Minimally classifying relative equilibria

Estimates on a minimal classification of relative equilibria in the planar n -body problem of celestial mechanics have been announced in [1], [2]. Our main theorem asserts that these estimates are actually met for any n ≧3 on an open set in IR inf+ sup n . For any n ≧4, this open set is proper.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43205/1/11005_2005_Article_BF01793953.pd

Rosette Central Configurations, Degenerate central configurations and bifurcations

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 $\pi/n$, and an additional particle of mass $m_0$ lies at the center of mass of the system. This system admits two mass parameters $\mu=m_0/m_1$ and \ep=m_2/m_1. We show that, as $\mu$ varies, if $n> 3$, there is a degenerate central configuration and a bifurcation for every \ep>0, while if $n=3$ there is a bifurcations only for some values of $\epsilon$.Comment: 16 pages, 6 figure

On the classification of N-point concentrating solutions for mean field equations and the critical set of the N-vortex singular Hamiltonian on the unit disk

Motivated by the analysis of the multiple bubbling phenomenon (Bartolucci et al. in Commun. Partial Differ. Equ. 29(7–8):1241–1265, 2004) for a singular mean field equation on the unit disk (Bartolucci and Montefusco in Nonlinearity 19:611–631, 2006), for any N≥3 we characterize a subset of the 2π/N-symmetric part of the critical set of the N-vortex singular Hamiltonian. In particular we prove that this critical subset is of saddle type. As a consequence of our result, and motivated by a recently posed open problem (Bartolucci et al. in Commun. Partial Differ. Equ. 29(7–8):1241–1265, 2004), we can prove the existence of a multiple bubbling sequence of solutions for the singular mean field equation