“Bifurcation of planar balanced configurations for the $n$-body problem in dimension four”
Martedì 17 Dicembre 2024, ore 17:00 - Aula 2BC60 - Giorgia Testolina (PhD Student 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 $d \leq 3$, such motions are produced only by central configurations. However, in dimension $d \geq 4$, 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 $C^2$ 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.
