Highly eccentric hip–hop solutions of the 2–body problem

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 ϵ , 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 ϵ ≠ 0 , the topological transversality persists and Brouwer’s fixed point theorem shows the existence of this kind of solutions in the full system.