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 Є, 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 Є ≠ 0, the topological transversality persists and Brouwer's fixed point theorem shows the existence of this kind of solutions in the full system.
CitationBarrabés, E. [et al.]. Highly eccentric hip-hop solutions of the 2N-body problem. "Physica. D, Nonlinear phenomena", 2010, vol. 239, núm. 3-4, p. 214-219.
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