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. | Turkish Journal of Mathematics Research Article Turk J Math (2014) 38: 576 – 585 ¨ ITAK ˙ c TUB ⃝ doi: Central configurations in the collinear 5-body problem Muhammad SHOAIB1,∗, Anoop SIVASANKARAN2 , Abdulrehman KASHIF1 1 Department of Mathematical Sciences, University of Ha’il, Ha’il, Saudi Arabia 2 Department of Applied Mathematics and Sciences, Khalifa University, Sharjah, United Arab Emirates Received: • Accepted: • Published Online: • Printed: Abstract: 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. Key words: Central configuration, n-body problem, inverse problem of central configuration 1. Introduction Central configurations are one of the most important and fundamental topics in the study of few-body problems. Therefore, few-body problems in general and central configurations in particular have attracted a lot of attention over the years [4],[5],[10]. Studies on the central configuration of n -body problems (with n ≥ 4 ) are limited due to the greater complexity of problems involving higher numbers of bodies. For n ≥ 4 , the main focus of the available .
