Relative equilibrium solutions in the four body problem

Simó, Carles

doi:10.1007/bf01228714

Cited by 128 publications

(92 citation statements)

References 11 publications

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“…We refer to this as the restricted four body problem (R4BP). There exist some preliminary studies of this problem in different versions, Simó (1978), Leandro (2006), Pedersen (1944) and Baltagiannis & Papadakis (2011b) studied the equilibrium points and their stability of this problem. Other authors have studied the case where the primaries form a collinear configuration.…”

Section: Accepted For Publication In Astrophysics and Space Sciencementioning

confidence: 99%

Periodic orbits in the restricted four-body problem with two equal masses

Burgos-Garcia¹,

Delgado²

2013

Astrophys Space Sci

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The restricted (equilateral) four-body problem consists of three bodies of masses m 1 , m 2 and m 3 (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral triangle in a rotating coordinate system. A massless fourth body moves under the Newtonian gravitation law due to the three primaries; as in the restricted three-body problem (R3BP), the fourth mass does not affect the motion of the three primaries. In this paper we explore symmetric periodic orbits of the restricted four-body problem (R4BP) for the case of two equal masses where they satisfy approximately the Routh's critical value. We will classify them in nine families of periodic orbits. We offer an exhaustive study of each family and the stability of each of them.

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Section: Accepted For Publication In Astrophysics and Space Sciencementioning

confidence: 99%

Periodic orbits in the restricted four-body problem with two equal masses

Burgos-Garcia¹,

Delgado²

2013

Astrophys Space Sci

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“…In 1975, Palmore [25] gave a simple example of a degenerate central configuration, an equilateral triangle of bodies with unit mass and a fourth body with mass (64 √ 3+81)/249 at the center of the triangle. Simó [28] showed how the number of 4-body central configurations with a given body in the interior of the triangle formed by the other bodies varies when the four masses vary. Xia [34] proved that in the n-body problem, the relative equilibria may be counted exactly for several 1 [5] was published a few weeks after the end of first world war.…”

Section: Introduction and Statementsmentioning

confidence: 99%

Finiteness of central configurations of five bodies in the plane

2012

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“…A numerical study for the number of central configurations for N = 4 and arbitrary masses was done by Simó in [39]. In a computer assisted proof Hampton and Moeckel [21] proved the finiteness of the number of central configurations for N = 4 and any choice of the masses.…”

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confidence: 99%

Bifurcation of Relative Equilibria of the (1+3)-Body Problem

Corbera

Cors

Llibre

et al. 2015

SIAM J. Math. Anal.

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Abstract. We study the relative equilibria of the limit case of the planar Newtonian 4-body problem when three masses tend to zero, the so-called (1+3)-body problem. Depending on the values of the infinitesimal masses the number of relative equilibria varies from ten to fourteen. Always six of these relative equilibria are convex and the others are concave. Each convex relative equilibrium of the (1+3)-body problem can be continued to a unique family of relative equilibria of the general 4-body problem when three of the masses are sufficiently small and every convex relative equilibrium for these masses belongs to one of these six families.

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Relative equilibrium solutions in the four body problem

Cited by 128 publications

References 11 publications

Periodic orbits in the restricted four-body problem with two equal masses

Periodic orbits in the restricted four-body problem with two equal masses

Finiteness of central configurations of five bodies in the plane

Bifurcation of Relative Equilibria of the (1+3)-Body Problem

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