Highly eccentric hip―hop solutions of the 2N―body problem - BARRABES, Esther; CORS, Josep M; PINYOL, Conxita; SOLER, Jaume (Physica. D, 2010)
We show the existence of families of hip-hop solutions in the equal-mass 2N-body problem which are close to highly eccentric planar elliptic homographic motions of 2N bodies plus small perpendicular non-harmonic oscillations. By introducing a parameter [epsilon (Porson)], 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 (Porson)][not equal to]0, the topological transversality persists and Brouwer's fixed point theorem shows the existence of this kind of solutions in the full system.