematics and Natural Sciences

## Abstract

In this paper, we study the inverse problem of collinear central configurations of a 5-body problem: given a collinear configuration *q* = (−*s* − 1, −1, *r*, 1, *t* + 1) of 5 bodies, does there exist positive masses to make the configuration central? Here we proved the following results: If *r* = 0 and *s* = *t* > 0, there always exist positive masses to make the configuration central and the masses are symmetrical such that *m*_{1} = *m*_{5}, *m*_{2} = *m*_{4}, and *m*_{3} is an arbitrary parameter. Specially if *r* = 0 and ** s=t=s⎯⎯**, the configuration

**is always a central configuration for any positive masses 0 <**

*q=(−s⎯⎯−1,−1,0,1,s⎯⎯+1)**m*

_{2}=

*m*

_{4}< ∞ when

*m*

_{1}=

*m*

_{5}are fixed at particular values, which only depend on s⎯⎯ and

*m*

_{3}. s⎯⎯ is the unique real root of a fifth order polynomial and numerically s⎯⎯≈1.396 812 289. If r = 0 and s ≠ t > 0, there also always exist positive masses to make the configuration central. For any

*r*∈ (0, 1) [or

*r*∈ (−1, 0)], there exist a set

*E*

_{14}(or

*E*

_{25}) in the first quadrant of st-plane where every configuration is a central configuration for some positive masses. However, no configuration in the complement of

*E*

_{14}(or

*E*

_{25}) is a central configuration for any positive masses.

## Publication Title

Journal of Mathematical Physics

## Volume

59

## Issue

5

## Recommended Citation

Davis, C.,
Geyer, S.,
Johnson, W.,
Xie, Z.
(2018). Inverse Problem of Central Configurations in the Collinear 5-Body Problem. *Journal of Mathematical Physics, 59*(5).

Available at: https://aquila.usm.edu/fac_pubs/15769