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

Abstract

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

Periodic oscillations in the restricted Hip-Hop 2N+1 body problem

“…We will follow the same set up as in the references [13] and [2]. To describe the motion of the primaries we consider Q 1 , Q 2 , .…”

“…In this paper we are considering also n-primaries but this time they do not move on a plane, they are periodic Hip-hop solutions of the n-body problem. The existence of these solutions were studied in [2,3] and more recently, in [13] we found the existence of families of periodic Hip-hop solutions for a 2N -body problem. A hip-hop solution satisfies i) All the bodies have the same mass.…”

Periodic oscillations in a 2N-body problem

“…The link https://youtu.be/RjlZpqDFsDM leads to a video showing some of these periodic solutions. Using a Bolzano argument, in [BCPS10] the authors show the periodicity of hip-hop solution with motion close to ellipses with high eccentricity. In [BCPS06] the authors use a Poincaré analytical continuation method to show the existence of double symmetric hip-hop solutions with trajectories close to circles.…”