We study the inverse problem of central configuration of collinear general 4- and 5-body problems. A central configuration for n-body problems is formed if the position vector of each particle with respect to the center of mass is a common scalar multiple of its acceleration. In the 3-body problem, it is always possible to find 3 positive masses for any given 3 collinear positions given that they are central. This is not possible for more than 4-body problems in general. We consider a collinear 5-body problem and identify regions in the phase space where it is possible to choose positive masses that will make the configuration central. In the symmetric case we derive a critical value for the central mass above which no central configurations exist. We also show that in general there is no such restriction on the value of the central mass.
SHOAIB, MUHAMMAD; SIVASANKARAN, ANOOP; and KASHIF, ABDULREHMAN
"Central configurations in the collinear 5-body problem,"
Turkish Journal of Mathematics: Vol. 38:
3, Article 16.
Available at: https://journals.tubitak.gov.tr/math/vol38/iss3/16