Bifurcation of planar balanced configurations for the n-body problem in dimension four - Giorgia Testolina, PhD at  Ruhr-Universität Bochum

Abstract: Simple periodic motions for the n-body problem, in which each body moves on a Keplerian elliptical orbit while the shape of the system remains constant (up to rotations and scalings), arise from specific mass arrangements. In dimension d ≤ 3, such motions are produced only by central configurations. However, in dimension d ≥ 4, the higher complexity of the orthogonal group allows to define a broader class of balanced configurations, which includes central configurations as a special case.

 In this talk we study the existence of bifurcation points along trivial branches of planar balanced configurations in dimension 4 and give a lower bound on their number. This follows from an abstract bifurcation result which shows that, for a continuous family of C2 functionals on a finite dimensional manifold, the non vanishing of the spectral flow of the associated family of Hessians along a trivial branch of critical points implies the existence of bifurcation points.