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

• Main Authors: Grebenikov, E.A., Gadomski, L., Prokopenya, A.N.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2007
• Online Access: http://dspace.nbuv.gov.ua/handle/123456789/7243

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