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This article is cited in 12 scientific papers (total in 12 papers)
Symmetry of the Restricted 4+1 Body Problem with Equal Masses
C. Vidala, A. A. Santosb a Departamento de Matemática, Facultad de Ciencias, Universidad del Bío Bío, Casilla 5-C, Concepción, VIII-Región, Chile
b Departamento de Matemática, Universidade Federal de Sergipe, Av. Marechal Rondon, s/n, Jardim Rosa Elze, São Cristóvão-SE, CEP. 49100-000, Brazil
Abstract:
We consider the problem of symmetry of the central configurations in the restricted 4+1 body problem when the four positive masses are equal and disposed in symmetric configurations, namely, on a line, at the vertices of a square, at the vertices of a equilateral triangle with a mass at the barycenter, and finally, at the vertices of a regular tetrahedron [1-3]. In these situations, we show that in order to form a non collinear central configuration of the restricted 4+1 body problem, the null mass must be on an axis of symmetry. In our approach, we will use as the main tool the quadratic forms introduced by A. Albouy and A. Chenciner [4]. Our arguments are general enough, so that we can consider the generalized Newtonian potential and even the logarithmic case. To get our results, we identify some properties of the Newtonian potential (in fact, of the function $\varphi(s) = -s^k$, with $k<0$) which are crucial in the proof of the symmetry.
Keywords:
$n$-body problem, central configurations, symmetry.
Received: 02.02.2006 Accepted: 11.11.2006
Citation:
C. Vidal, A. A. Santos, “Symmetry of the Restricted 4+1 Body Problem with Equal Masses”, Regul. Chaotic Dyn., 12:1 (2007), 27–38
Linking options:
https://www.mathnet.ru/eng/rcd609 https://www.mathnet.ru/eng/rcd/v12/i1/p27
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