Bifurcations of Two Symmetric Classes of Central Configurations

We consider the bifurcations of symmetric classes of central configurations formed by four and five bodies, namely the centered equilateral triangle and the centered regular tetrahedron. Results in the current literature have been obtained for a varying central mass and the Newtonian potential, or for a varying vertex mass and homogeneous potentials with exponents less than − 1. For the centered equilateral triangle we classify all the symmetric central configurations obtained when the central mass increases through its degenerate value for exponents less than − 1/5. For the centered regular tetrahedron, we vary two and three of the masses at the vertices in the same manner and classify all the new central configurations for exponents less than or equal to − 1. We show that all such central configurations must be symmetric, that is, there are no non-symmetric central configurations bifurcating from the degenerate centered regular tetrahedron when two or three of the masses at the vertices are varied equally.