Number of central configurations and singular surfaces in the mass space in the collinear four-body problem (Q2884391)

The authors consider the problem of the number of collinear central configurations in the \(N\)-body problem, where the \(N\) bodies are located on the \(x\)-axis. For any given mass vector, the problem is to determine the number of geometric equivalence classes of the \(N\)-body collinear central configurations, where two configurations are said to be geometrically equivalent if they are similar modulo translations, dilations, rotations and permutations. It is shown that there exists a parametric surface in the mass space, which decreases the total number of collinear central configurations. Within this set and for mutually distinct masses the number of central configurations is 11.