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

Abstract

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 satisfies µ ∈ (0, µ 0 ), where µ 0 is0 a sufficiently small number, and µ = µ j , j = = 1, 2, 3. We have also proved that for µ = µ 1 and µ = µ 3 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.Розглядається кругова зрiзана проблема ньютонiвської динамiки для чотирьох тiл. Отримано нелiнiйнi алгебраїчнi рiвняння, що визначають рiвноважнi розв'язки вiдносно обертаючої сис-теми вiдлiку, та знайдено шiсть рiвноважних конфiгурацiй системи. При вивченнi стiйкостi рiвноважних розв'язкiв доведено, що радiальнi рiвноважнi розв'язки є нестiйкими, проте бiсек-торнi рiвноважнi розв'язки є стiйкими за Ляпуновим, якщо параметр маси µ ∈ (0, µ 0 ), де µ 0 -досить мале число, i µ = µ j , j = 1, 2, 3. Також доведено, що для µ = µ 1 та µ = µ 3 умови ре-зонансу вiдповiдно третього та четвертого порядкiв виконано, i для цих значень µ бiсекторнi рiвноважнi розв'язки є вiдповiдно нестiйкими та стiйкими за Ляпуновим. Усi символьнi та чис-ловi обчислення виконано за допомогою системи комп'ютерної алгебри "Математика" .

Stability of the planar equilibrium solutions of a restricted 1 + N body problem

“…The case μ d where the fourth-order terms cannot decide the stability is discussed in Section 5 together with the special cases at μ 3 , μ 2 and μ 1 . When N = 2, our formulas simplify further and we compare our findings with those of Grebenikov et al in [3] in most sections.…”

“…Our investigation started with trying to verify and extend the work of Grebenikov. We discovered that the main facts given in [3] are correct, but that they have an error in their symbolic computations as the graph in their Fig. 3 does not match ours.…”

“…al, [3], who tried to show with the help of KAM theory that a linearly stable equilibrium point is also stable in the Lyapunov sense. Our investigation started with trying to verify and extend the work of Grebenikov.…”

“…We use the Lie transformation of Deprit, which is more easily automated and less error prone than the Birkhoff transformations, which were used in [3]. Since a Birkhoff transformation is obtained from a generating function, which mixes old and new variables, one has to solve first a set of equations to get the desired transformation.…”

“…We will not follow the notation used in [3] but instead employ a more meaningful and common notation. For us μ r refers to the value of μ where we have r : 1 resonance in the linearized system.…”