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

Authors

Candice Davis, University of Southern Mississippi
Scott Geyer, University of Southern Mississippi
William Johnson, University of Southern Mississippi
Zhifu Xie, University of Southern MississippiFollow

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 m₁ = m₅, m₂ = m₄, and m₃ is an arbitrary parameter. Specially if r = 0 and s=t=s⎯⎯, the configuration q=(−s⎯⎯−1,−1,0,1,s⎯⎯+1) is always a central configuration for any positive masses 0 < m₂ = m₄ < ∞ when m₁ = m₅ are fixed at particular values, which only depend on s⎯⎯ and m₃. 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 E₁₄ (or E₂₅) 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 E₁₄ (or E₂₅) 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