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 H3H3 is constructed in the mass space (m1,m2,m3)∈(R+)3(m1,m2,m3)∈(R+)3. The construction of H3H3 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=23!/2−1=2, m1m1, m2m2, and m3m3 are mutually distinct if m∈H3m∈H3. 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 H3H3 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.