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Rosette central configurations, degenerate central configurations and bifurcations. (English) Zbl 1175.70013
Summary: In this paper we find a class of new degenerate central configurations and bifurcations in the Newtonian n-body problem. In particular we analyze the Rosette central configurations, namely a coplanar configuration where n particles of mass m 1 lie at the vertices of a regular n-gon, n particles of mass m 2 lie at the vertices of another n-gon concentric with the first, but rotated of an angle π/n, and an additional particle of mass m 0 lies at the center of mass of the system. This system admits two mass parameters μ=m 0 /m 1 and ϵ=m 2 /m 1 . We show that, as μ varies, if n>3, there is a degenerate central configuration and a bifurcation for every ϵ>0, while if n=3 there is a bifurcation only for some values of ϵ.
MSC:
70F10n-body problems
References:
[1]
[2]
[3]
[4]Hampton, M. and Moeckel, R.: 2006, ’Finiteness of relative equilibria of the four-body problem’, Inv. Math., to appear.
[5] · Zbl 0328.57012 · doi:10.1090/S0002-9904-1975-13794-3
[6] · Zbl 0332.58008 · doi:10.2307/1970964
[7] · Zbl 1138.70008 · doi:10.1007/s10569-004-1991-2
[8] · Zbl 0203.26102 · doi:10.1007/BF01389805
[9] · doi:10.1007/BFb0068618