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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Vydané v:Physica. D Ročník 239; číslo 3-4; s. 214 - 219
Hlavní autori: BARRABES, Esther, CORS, Josep M, PINYOL, Conxita, SOLER, Jaume
Médium: Journal Article Publikácia
Jazyk:English
Vydavateľské údaje: Amsterdam Elsevier 01.02.2010
Predmet:
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
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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 [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.
AbstractList 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.
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.
Author PINYOL, Conxita
CORS, Josep M
SOLER, Jaume
BARRABES, Esther
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Issue 3-4
Keywords Topological method
Fixed point theorem
Eccentricity
Hip-hop solutions
Spatial 2N body problem
Periodic solution
Hill equation
Non linear phenomenon
Rotating frame
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Snippet 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...
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...
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SubjectTerms 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
Title Highly eccentric hip―hop solutions of the 2N―body problem
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