Dziobek’s configurations in restricted problems and bifurcation

Abstract

We consider some questions on central configurations of five bodies in space. In the first one, we get a general result of symmetry for the restricted problem of n+1 bodies in dimension n-1. After that, we made the calculation of all c.c. for n=4. In our second result, we extend a theorem of symmetry due to [Albouy, A. and Libre, I.: 2002, Contemporary Math. 292, 1-16] on non-convex central configurations with 4 unit masses and an infinite central mass. We obtain similar results in the case of a big, but finite central mass. Finally, we continue the study by [Schmidt, D.S.: 1988, Contemporary Math. 81 ] of the bifurcations of the configuration with four unit masses located at the vertices of a equilateral tetrahedron and a variable mass at the barycenter. Using Liapunov-Schmidt reduction and a result on bifurcation equations, which appear in [Golubitsley, M., Stewart, L. and Schaeffer, D.: 1988, Singularties and Groups in Bifurcation Theory, Vol. II, Springer-Verlag, New York], we show that there exist indeed seven families of central configurations close to a regular tetrahedron parameterized by the value of central mass.