Central configurations of the collinear three-body problem and singular surfaces in the mass space

This Letter is to provide a new approach to study the phenomena of degeneracy of the number of the collinear central configurations under geometric equivalence. A direct and simple explicit parametric expression of the singular surface H 3 is constructed in the mass space ( m 1 , m 2 , m 3 ) ∈ ( R + ) 3 . The construction of H 3 is from an inverse respective, that is, by specifying positions for the bodies and then determining the masses that are possible to yield a central configuration. It reveals the relationship between the phenomena of degeneracy and the inverse problem of central configurations. We prove that the number of central configurations is decreased to 3 ! / 2 − 1 = 2 , m 1 , m 2 , and m 3 are mutually distinct if m ∈ H 3 . Moreover, we know not only the number of central configurations but also what the nonequivalent central configurations are. ► Provide a new method to study the degeneracy of number of CC. ► Results advanced the understanding of number of central configurations. ► Singular mass surface H 3 is given by a direct and simple parametric expression. ► The proof only requires some basic knowledge of linear algebra. ► The method can be applied to some other collinear n-body problem.