Inverse Problem of Central Configurations in the Collinear 5-Body Problem
Mathematics and Natural Sciences
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.
Journal of Mathematical Physics
(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