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Nonlinear Oscillations

, Volume 10, Issue 1, pp 62–77 | Cite as

Studying the stability of equilibrium solutions in a planar circular restricted four-body problem

  • E. A. Grebenikov
  • L. Gadomski
  • A. N. Prokopenya
Article

Abstract

The Newtonian circular restricted four-body problem is considered. We obtain nonlinear algebraic equations determining equilibrium solutions in the rotating frame and find six possible equilibrium configurations of the system. Studying the stability of equilibrium solutions, we prove that the radial equilibrium solutions are unstable, while the bisector equilibrium solutions are stable in Lyapunov’s sense if the mass parameter satisfies the conditions µ ∈ (0, µ0, where µ0 is a sufficiently small number, and µ ≠ µj, j = 1, 2, 3. We also prove that, for µ = µ1 and µ = µ3, the resonance conditions of the third order and the fourth order, respectively, are satisfied and, for these values of µ, the bisector equilibrium solutions are unstable and stable in Lyapunov’s sense, respectively. All symbolic and numerical calculations are done with the Mathematica computer algebra system.

Keywords

Hamiltonian System Equilibrium Solution Canonical Transformation Celestial Mechanic Circular Restricted Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • E. A. Grebenikov
    • 1
  • L. Gadomski
    • 2
  • A. N. Prokopenya
    • 3
  1. 1.Computing CenterRussian Academy of SciencesMoscowRussia
  2. 2.University of PodlasieSiedlcePoland
  3. 3.Brest State Technical UniversityBrestBelarus

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