On the central configurations in the spatial 5-body problem with four equal masses

Abstract

We analyze the families of central configurations of the spatial 5-body problem with four masses equal to 1 when the fifth mass m varies from 0 to $$+\infty $$ . In particular we continue numerically, taking m as a parameter, the central configurations (which all are symmetric) of the restricted spatial ( $$4+1$$ )-body problem with four equal masses and $$m=0$$ to the spatial 5-body problem with equal masses (i.e. $$m=1$$ ), and viceversa we continue the symmetric central configurations of the spatial 5-body problem with five equal masses to the restricted ( $$4+1$$ )-body problem with four equal masses. Additionally we continue numerically the symmetric central configurations of the spatial 5-body problem with four equal masses starting with $$m=1$$ and ending in $$m=+\infty $$ , improving the results of Alvarez-Ramirez et al. (Discrete Contin Dyn Syst Ser S 1: 505–518, 2008). We find four bifurcation values of m where the number of central configuration changes. We note that the central configurations of all continued families varying m from 0 to $$+\infty $$ are symmetric.