Central configurations for the planar Newtonian four-body problem

Piña, E.; Lonngi, P.

doi:10.1007/s10569-010-9291-5

Cited by 18 publications

(22 citation statements)

References 17 publications

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“…Also considering the particles endowed with masses and charges, Alfaro and Perez-Chavela (2002) proved the existence of a continuum of central configurations in a particular 4-body problem. Other recent papers on central configurations are due to Corbera, Cors and Llibre (2010), , Gidea and Llibre (2010), Piña and Lonngi (2010), ... Recently the first and second author of this paper gave new examples of stacked central configurations of the 5-bodies which, as the ones studied by Hampton (2005), have three bodies in the vertices of an equilateral triangle, but the other two are on the perpendicular bisector (Llibre and Mello 2008).…”

Section: Introductionmentioning

confidence: 93%

New stacked central configurations for the planar 5-body problem

Llibre

Mello

Pérez‐Chavela³

2011

Celest Mech Dyn Astr

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Abstract. A stacked central configuration in the n-body problem is one that has a proper subset of the n-bodies forming a central configuration. In this paper we study the case where three bodies with masses m1, m2, m3 (bodies 1, 2, 3) form an equilateral central configuration, and the other two with masses m4, m5 are symmetric with respect to the mediatrix of the segment joining 1 and 2, and they are above the triangle generated by {1, 2, 3}. We show the existence and non-existence of this kind of stacked central configurations for the planar 5-body problem.

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Section: Introductionmentioning

confidence: 93%

New stacked central configurations for the planar 5-body problem

Llibre

Mello

Pérez‐Chavela³

2011

Celest Mech Dyn Astr

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“…The four-body problem. Although unsolved problems remain, for the Newtonian (A = 3) and vortex case (A = 2) of the four-body problem the central configurations are well understood, with many particular results for configurations with some special symmetry or other properties [77,39,86,49,15,104,54,84,55,61,75,108,9,62,110,128,35,143,11,18,63,43,37,46,30,122]. The equal mass case is especially well characterized.…”

Section: 2mentioning

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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“…In order to prove that these equations are equivalent to the equations defining the central configurations, we need the planar conditions [7], [8]…”

Section: Central Configurations Of 4 Massesmentioning

confidence: 99%

Central Configurations of Four Bodies with Gravitational Charges of Both Signs

Piña

Lonngi

2011

Qual. Theory Dyn. Syst.

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The Newtonian Four-Body Problem with positive and negative gravitational charges is studied from a physical point of view that considers inertial masses that are always positive and positive or negative charges in the gravitational interaction forces. The difference with respect to the differential equation of motion used in several mathematical papers is stressed. Formulation of the modified Dziobek equations is developed. We prove that four masses with one charge of different sign lead to a concave central configuration with the charge of different sign in the convex hull of the other three. For two charges of one sign and two charges of the opposite sign, we prove that the central configuration is convex with the two charges of the same sign on each diagonal. An algorithm used for computing the six distances between particles of a planar Four-Body central configuration, presented previously for positive gravitational masses, is adapted to compute physical central configurations with charges of different sign.

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Central configurations for the planar Newtonian four-body problem

Cited by 18 publications

References 17 publications

New stacked central configurations for the planar 5-body problem

New stacked central configurations for the planar 5-body problem

Planar N-body central configurations with a homogeneous potential

Central Configurations of Four Bodies with Gravitational Charges of Both Signs

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