On Co-Circular Central Configurations in the Four and Five Body-Problems for Homogeneous Force Law

  • Martha Alvarez-Ramírez
  • Alan Almeida Santos
  • Claudio Vidal
Article

Abstract

We study the central configurations (cc for short) for four masses arranged on a common circle (called co-circular cc) in two different situations, namely with no mass inside and later adding a fifth mass at the center of the circle. In the former, we focus the kite shape configurations by proving the existence of a one-parameter family of cc which goes from the kite containing an equilateral triangle up to the square shape. After, by putting a fifth mass at the center, we feature the planar cc of five bodies as a tensor of corange two see, “Albouy and Chenciner (Invent Math 131:151–184, 1998)” and we prove that cc is stacked see, “Hampton (Nonlinearity 18:2299–2304, 2005b)” in a such way that the center of mass of the four bodies should be the center of the circle. We emphasize that our approach includes not only the Newtonian force law, but the homogeneous ones with exponent \(a\le -1\).

Keywords

\(N\)-body problem Central configurations Co-circular configurations Homogeneous force law 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Martha Alvarez-Ramírez
    • 1
  • Alan Almeida Santos
    • 2
  • Claudio Vidal
    • 3
  1. 1.Departamento de MatemáticasUniversidad Autónoma Metropolitana-IztapalapaMexicoMexico
  2. 2.Departamento de Matemática, Campus Professor Alberto CarvalhoUniversidade Federal de SergipeItabaianaBrazil
  3. 3.Departamento de Matemática, Facultad de CienciasUniversidad del Bío BíoConcepcionChile

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