Central Configurations of the 5-Body Problem with Four Infinitesimal Particles
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Jian Chen
· Mingfang Yang
Central Configurations of the 5-Body Problem with Four Infinitesimal Particles
Received: 7 April 2020 / Accepted: 15 July 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract We study central configurations of 5-body problem with one dominant particle and four infinitesimal particles. In 2013, Oliveira (Celest Mech Dyn Astron 116:11–20, 2013) showed that the configurations are symmetric when two infinitesimal particles are diametrically opposite. Moreover, in the case of these two particles have the same mass he proved that the number of central configurations is one or two. In this paper, we provide criteria for the number of the central configurations in the general case where these two particles are unequal mass. Mathematics Subject Classification
70F10 · 70F15
1 Introduction For Newtonian N -body problem, the motion is governed by the following equations [11,12]: m k q¨k =
∂U (q) m k m j (q j − qk ) = , k = 1, 2, . . . , N ∂qk q j − qk 3
(1.1)
j =k
where qk = (xk , yk , z k ) ∈ R3 is the position of the kth particle with mass m k > 0 and U (q) is the Newtonian potential m m k j , q = (q1 , q2 , . . . , q N ) ∈ R3N U (q) = (1.2) q j − qk 1≤k< j≤N
A configuration q = (q1 , . . . , qn ) forms a central configuration if there exists some positive constant λ, called Lagrangian multiplier, such that − λ(qk − C) =
N j=1, j =k
m j (q j − qk ) , k = 1, 2, . . . , N |q j − qk |3
(1.3)
N where C = M1 k=1 m k qk is the center of masses which can be fixed at the origin in the inertial coordinate N system and M = k=1 m k is the total masses. In this work, we study the so-called 1 + N body problem where one particle is dominant and the other N particles are infinitesimal. This problem was first proposed by Maxwell [7] when he tried to construct a model for Saturns rings. J. Chen (B) · M. Yang School of Science, Southwest University of Science and Technology, Mianyang 621010, Sichuan, China
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J. Chen, M. Yang
In the case of the N infinitesimal particles have equal masses, Hall [6] proved that the dominant particle must be at the center of a circle which passes through the infinitesimal particles and he showed that the configuration which the infinitesimal particles are at the vertices of a regular polygon is the unique central configuration if N ≥ e27000 . Casasayas, Llibre and Nunes [2] improved this result to N ≥ e73 . Cors, Llibre and Olle´ [4] proved that there is only one central configuration if N ≥ 9 by numerical calculation and in the case of N = 4, they proved that there are only three symmetric central configurations. Albouy and Fu [1] proved that any central configurations of the 1 + 4 body problem must be symmetric. When the infinitesimal particles are not necessarily equal, Moeckel [8] provided a criterion for the linear stability of central configurations. Renner and Sicardy [10] studied the inverse problem and the linear stability. For N = 3, Corbera, Cors and Llibre [3] found two diffe
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