Central Configurations of the Planar Coorbitalsatellite Problem

Cors, Josep M.; Llibre, Jaume; Ollé, Mercè

doi:10.1023/b:cele.0000043569.25307.ab

Cited by 33 publications

(40 citation statements)

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“…When n is small and the small masses are equal, in (Cors, Llibre and Ollé, 2004) the authors obtain numerically that the 1 + n-gon is the only configuration when n ≥ 9. In the case n = 4 they proved that there are only three symmetric central configurations.…”

Section: Introductionmentioning

confidence: 94%

On the central configurations of the planar 1 + 3 body problem

Corbera

Cors

Llibre

2010

Celest Mech Dyn Astr

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Abstract. We consider the Newtonian 4 body problem in the plane with a dominat mass M . We study the planar central configurations of this problem when the remaining masses are infinitesimal. We obtain two different classes of central configurations depending on the mutual distances between the infinitesimal masses. Both classes exhibit symmetric and non-symmetric configurations. And when two infinitesimal masses are equal, with the help of extended precision arithmetics, we provide evidence that the number of central configurations varies from five to seven.

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Section: Introductionmentioning

confidence: 94%

On the central configurations of the planar 1 + 3 body problem

Corbera

Cors

Llibre

2010

Celest Mech Dyn Astr

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“…(4) Extend any of these results to non-equal masses; even a perturbative analysis near the equal mass case would be a significant advance. It may also be relatively easy to extend to restricted problems (where some of the masses are infinitesimal compared to others), which already have a rich literature of results in the Newtonian case [79,124,65,106,107,49,15,142,97,34,75,123,125,18,60]. (5) Derive equations, or a combinatorial/linear-algebraic framework, for central configurations in the limiting case of A → ∞.…”

Section: Future Directionsmentioning

confidence: 99%

Planar N-body central configurations with a homogeneous potential

Hampton

2019

Celest Mech Dyn Astr

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Central configurations give rise to self-similar solutions to the Newtonian N -body problem, and play important roles in understanding its complicated dynamics. Even the simple question of whether or not there are finitely many planar central configurations for N positive masses remains unsolved in most cases. Considering central configurations as critical points of a function f , we explicity compute the eigenvalues of the Hessian of f for all N for the point vortex potential for the regular polygon with equal masses. For homogeneous potentials including the Newtonian case we compute bounds on the eigenvalues for the regular polygon with equal masses, and give estimates on where bifurcations occur. These eigenvalue computations imply results on the Morse indices of f for the regular polygon. Explicit formulae for the eigenvalues of the Hessian are also given for all central configurations of the equal mass 4-body problem with a homogeneous potential. Classic results on collinear central configurations are also generalized to the homogeneous potential case. Numerical results, conjectures, and suggestions for future work in the context of a homogeneous potential are given.

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“…Many different aspects of coorbiting satellites have been studied, for instance planar central configurations and relative equilibria [12,13], or the dynamics of ring systems conformed by several satellites [14,15]. Nevertheless, the exchange character for N-body systems with N > 3 has not been contemplated.…”

Section: Introductionmentioning

confidence: 99%