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

We show the existence of families of hip–hop solutions in the equal–mass 2NN–body problem which are close to highly eccentric planar elliptic homographic motions of 2NN bodies plus small perpendicular non–harmonic oscillations. By introducing a parameter ϵϵ, 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≠0ϵ≠0, the topological transversality persists and Brouwer’s fixed point theorem shows the existence of this kind of solutions in the full system.