Highly eccentric hip―hop solutions of the 2N―body problem

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...

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Bibliographic Details
Published in:Physica. D Vol. 239; no. 3-4; pp. 214 - 219
Main Authors: BARRABES, Esther, CORS, Josep M, PINYOL, Conxita, SOLER, Jaume
Format: Journal Article Publication
Language:English
Published: Amsterdam Elsevier 01.02.2010
Subjects:
70 Mechanics of particles and systems
70F Dynamics of a system of particles, including celestial mechanics
Classificació AMS
Enginyeria mecànica
Exact sciences and technology
Many-body problem
Oscillations
Oscil·lacions
Physics
Problema dels cossos múltiples
Àrees temàtiques de la UPC
Topological method
Fixed point theorem
Eccentricity
Hip-hop solutions
Spatial 2N body problem
Periodic solution
Hill equation
Non linear phenomenon
Rotating frame
ISSN:0167-2789
Online Access:Get full text
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Summary: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.
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ISSN:0167-2789
DOI:10.1016/j.physd.2009.10.019