Positive mass of $$k+l$$ -Moulton configuration

Abstract

For given k bodies of collinear central configuration of Newtonian k-body problem, we ask whether one can add other l bodies at the same time on the line without changing the configuration and motion of the initial bodies so that the total k \(+\) l bodies provide a central configuration. We call it k+l-Moulton configuration. We find the following. When l < k \(+\) 1, there exist only zero-mass solutions, masses of added bodies are all zero that means infinitesimal mass. When l \(=\) k \(+\) 1, we show the existence of k+l-Moulton configuration where masses are non-negative given as a one parameter family, \({\mathbf {m_{B}}}={\mathbf {m_{B_{0}}}}\) t, t \(\ge \) 0. Then there exist not only zero-mass but also positive-mass solutions whose masses are all positive. Moreover when l > k \(+\) 1, there is not zero-mass solution because one cannot put more than one body in an interval which is separated by initial k bodies. Then maximum number of added bodies is k \(+\) 1 at once in zero-mass solutions.