Regular and Chaotic Dynamics

, Volume 12, Issue 1, pp 27–38 | Cite as

Symmetry of the restricted 4 + 1 body problem with equal masses

  • A. A. Santos
  • C. Vidal
Research Articles

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 ϕ(s) = −sk, with k < 0) which are crucial in the proof of the symmetry.

MSC2000 numbers

37C75 34D20 34A25 

Keywords

n-body problem central configurations symmetry 

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Copyright information

© Pleiades Publishing, Ltd. 2007

Authors and Affiliations

  • A. A. Santos
    • 1
  • C. Vidal
    • 2
  1. 1.Departamento de MatemáticaUniversidade Federal de SergipeSāo Cristóvāo-SE, CEPBrazil
  2. 2.Departamento de Maternática, Facultad de CienciasUniversidad del Bío BíoConcepción, VIII-RegiónChile

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