Planar symmetric concave central configurations in Newtonian four-body problems

Deng, Chunhua; Zhang, Shiqing

doi:10.1016/j.geomphys.2014.05.016

Cited by 7 publications

(6 citation statements)

References 24 publications

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“…Consider four positive point masses m 0 , m 1 , m 2 , and m 3 having position vectors r i and interbody distances r ij . For a general four-body setup, equation (1) gives the following six central configuration equations when n � 4:…”

Section: Equations Of Motionmentioning

confidence: 99%

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Central Configurations and Action Minimizing Orbits in Kite Four-Body Problem

Benhammouda

Mansur

Shoaib

et al. 2020

Advances in Astronomy

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In the current article, we study the kite four-body problems with the goal of identifying global regions in the mass parameter space which admits a corresponding central configuration of the four masses. We consider two different types of symmetrical configurations. In each of the two cases, the existence of a continuous family of central configurations for positive masses is shown. We address the dynamical aspect of periodic solutions in the settings considered and show that the minimizers of the classical action functional restricted to the homographic solutions are the Keplerian elliptical solutions. Finally, we provide numerical explorations via Poincaré cross-sections, to show the existence of periodic and quasiperiodic solutions within the broader dynamical context of the four-body problem.

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Section: Equations Of Motionmentioning

confidence: 99%

“…To understand the dynamics presented by a total collision of the masses or the equilibrium state of a rotating system, we are led to the concept of central configurations. A configuration of n bodies is central if the acceleration of each body is a scalar multiple of its position [1][2][3][4]. Let r i ∈ R 2 and m i , i � 1, .…”

Section: Introductionmentioning

confidence: 99%

Central Configurations and Action Minimizing Orbits in Kite Four-Body Problem

Benhammouda

Mansur

Shoaib

et al. 2020

Advances in Astronomy

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“…It can be used to fnd simple or special solutions to the n-body problem since the geometry formed by the arrangement of the primaries remains constant for all time (cf. Saari [1]; Moeckel [2]; Farantos [3]; Deng and Zhang [4]; MacMillan and Bartky [5]; Sim [6]; Llibre and Mello [7]; Papadakis and Kanavos [8]).…”

Section: Introductionmentioning

confidence: 99%

Restricted Concave Kite Five-Body Problem

Kashif

Shoaib²

2023

Advances in Astronomy

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The restricted concave kite five-body problem is a problem in which four positive masses, called the primaries, rotate in the concave kite configuration with a mass at the center of the triangle formed by three of the primaries. The fifth body has negligible mass and does not influence the motion of the four primaries. It is assumed that the fifth mass is in the same plane of the primaries and that the masses of the primaries are m 1 , m 2 , m 3 , and m 4 , respectively. Three different types of concave kite configurations are considered based on the masses of the primaries. In case I, one pair of primaries has equal masses; in case II, two pairs of primaries have equal masses; in case III, three of the primaries have equal masses. For all three cases, the regions of central configuration are obtained using both analytical and numerical techniques. The existence and uniqueness of equilibrium positions of the infinitesimal mass are investigated in the gravitational field of the four primaries. It is numerically confirmed that none of the equilibrium points are linearly stable. The Jacobian constant C is used to investigate the regions of possible motion of the infinitesimal mass.

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“…Albouy, Fu, and Su provided the necessary and sufficient condition for a planar convex four-body central configuration be symmetric with respect to one of its diagonals [2]. Problems involving existence or enumeration of symmetric central configurations satisfying some geometrical constraints were considered for many researchers (See for instance [4], [6], [10], [16], [22], and [29]). Montaldi proved that there is a central configuration for every choice of a symmetry type and symmetric choice of mass [20].…”

Section: Introductionmentioning

confidence: 99%

Generic Finiteness for a Class of Symmetric Planar Central Configurations of the Six-Body Problem and the Six-Vortex Problem

Dias

Pan

2019

J Dyn Diff Equat

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A symmetric planar central configuration of the Newtonian six-body problem x is called cross central configuration if there are precisely four bodies on a symmetry line of x. We use complex algebraic geometry and Groebner basis theory to prove that for a generic choice of positive real masses m1, m2, m3, m4, m5 = m6 there is a finite number of cross central configurations. We also show one explicit example of a configuration in this class. A part of our approach is based on relaxing the output of the Groebner basis computations. This procedure allows us to obtain upper bounds for the dimension of an algebraic variety. We get the same results considering cross central configurations of the six-vortex problem.

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Planar symmetric concave central configurations in Newtonian four-body problems

Cited by 7 publications

References 24 publications

Central Configurations and Action Minimizing Orbits in Kite Four-Body Problem

Central Configurations and Action Minimizing Orbits in Kite Four-Body Problem

Restricted Concave Kite Five-Body Problem

Generic Finiteness for a Class of Symmetric Planar Central Configurations of the Six-Body Problem and the Six-Vortex Problem

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