Symbolic-Numerical Analysis of the Relative Equilibria Stability in the Planar Circular Restricted Four-Body Problem

Abstract

We study the stability of relative equilibrium positions in the planar circular restricted four-body problem formulated on the basis of the Euler collinear solution of the three-body problem. The stability problem is solved in a strict nonlinear formulation in the framework of the KAM theory. We obtained algebraic equations determining the equilibrium positions and showed that there are 18 different equilibrium configurations of the system for any values of the two system parameters μ _1, μ_2. Canonical transformation of Birkhoff’s type reducing the Hamiltonian of the system to the normal form is constructed in a general symbolic form. Combining symbolic and numerical calculations, we showed that only 6 equilibrium positions are stable in Lyapunov’s sense if parameters μ _1 and μ _2 are sufficiently small, and the corresponding points in the plane O\mu _1\mu _2 belong to the domain bounded by the second order resonant curve. It was shown also that the third order resonance results in instability of the equilibrium positions while in case of the fourth order resonance, either stability or instability can take place depending on the values of parameters μ _1 and μ _2. All relevant symbolic and numerical calculations are done with the aid of the computer algebra system Wolfram Mathematica.