Nonlinear Stability Analysis of Equilibrium Solutions in Restricted Four-body Problem

It is well-known that the stability problem of relative equilibria in the N-body problem is one the most difficult and intricate one. The solution of such problem in the case of planar circular restricted three-body problem has been fully obtained only after 200 years while the triangular equilateral configuration was discovered in 1772 by Lagrange. Stability analysis in strict nonlinear formulation assumes certain steps while studying equilibrium solutions of autonomous Hamiltonian system of differential equations with two degrees of freedom [1]. One of important and most labour-intensive tasks is a reduction of Hamiltonian system to normal form what allows to apply theorems on stability from KAM-theory [1] in the end. This paper is devoted to the circular restricted four-body problem, formulated on the basis of Lagrange’s triangular solutions [2]. We have constructed the chain of canonical changes of variables based on generating function approach, known as Birkhoff’s transformation [3], in order to obtain the normal form up to the sixth order of Hamiltonian function that describes the motion of infinitesimal body in the considered restricted four-body problem. Corresponding theorems on Lyapunov stability of equilibrium solutions in considered problem have been proved. The influence of resonances on the stability of the system have been examined in detail. All necessary numeric computations and symbolic transformations are done with computer algebra system Mathematica.