TITLE:
Highly eccentric hip--hop solutions of the 2$N$--body problem

AUTHORS:
Esther Barrabes, Josep M. Cors, Conxita Pinyol i Jaume Soler
E-mails: barrabes@ima.udg.edu, cors@epsem.upc.edu, conxita.pinyol@uab.cat, jaume.soler.villanueva@upc.edu


ABSTRACT:
We show the existence of families of hip--hop solutions in the
equal--mass 2$N$--body problem which are close to highly eccentric
planar elliptic homographic motions of 2$N$ bodies plus small
perpendicular non--harmonic oscillations. By introducing a
parameter $\epsilon$, the homographic motion and the small
amplitude oscillations can be uncoupled into a purely Keplerian
homographic motion of fixed period and a vertical oscillation
described by a Hill type equation. Small changes in the
eccentricity induce large variations in the period of the
perpendicular oscillation and give rise, via a Bolzano argument,
to resonant periodic solutions of the uncoupled system in a
rotating frame. For small $\epsilon \neq 0$, the topological
transversality persists and Brouwer's fixed point theorem shows
the existence of this kind of solutions in the full system.