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

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...

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Дата:	2007
Автори:	Grebenikov, E.A., Gadomski, L., Prokopenya, A.N.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2007
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/7243
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Цитувати:	Studying the stability of equilibrium solutions in the planar circular restricted four-body problem / E.A. Grebenikov, L. Gadomski, A.N. Prokopenya // Нелінійні коливання. — 2007. — Т. 10, № 1. — С. 66-82. — Бібліогр.: 17 назв. — англ.

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spelling	irk-123456789-72432010-03-29T12:01:16Z Studying the stability of equilibrium solutions in the planar circular restricted four-body problem Grebenikov, E.A. Gadomski, L. Prokopenya, A.N. 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. Розглядається кругова зрiзана проблема ньютонiвської динамiки для чотирьох тiл. Отримано нелiнiйнi алгебраїчнi рiвняння, що визначають рiвноважнi розв’язки вiдносно обертаючої системи вiдлiку, та знайдено шiсть рiвноважних конфiгурацiй системи. При вивченнi стiйкостi рiвноважних розв’язкiв доведено, що радiальнi рiвноважнi розв’язки є нестiйкими, проте бiсекторнi рiвноважнi розв’язки є стiйкими за Ляпуновим, якщо параметр маси μ належить (0, μ0), де μ0 - досить мале число, і μ ≠ μj , j = 1, 2, 3. Також доведено, що для μ = μ1 та μ = μ3 умови резонансу вiдповiдно третього та четвертого порядкiв виконано, iдля цих значень µ бiсекторнi рiвноважнi розв’язки є вiдповiдно нестiйкими та стiйкими за Ляпуновим. Усi символьнi та числовi обчислення виконано за допомогою системи комп’ютерної алгебри "Математика". 2007 Article Studying the stability of equilibrium solutions in the planar circular restricted four-body problem / E.A. Grebenikov, L. Gadomski, A.N. Prokopenya // Нелінійні коливання. — 2007. — Т. 10, № 1. — С. 66-82. — Бібліогр.: 17 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/7243 517.9 en Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	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.
format	Article
author	Grebenikov, E.A. Gadomski, L. Prokopenya, A.N.
spellingShingle	Grebenikov, E.A. Gadomski, L. Prokopenya, A.N. Studying the stability of equilibrium solutions in the planar circular restricted four-body problem
author_facet	Grebenikov, E.A. Gadomski, L. Prokopenya, A.N.
author_sort	Grebenikov, E.A.
title	Studying the stability of equilibrium solutions in the planar circular restricted four-body problem
title_short	Studying the stability of equilibrium solutions in the planar circular restricted four-body problem
title_full	Studying the stability of equilibrium solutions in the planar circular restricted four-body problem
title_fullStr	Studying the stability of equilibrium solutions in the planar circular restricted four-body problem
title_full_unstemmed	Studying the stability of equilibrium solutions in the planar circular restricted four-body problem
title_sort	studying the stability of equilibrium solutions in the planar circular restricted four-body problem
publisher	Інститут математики НАН України
publishDate	2007
url	http://dspace.nbuv.gov.ua/handle/123456789/7243
citation_txt	Studying the stability of equilibrium solutions in the planar circular restricted four-body problem / E.A. Grebenikov, L. Gadomski, A.N. Prokopenya // Нелінійні коливання. — 2007. — Т. 10, № 1. — С. 66-82. — Бібліогр.: 17 назв. — англ.
work_keys_str_mv	AT grebenikovea studyingthestabilityofequilibriumsolutionsintheplanarcircularrestrictedfourbodyproblem AT gadomskil studyingthestabilityofequilibriumsolutionsintheplanarcircularrestrictedfourbodyproblem AT prokopenyaan studyingthestabilityofequilibriumsolutionsintheplanarcircularrestrictedfourbodyproblem
first_indexed	2025-07-02T10:06:47Z
last_indexed	2025-07-02T10:06:47Z
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fulltext	UDC 517 . 9 STUDYING THE STABILITY OF EQUILIBRIUM SOLUTIONS IN THE PLANAR CIRCULAR RESTRICTED FOUR-BODY PROBLEM ВИВЧЕННЯ СТIЙКОСТI РIВНОВАЖНИХ РОЗВ’ЯЗКIВ У ПЛАНАРНIЙ КРУГОВIЙ ЗРIЗАНIЙ ПРОБЛЕМI ЧОТИРЬОХ ТIЛ E. A. Grebenikov Computing Center Rus. Acad. Sci. Vavilova Str., 40, 119991, Moscow, Russia e-mail: greben@ccas.ru L. Gadomski Univ. Podlasie May 3 Str., 54, 08-110, Siedlce, Poland e-mail: legad@ap.siedlce.pl A. N. Prokopenya Brest State Techn. Univ. Moskowskaya Str., 267, 224017, Brest, Belarus e-mail: prokopenya@brest.by 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, µ0), where µ0 is0 a sufficiently small number, and µ 6= 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. 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, µ0), де µ0 — досить мале число, i µ 6= µj , j = 1, 2, 3. Також доведено, що для µ = µ1 та µ = µ3 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”. Introduction. In the fifties-sixties of the last century one of the authors of the present paper had a happy occasion to attend lectures on celestial mechanics given by the outstanding sci- entist, bright representative of the world famous Ukrainian Mathematical School, academician Yuri Mitropolskii. Those lectures were devoted to analytical and qualitative investigations of c© E. A. Grebenikov, L. Gadomski, A. N. Prokopenya, 2007 66 ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1 STUDYING THE STABILITY OF EQUILIBRIUM SOLUTIONS IN THE PLANAR CIRCULAR RESTRICTED . . . 67 the famous problem in astronomy and classic dynamics, namely, the Newtonian problem of three and more bodies. It was noted that Gauss, together with Laplace, was one of the first who demonstrated efficiency of the averaging methods in developing the analytical theories for planets motion and investigated the general problem of integrability of differential equations in the Newtonian three-body problem on the basis of these methods. It followed from the lectures that Gauss proposed a method of constructing a general integral for the averaged systems of differential equations in the restricted three-body problem long before "the theorems on cyclic coordinates"appeared (in the planar case of the problem a single averaging in one fast variable is used while in the spacial case double averaging is necessary). General integrals for the mentioned differential equations are expressed via complex impli- cit functions of phase variables and here a problem, being topical for all times, arises, — how to construct a general solution of differential equations in celestial mechanics or how to obtain phase variables in the form of explicit functions of time? This means that it is necessary to develop constructive methods for obtaining coordinates as explicit functions of time if we are investigating a motion of some concrete object in asymptotically large time intervals. Remarkable lectures of Yu. Mitropolskii convinced us that constructive methods of nonli- near analysis were and will be in the future used for solving the problems of celestial mechanics and astrodynamics. And the present paper may be considered as an illustration of this statement. It is well-known that for the two-body problem, first stated and solved by Newton, a comple- te and general solution exist. Using this solution it is natural to study the motion of a third parti- cle of infinitesimal mass under the influence of the gravitational attraction of two massive parti- cles moving in circular or elliptic orbits about their center of mass. This model is a special case of the three-body problem and is known as the restricted three-body problem [1]. It is very interes- ting from the theoretical point of view and is widely used in astronomy and cosmic dyna- mics. Many eminent mathematicians have investigated it in the last three centuries and a great progress in this field has been achieved [2]. This problem stimulated development of new quali- tative, analytical and asymptotic methods for studying nonlinear Hamiltonian systems [3 – 5]. Nevertheless, the development of the stability theory of Hamiltonian systems is not completed yet and investigations in this field are very topical. In [6 – 8] a new class of the exact particular solutions in the planar Newtonian many-body problem was found. On this basis two new dynamical models were proposed which are known as Newtonian restricted many-body problems [9, 10]. They seem to be not integrable in general. Hence, similarly to the case of the restricted three-body problem, one can start their studying from seeking exact particular solutions of the equations of motion and investigating their stabi- lity. In the simplest case of four interacting particles it was shown [11] that there exist six equi- librium solutions but only two of them are stable in linear approximation. In the present paper the stability problem is resolved in a strict nonlinear formulation. It should be emphasized that this problem is very complicated and it can be solved only on the basis of the KAM-theory [12 – 14]. Besides, very cumbersome symbolic calculations are involved, which can be reasonably done only with a modern computer software. Here all symbolic and numeric calculations are done with the computer algebra system Mathematica [15]. Equations of motion and their equilibrium solutions. Let two particles P1, P2 having equal masses m move uniformly in a circular orbit about their common center of mass, where the third particle P0 of mass m0 rests. The orbit is situated in the Oxy plane of the barycentric inertial frame of reference and its center is in the origin. The particles attract each other according to ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1 68 E. A. GREBENIKOV, L. GADOMSKI, A. N. PROKOPENYA the Newtonian law of gravitation and form symmetric with respect to the origin configuration at any instant of time (see Fig. 1). The corresponding solution of the three-body problem is well-known [1] and angular velocity of the particles is given by dν dt ≡ ω0 = ( Gm0(4 + µ) 4r3 0 )1/2 , (1) where G is the gravity constant, ν is the polar angle, r0 is the radius of the circle and the mass parameter µ = m/m0. Fig. 1. Geometrical configuration of the system. Let us consider the motion of the particle P3 of infinitesimal mass m1 in the gravitational field generated by the particles P0, P1, P2. Denoting their polar coordinates by ρ, ϕ, we can write the Lagrangian of the system in the form L = m1 2 (ρ̇2 + ρ2ϕ̇2) + Gm1 ( m0 r + m r1 + m r2 ) , (2) where r = ρ, rj = √ ρ2 + ρ2 j − 2ρρj cos(ϕ− ϕj), j = 1, 2, are the distances between the particle P3 and the particles P0, P1, P2, respectively, and the dot denotes the derivative d/dt. Obviously, it is convenient to analyze the motion of the particle P3 in the frame of reference rotating about Oz axis with the angular velocity ω0. Taking into account (1), (2), using the polar angle ν = ω0t as a new independent variable and doing some standard transformations, we can easily obtain the Hamiltonian of the system, H = 1 2 ( p2 ρ + p2 ϕ ρ2 − 2pϕ ) − − 4 4 + µ ( 1 ρ + µ√ 1 + ρ2 − 2ρ cos ϕ + µ√ 1 + ρ2 + 2ρ cos ϕ ) , (3) ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1 STUDYING THE STABILITY OF EQUILIBRIUM SOLUTIONS IN THE PLANAR CIRCULAR RESTRICTED . . . 69 where pρ, pϕ are momenta canonically conjugated to the coordinates ρ, ϕ. Note that in the rotating frame, the particles P0, P1, P2 rest in the points (0, 0), (−1, 0), (1, 0), respectively. With the Hamiltonian (3) the equations of motion of the particle P3 can be written as dρ dν = pρ , dϕ dν = pϕ ρ2 − 1, dpρ dν = p2 ϕ ρ3 − 4 4 + µ ( 1 ρ2 + µ(ρ− cos ϕ) (1 + ρ2 − 2ρ cos ϕ)3/2 + µ(ρ + cos ϕ) (1 + ρ2 + 2ρ cos ϕ)3/2 ) , (4) dpϕ dν = 4µρ 4 + µ ( 1 (1 + ρ2 + 2ρ cos ϕ)3/2 − 1 (1 + ρ2 − 2ρ cos ϕ)3/2 ) sinϕ. The equilibrium solutions of system (4) are determined from the condition ρ = R = = const, ϕ = β = const, pρ = const, pϕ = const. In this case equations (4) take the form pρ = 0 , pϕ = R2, R− 4 (4 + µ)R2 − 4µ 4 + µ ( R− cos β (1 + R2 − 2R cos β)3/2 + R + cos β (1 + R2 + 2R cos β)3/2 ) = 0, (5) ( 1 (1 + R2 + 2R cos β)3/2 − 1 (1 + R2 − 2R cos β)3/2 ) sinβ = 0. One can readily see that the last equation of (5) is satisfied for any R if β = 0, π/2, π, 3π/2. In terms of [10], we’ll call the equilibrium positions located on the rays β = 0 and β = π the radial equilibrium solutions. The corresponding values of R are obtained as roots of the equation (4 + µ)R 4 = 1 R2 + µ (1 + R)2 − µ(1−R) \|1−R\|3 . (6) The equilibrium positions on the rays β = π/2 and β = 3π/2 will be called the bisector equili- brium solutions. They are determined as roots of the equation (4 + µ)R 4 = 1 R2 + 2Rµ (1 + R)3/2 . (7) Equations (6), (7) are nonlinear and can not be solved analytically. But with the system Mathematica we can find their roots numerically with arbitrary precision for any value of the parameter µ. In the case of µ = 0 equation (6) has only one root R = 1. But for any µ > 0 there is one root in the interval 0 < R1 < 1 and another one R2 > 1 (points N1, N3 and N2, N4 in Fig. 2). They tend to the limits R1 = 0 and R2 = 2, 39681, respectively, as µ → ∞. Equation (7) also has one root R = 1 (points S1, S2 in Fig. 2) if µ = 0 which tends to the limit R = √ 3 as µ → ∞. Note that this limit corresponds to the famous Lagrange triangular solution for the three-body problem [1, 2]. Thus, there exist six equilibrium positions of the particle P3 in the rotating frame. They determine circular trajectories of this particle in the barycentric inertial frame of reference and, hence, there exist six equilibrium solutions in the restricted four-body problem. ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1 70 E. A. GREBENIKOV, L. GADOMSKI, A. N. PROKOPENYA Fig. 2. Equilibrium positions of the partic- le P3 in the Oxy plane. Stability analysis in linear approximation. To study the stability of equilibrium solutions let us make the following canonical transformation: ρ → R + u, ϕ → β + γ R , Pρ → v, Pϕ → R(R + ω). (8) Then considering the functions u(ν), γ(ν), v(ν), ω(ν) as small perturbations of the equilibrium solutions, we can expand the Hamiltonian (3) in the Taylor series in powers of u, γ, v and ω and neglect all terms of the third and higher orders. The first term in the obtained expansi- on depends only on the independent variable ν and it may be omitted because it does not influence the equations of motion. The the first order term in perturbations is equal to zero as a consequence of the equations for equilibrium solutions (5) – (7). So we obtain the Hamiltonian in the form H2 = 1 2 ( v2 + ω2 − 4uω + au2 + bγ2 ) , (9) where a = 1 + 8µ 4 + µ ( 2 (R2 + 1)3/2 − 1 (1−R)3 − 1 (R + 1)3 ) , b = 8µ(3 + R2) (4 + µ)(1−R2)3 , (10) a = 1 + 8µ 4 + µ ( 2 (R2 + 1)3/2 − 1 (R− 1)3 − 1 (R + 1)3 ) , b = 8µ(1 + 3R2) (4 + µ)R(R2 − 1)3 (11) for the radial equilibrium solutions N1, N3 and N2, N4, respectively, and a = 1 + 24µ (4 + µ)(1 + R2)5/2 , b = − 24µ (4 + µ)(1 + R2)5/2 (12) in the case of bisector equilibrium solutions S1, S2. With the Hamiltonian (9) we obtain the equations of the disturbed motion in the form du dν = ∂H2 ∂v = v, dv dν = −∂H2 ∂u = 2ω − au, (13) dγ dν = ∂H2 ∂ω = ω − 2u, dω dν = −∂H2 ∂γ = −bγ. ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1 STUDYING THE STABILITY OF EQUILIBRIUM SOLUTIONS IN THE PLANAR CIRCULAR RESTRICTED . . . 71 One can readily see that we have obtained a system of four linear differential equations of the first order with constant coefficients. Behavior of its solutions is determined by the corresponding characteristic exponents which can easily be found and may be written in the form λ = ±iσ1,2, (14) where i is the imaginary unit (i2 = −1) and σ1,2 = 1√ 2 ( a + b± √ a2 + 16b− 2ab + b2 )1/2 . (15) Using the system Mathematica, we have investigated characteristic exponents (14) as functi- ons of the parameter µ numerically. It turned out that for all radial equilibrium solutions there is one characteristic exponent (14) with a positive real part for any µ from the interval 0 ≤ µ < < ∞. So, according to the Liapunov’s theorem on linearized stability [16], we can conclude that radial equilibrium solutions of the circular restricted problem of four bodies are unstable. The same is true for the bisector equilibrium solutions if the parameter µ is large enough. But our calculations have shown that there is the value µ0 = 0, 0853217 such that for 0 < µ < µ0 (16) characteristic exponents (14) are different pure imaginary numbers. Note that for such values of the parameter µ, expression (15) takes the form σ1,2 = 1√ 2 ( 1± √ 1 + 12b + 4b2 )1/2 , (17) where the parameter b is defined in (12) and the imaginary parts σ1, σ2 of the characteristic exponents satisfy the following inequalities: 0 < σ2 < 1√ 2 < σ1 < 1. (18) Thus, the bisector equilibrium solutions are stable in linear approximation if the parameter µ belongs to the interval (16). Normalizing the third order term in the Hamiltonian. It is well-known that the stabi- lity problem for a Hamiltonian system of differential equations belongs to the critical case, in Liapunov’s sense [16], and it can be resolved only in a strict nonlinear formulation. The most general approach for studying such systems is the Poincare method of normal forms that was used successfully in solving many problems of nonlinear mechanics [3]. According to this method we have to construct the Birkhoff canonical transformation [17] that reduces the Hami- ltonian function to some simplest form when the equations of motion can be solved. In a neigh- borhood of the bisector equilibrium, solutions the Hamiltonian (3) can be expanded in a Taylor series in powers of the perturbations u, γ, v and ω and can be represented in the form H = H2 + H3 + H4 + . . . , (19) ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1 72 E. A. GREBENIKOV, L. GADOMSKI, A. N. PROKOPENYA where the quadratic part H2 is defined in (9) and H3 = −6 + 2b + (6 + 7b)R2 6R(1 + R2) u3 + 3u2ω R − uω2 R − (2− 3R2)b 2R(1 + R2) uγ2, (20) H4 = 36(1 + R2)2 + b(8 + 21R2 + 48R4) 24R2(1 + R2)2 u4 + b 24 ( 4 R2 − 35 (1 + R2)2 ) γ4− − 4u3ω R2 + 3u2ω2 2R2 − (2− 21R2 + 12R4)b 4R2(1 + R2)2 u2γ2. (21) Thus, we have to normalize successively the terms H2, H3, . . . in the expansion (19). First of all, let us normalize the quadratic part H2 of the Hamiltonian. Using the algorithm proposed in [2], we construct a canonical transformation of the form u = 2c1p1 + 2c2p2, γ = −c1σ1(b + σ2 2) b q1 + c2σ2(b + σ2 1) b q2, (22) v = −2c1σ1q1 + 2c2σ2q2, ω = c1(b + σ2 2)p1 + c2(b + σ2 1)p2, where c1 = ( σ1 (σ2 1 − σ2 2)(3− b + σ2 1) )1/2 , c2 = ( σ2 (σ2 1 − σ2 2)(3− b + σ2 2) )1/2 . Then the quadratic part H2 of the Hamiltonian (3) takes a normal form, H2 = 1 2 ( σ1(p2 1 + q2 1)− σ2(p2 2 + q2 2) ) . (23) Here p1, q1 and p2, q2 are two pairs of canonically conjugated variables. Substituting (22) into (20) we can rewrite the third order term H3 in the form H3 = ∑ i+j+k+l=3 h (3) ijklq i 1q j 2p k 1p l 2, (24) ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1 STUDYING THE STABILITY OF EQUILIBRIUM SOLUTIONS IN THE PLANAR CIRCULAR RESTRICTED . . . 73 where all non-zero terms h (3) ijkl are given by h (3) 0030 = −2c3 1 3R ( 2b(R2 − 4) 1 + R2 − 3 + 3b2 + 6(2− b)σ2 1 + 3σ4 1 ) , h (3) 0003 = −2c3 2 3R ( 2b(R2 − 4) 1 + R2 − 3 + 3b2 + 6(2− b)σ2 2 + 3σ4 2 ) , h (3) 0012 = 2c1c 2 2 R ( 3 + 8b− 3b2 + 3R2 − 2bR2 − 3b2R2 1 + R2 + 4(b− 2)σ2 2 − σ4 2 + 2σ2 1(b− 2− σ2 2) ) , h (3) 0021 = 2c2 1c2 R ( 3 + 8b− 3b2 + 3R2 − 2bR2 − 3b2R2 1 + R2 + 2(b− 2)σ2 2 − σ4 1 + 2σ2 1(2b− 4− σ2 2) ) , (25) h (3) 2010 = c3 1(3R2 − 2)σ2 1(b + σ2 2) 2 bR(1 + R2) , h (3) 2001 = c2 1c2(3R2 − 2)σ2 1(b + σ2 2) 2 bR(1 + R2) , h (3) 0210 = c1c 2 2(3R2 − 2)σ2 2(b + σ2 1) 2 bR(1 + R2) , h (3) 0201 = c3 2(3R2 − 2)σ2 2(b + σ2 1) 2 bR(1 + R2) , h (3) 1110 = 2(2− 3R2)c2 1c2σ1σ2(b + σ2 1)(b + σ2 2) bR(1 + R2) , h (3) 1101 = 2(2− 3R2)c1c 2 2σ1σ2(b + σ2 1)(b + σ2 2) bR(1 + R2) . The second step in normalizing the Hamiltonian is to find a canonical transformation such that the third order term H3 would be cancelled. Following Birkhoff [17], we can try to find the corresponding generating function in the form S(p̃1, p̃2, q1, q2) = q1p̃1 + q2p̃2 + ∑ i+j+k+l=3 s (3) ijklq i 1q j 2p k 1p l 2. (26) Then the new momenta p̃1, p̃2 and the coordinates q̃1, q̃2 are determined by the following relati- onships: q̃1 = ∂S ∂p̃1 , q̃2 = ∂S ∂p̃2 , p1 = ∂S ∂q1 , p2 = ∂S ∂q2 . (27) These relationships are just equations with respect to the old canonical variables q1, q2, p1, p2 which are analytic functions in neighborhood of the point q̃1 = q̃2 = p̃1 = p̃2 = 0 if q̃1, q̃2, p̃1 and p̃2 are sufficiently small. Hence, substituting (26) into (27) and taking into account the ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1 74 E. A. GREBENIKOV, L. GADOMSKI, A. N. PROKOPENYA terms up to the second order in q̃1, q̃2, p̃1, p̃2, we can rewrite it in the form q1 = q̃1 − s (3) 0012p̃ 2 2 − 2s (3) 0021p̃1p̃2 − 3s (3) 0030p̃ 2 1 − s (3) 0111p̃2q̃2 − 2s (3) 0120p̃1q̃2 − s (3) 0210q̃ 2 2− − s (3) 1011p̃2q̃1 − 2s (3) 1020p̃1q̃1 − s (3) 1110q̃1q̃2 − s (3) 2010q̃ 2 1, q2 = q̃2 − 3s (3) 0003p̃ 2 2 − 2s (3) 0012p̃1p̃2 − s (3) 0021p̃ 2 1 − 2s (3) 0102p̃2q̃2 − s (3) 0111p̃1q̃2 − s (3) 0201q̃ 2 2− − 2s (3) 1002p̃2q̃1 − s (3) 1011p̃1q̃1 − s (3) 1101q̃1q̃2 − s (3) 2001q̃ 2 1, (28) p1 = p̃1 + s (3) 1002p̃ 2 2 + s (3) 1011p̃1p̃2 + s (3) 1020p̃ 2 1 + s (3) 1101p̃2q̃2 + s (3) 1110p̃1q̃2 + s (3) 1200q̃ 2 2+ + 2s (3) 2001p̃2q̃1 + 2s (3) 2010p̃1q̃1 + 2s (3) 2100q̃1q̃2 + 3s (3) 3000q̃ 2 1, p2 = p̃2 + s (3) 0102p̃ 2 2 + s (3) 0111p̃1p̃2 + s (3) 0120p̃ 2 1 + 2s (3) 0201p̃2q̃2 + 2s (3) 0210p̃1q̃2 + 3s (3) 0300q̃ 2 2+ + s (3) 1101p̃2q̃1 + s (3) 1110p̃1q̃1 + 2s (3) 1200q̃1q̃2 + s (3) 2100q̃ 2 1. Now we can substitute (28) into (19) and expand the Hamiltonian H in a Taylor series in powers of q̃1, q̃2, p̃1, p̃2. One can readily check that the second order term H̃2 in this expansion will have the form (23) while the third order term H̃3 will be a sum of twenty terms of the form h̃ (3) ijklq̃ i 1q̃ j 2p̃ k 1 p̃ l 2, i + j + k + l = 3. Coefficients h̃ (3) ijkl can be divided into three independent groups. The first group is h̃ (3) 0012 = h (3) 0012 + s (3) 1002σ1 − s (3) 0111σ2, h̃ (3) 0210 = h (3) 0210 + s (3) 1200σ1 + s (3) 0111σ2, (29) h̃ (3) 1101 = h (3) 1101 − s (3) 0111σ1 + 2s (3) 1002σ2 − 2s (3) 1200σ2 and corresponds to the coefficients of p̃1p̃ 2 2, p̃1q̃ 2 2 , q̃1q̃2p̃2 in the expression for H̃3. Coefficients of q̃2p̃1p̃2, q̃1p̃ 2 2, q̃1q̃ 2 2 in H̃3 form the second group and are given by h̃ (3) 0111 = s (3) 1101σ1 + 2s (3) 0012σ2 − 2s (3) 0210σ2, h̃ (3) 1002 = −s (3) 0012σ1 − s (3) 1101σ2, (30) h̃ (3) 1200 = −s (3) 0210σ1 + s (3) 1101σ2. ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1 STUDYING THE STABILITY OF EQUILIBRIUM SOLUTIONS IN THE PLANAR CIRCULAR RESTRICTED . . . 75 The rest fourteen coefficients h̃ (3) ijkl form the third group h̃ (3) 0003 = h (3) 0003 − s (3) 0102σ2, h̃ (3) 0021 = h (3) 0021 + s (3) 1011σ1 − s (3) 0120σ2, h̃ (3) 0030 = h (3) 0030 + s (3) 1020σ1, h̃ (3) 0102 = 3s (3) 0003σ2 − 2s (3) 0201σ2, h̃ (3) 0120 = s (3) 1110σ1 + s (3) 0021σ2, h̃ (3) 0201 = h (3) 0201 + 2s (3) 0102σ2 − 3s (3) 0300σ2, h̃ (3) 0300 = s (3) 0201σ2, h̃ (3) 1011 = −2s (3) 0021σ1 + 2s (3) 2001σ1 − s (3) 1110σ2, (31) h̃ (3) 1020 = −3s (3) 0030σ1 + 2s (3) 2010σ1, h̃ (3) 1110 = h (3) 1110 − 2s (3) 0120σ1 + 2s (3) 2100σ1 + s (3) 1011σ2, h̃ (3) 2001 = h (3) 2001 − s (3) 1011σ1 − s (3) 2100σ2, h̃ (3) 2010 = h (3) 2010 − 2s (3) 1020σ1 + 3s (3) 3000σ1, h̃ (3) 2100 = −s (3) 1110σ1 + s (3) 2001σ2, h̃ (3) 3000 = −s (3) 2010σ1. We can consider relationships (29) – (31) as the equations determining unknown coefficients s (3) ijkl of canonical transformation (28). Recall that we would like to find coefficients s (3) ijkl such that the third order term H̃3 cancells or all the coefficients h̃ (3) ijkl are equal to zero. One can readily see that the system of three equations (29) determines coefficients s (3) 0111, s (3) 1002, s (3) 1200. The determinant of this system is equal to σ1(4σ2 2 − σ2 1). Hence, if σ1 6= 0 and the condition σ1 ± 2σ2 6= 0 (32) is fulfilled, system (29) has a solution. In this case, we can set h̃ (3) ijkl = 0 in (29) and find s (3) 1002 = 1 σ3 1 − 4σ1σ2 2 (h(3) 1101σ1σ2 − h (3) 0012σ 2 1 + 2 ( h (3) 0012 + h (3) 0210)σ 2 2 ) , s (3) 0111 = 1 σ2 1 − 4σ2 2 ( h (3) 1101σ1 + 2(h(3) 0210 − h (3) 0012)σ2 ) , (33) s (3) 1200 = − 1 σ3 1 − 4σ1σ2 2 ( h (3) 1101σ1σ2 + h (3) 0210σ 2 1 − 2(h(3) 0012 + h (3) 0210)σ 2 2 ) . The determinant of system (30) for the coefficients s (3) 0012, s(3) 0210, s(3) 1101 is equal to σ1(σ2 1−4σ2 2). Again, supposing that conditions (32) are fulfilled and setting h̃ (3) ijkl = 0 in (30), we obtain s (3) 0012 = s (3) 0210 = s (3) 1101 = 0. (34) The determinant of system (31) is also easily found and equal to 81σ4 1σ 6 2(4σ2 1 − σ2 2) 2. If σ1 6= 0, σ2 6= 0 and 2σ1 ± σ2 6= 0, (35) ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1 76 E. A. GREBENIKOV, L. GADOMSKI, A. N. PROKOPENYA we can set h̃ (3) ijkl = 0 in (31) and find s (3) 0300 = 1 3σ2 (2h (3) 0003 + h (3) 0201), s (3) 0102 = 1 σ2 h (3) 0003, s (3) 1020 = − 1 σ1 h (3) 0030, s (3) 0120 = 1 4σ2 1σ2 − σ3 2 (h(3) 1110σ1σ2 − h (3) 0021σ 2 2 + 2(h(3) 0021 + h (3) 2001)σ 2 1), s (3) 1011 = 1 4σ2 1 − σ2 2 (h(3) 1110σ2 − 2(h(3) 0021 − h (3) 2001)σ1), (36) s (3) 2100 = − 1 4σ2 1σ2 − σ3 2 (h(3) 1110σ1σ2 + h (3) 2001σ 2 2 − 2(h(3) 0021 + h (3) 2001)σ 2 1), s (3) 3000 = − 1 3σ1 (2h (3) 0030 + h (3) 2010), s (3) 1110 = s (3) 2010 = 0, s (3) 0003 = s (3) 0030 = s (3) 0021 = s (3) 0201 = s (3) 2001 = 0. It should be emphasized that the coefficients s (3) ijkl have form (33), (34), (36) only if the condi- tions (32), (35) are fulfilled. Note that these inequalities mean the absence of the third order resonances in the system [3]. In this case there exists canonical transformation (28) with coeffi- cients (33), (34), (36) such that the third order term H̃3 in the Hamiltonian (19) is cancelled. Analyzing expressions (17), one can easily find that there is only one value of the parameter µ in the interval (16), namely, µ1 = 0, 0529422, where the condition σ1 − 2σ2 = 0 is fulfilled. Hence, with canonical transformation (28) the third order term in the expansion (19) of the Hamiltonian (3) is cancelled for any value of the parameter µ in the interval (16), except for µ = µ1. Now let us consider the case µ = µ1 when the third order resonance takes place. The determinant of system (31) is not equal to zero and, hence, we again obtain solution (36) if all coefficients h̃ (3) ijkl = 0. On the contrary, determinants of the systems (29), (30) become equal to zero. Nevertheless, system (30) has the same solution (34) for h̃ (3) ijkl = 0 because it doesn’t contain the coefficients h (3) ijkl. But we can not set h̃ (3) ijkl = 0 in system (29) because it will not have solutions at all. It means that in the case of the third order resonance we can not cancel all terms in H̃3 with canonical transformation (28). We can try only to reduce H̃3 to a form such that theorem of Markeev [2], for example, on the stability of Hamiltonian system under the third order resonance would be applied. Setting h̃ (3) 0012 = −h̃ (3) 0210 = −1 2 h̃ (3) 1101 = B 2 √ 2 ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1 STUDYING THE STABILITY OF EQUILIBRIUM SOLUTIONS IN THE PLANAR CIRCULAR RESTRICTED . . . 77 in system (29), we obtain its solution in the form s (3) 1002 = − 1 2σ2 ( h (3) 0012 + h (3) 0210 + 2σ2s (3) 1200 ) , s (3) 0111 = − 1 4σ2 ( h (3) 0012 + 3h (3) 0210 − h (3) 1101 + 8σ2s (3) 1200 ) , B = 1√ 2 ( h (3) 0012 − h (3) 0210 − h (3) 1101 ) , where s (3) 1200 is an arbitrary constant. Then the Hamiltonian (19) is reduced to the form H̃ = σ2(p̃2 1 + q̃2 1)− 1 2 σ2(p̃2 2 + q̃2 2) + B 2 √ 2 (p̃1p̃ 2 2 − p̃1q̃ 2 2 − 2p̃2q̃1q̃2) + H̃4 + . . . . (37) And using the standard canonical transformation q̃1 = √ 2τ1 sin ϕ1, p̃1 = √ 2τ1 cos ϕ1, (38) q̃2 = √ 2τ2 sin ϕ2, p̃2 = √ 2τ2 cos ϕ2, we rewrite (37) as H = 2σ2τ1 − σ2τ2 + Bτ2 √ τ1 cos(ϕ1 + 2ϕ2) + H∗ 4 (ϕ1, ϕ2, τ1, τ2) + . . . , (39) where B = −0, 365822 6= 0 for µ = µ1. Now on the basis of theorem of Markeev [2] we can make the following conclusion. Theorem 1. The bisector equilibrium solutions of the circular restricted problem of four bodies are unstable for the parameter µ = µ1 in the interval (16) when the third order resonance takes place. Normalizing the fourth order term in the Hamiltonian. Let us suppose further that µ ∈ ∈ (0, µ0) but µ 6= µ1. Substituting successively (22) and (28) with coefficients (33), (34), (36) into (19) and expanding it in the Taylor series in powers of q̃1, q̃2, p̃1, p̃2, we obtain a new Hamiltonian in the form H̃ = H̃2 + H̃4 + . . . , (40) where the second order term, H̃2 = 1 2 ( σ1(p̃2 1 + q̃2 1)− σ2(p̃2 2 + q̃2 2) ) , (41) is normalized, the third order term H̃3 is absent and the fourth order term H̃4 may be written as H̃4 = ∑ i+j+k+l=4 h̃ (4) ijklq̃ i 1q̃ j 2p̃ k 1 p̃ l 2. (42) ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1 78 E. A. GREBENIKOV, L. GADOMSKI, A. N. PROKOPENYA The sum (42) contains 19 non-zero terms but the expressions for the coefficients h̃ (4) ijkl are quite cumbersome and we do not write them here. Again we can try to find the function S(p∗1, p ∗ 2, q̃1, q̃2) = q̃1p ∗ 1 + q̃2p ∗ 2 + ∑ i+j+k+l=4 s (4) ijklq̃ i 1q̃ j 2p ∗k 1 p∗l2 , (43) generating a canonical transformation that reduces the fourth order term H̃4 to its simplest form. Then new momenta p∗1, p ∗ 2 and coordinates q∗1, q ∗ 2 are determined by q∗1 = ∂S ∂p∗1 , q∗2 = ∂S ∂p∗2 , p̃1 = ∂S ∂q̃1 , p̃2 = ∂S ∂q̃2 . (44) Resolving (44) with respect to the old canonical variables q̃1, q̃2, p̃1, p̃2 in a neighborhood of the point q∗1 = q∗2 = p∗1 = p∗2 = 0 and substituting the solution into (40), we expand the Hamiltonian H̃ in a Taylor series in powers of q∗1 , q∗2 , p∗1, p∗2. Obviously, the second order term H∗ 2 in this expansion again has the form (41), the third order term H∗ 3 is absent and the fourth order term H∗ 4 is a sum of 35 terms of the form h ∗(4) ijkl q̃ i 1q̃ j 2p̃ k 1 p̃ l 2, i + j + k + l = 4. The coefficients h ∗(4) ijkl are again divided into three independent groups. The first group corresponds to the coefficients of p∗1p ∗3 2 , q∗22 p∗1p ∗ 2, q∗1q ∗ 2p ∗2 2 , q∗1q ∗3 2 in the expression for H∗ 4 and is given by h ∗(4) 0013 = h̃ (4) 0013 + s (4) 1003σ1 − s (4) 0112σ2, h ∗(4) 0211 = h̃ (4) 0211 + s (4) 1201σ1 + 2s (4) 0112σ2 − 3s (4) 0310σ2, (45) h ∗(4) 1102 = h̃ (4) 1102 − s (4) 0112σ1 + 3s (4) 1003σ2 − 2s (4) 1201σ2, h ∗(4) 1300 = h̃ (4) 1300 − s (4) 0310σ1 + s (4) 1201σ2. Relationships (45) form a system of equations determining the coefficients s (4) 1003, s (4) 0112, s (4) 1201, s (4) 0310 in the expansion (43). The determinant of this system is equal to (σ2 1 − σ2 2)(σ 2 1 − 9σ2 2) and is not equal to zero if σ1 6= σ2 and σ1 ± 3σ2 6= 0. (46) Hence, if conditions (46) are fulfilled, which means the absence of the fourth order resonances, system (45) will have a nontrivial solution in the case of h ∗(4) ijkl = 0 and the corresponding terms in H∗ 4 can be cancelled. The second group consists of the coefficients of q∗2p ∗ 1p ∗2 2 , p∗1q ∗3 2 , q∗1p ∗3 2 , q∗1q ∗2 2 p∗2 in H∗ 4 and can ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1 STUDYING THE STABILITY OF EQUILIBRIUM SOLUTIONS IN THE PLANAR CIRCULAR RESTRICTED . . . 79 be written as h ∗(4) 0112 = s (4) 1102σ1 + 3s (4) 0013σ2 − 2s (4) 0211σ2, h ∗(4) 0310 = s (4) 1300σ1 + s (4) 0211σ2, h ∗(4) 1003 = −s (4) 0013σ1 − s (4) 1102σ2, h ∗(4) 1201 = −s (4) 0211σ1 + 2s (4) 1102σ2 − 3s (4) 1300σ2. This system is similar to equations (30) and has a trivial solution, s (4) 1102 = s (4) 0013 = s (4) 0211 = s (4) 1300 = 0, in the case h ∗(4) ijkl = 0 despite its determinant becoming equal to zero for σ1 ± 3σ2 = 0. The other twenty seven coefficients h ∗(4) ijkl in the expression H∗ 4 form the third group but we do not write it here. This group is just a system of equations determining the rest twenty seven coefficients s (4) ijkl in the expansion (43). It should be emphasized that the determinant of this system is equal to zero and it will not have any solutions in the case h ∗(4) ijkl = 0. Thus, the fourth order term H∗ 4 can not be cancelled entirely in the expansion (40). Nevertheless, analyzing the system, we have shown that if, in addition to inequalities (32), (35), (46), the conditions σ1 6= 0, σ2 6= 0, σ1 ± σ2 6= 0, 3σ1 ± σ2 6= 0, (47) are fulfilled, the fourth order term H̃4 is reduced to the form 7H∗ 4 = 1 4 ( c20(p∗21 + q∗21 )2 + c11(p∗21 + q∗21 )(p∗22 + q∗22 ) + c02(p∗22 + q∗22 )2 ) . (48) Then, using the standard canonical transformation (38), we rewrite the Hamiltonian (40) as H = σ1τ1 − σ2τ2 + c20τ 2 1 + c11τ1τ2 + c02τ 2 2 + H∗ 5 (ϕ1, ϕ2, τ1, τ2) + . . . . (49) Now the Arnold – Moser theorem [13, 14] can be applied; it states that in the case of absence of the resonances up to the fourth order, stability of the bisector equilibrium solutions depends on the value of the parameter f = c20σ 2 2 + c11σ1σ2 + c02σ 2 1. Since the expessions for the coefficients c20, c11, c02 are very complicated we do not write them here. But using the system Mathematica, we can easily plot the curve f = f(µ) in the Oµf plane (see Fig. 3). Now we see that there is only one point µ2 = 0, 0502039 in the interval (16) where f = 0. There are also two values of µ, namely, µ1 = 0, 0529423 and µ3 = 0, 0291011, corresponding to the cases of resonance of the third and the fourth orders, respectively. Hence, on the basis of the Arnold – Moser theorem we can make the following conclusion. ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1 80 E. A. GREBENIKOV, L. GADOMSKI, A. N. PROKOPENYA Fig. 3. Parameter f as a function of µ. Theorem 2. The bisector equilibrium solutions of the circular restricted problem of four bodies are stable in Liapunov’s sense for any values of the parameter µ from the interval (16), except for three points, namely, µ1, µ2 and µ3. Let us consider the case µ = µ3 when the fourth order resonance takes place, i.e., the condition σ1 − 3σ2 = 0 is fulfilled. Now the determinant of system (45) becomes equal to zero and has not any solutions for h ∗(4) ijkl = 0. It means that we can not cancel the terms h ∗(4) 0013p ∗ 1p ∗3 2 , h ∗(4) 0211q ∗2 2 p∗1p ∗ 2, h ∗(4) 1102q ∗ 1q ∗ 2p ∗2 2 , h ∗(4) 1300q ∗ 1q ∗3 2 in the expression for H∗ 4 . But we can try to reduce H∗ 4 to form such a that the theorem of Markeev [2] on the stability of a Hamiltonian system under the fourth order resonance would be applied. Setting h ∗(4) 0013 = −1 3 h ∗(4) 0211 = −1 3 h ∗(4) 1102 = h ∗(4) 1300 = B 4 in system (45), we obtain its solution in the form s (4) 1003 = − 1 24σ2 (9h̃ (4) 0013 + 3h̃ (4) 0211 − h̃ (4) 1102 − 3h̃ (4) 1300 + 8σ2s (4) 1201), s (4) 0112 = − 1 4σ2 (h̃(4) 0013 + h̃ (4) 0211 − h̃ (4) 1102 − h̃ (4) 1300 + 4σ2s (4) 1201), s (4) 0310 = − 1 24σ2 (h̃(4) 0013 − h̃ (4) 0211 − h̃ (4) 1102 − 7h̃ (4) 1300 − 8σ2s (4) 1201), B = 1 2 (h̃(4) 0013 − h̃ (4) 0211 − h̃ (4) 1102 + h̃ (4) 1300), ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1 STUDYING THE STABILITY OF EQUILIBRIUM SOLUTIONS IN THE PLANAR CIRCULAR RESTRICTED . . . 81 where s (4) 1201 is an arbitrary constant. Then the Hamiltonian (40) is reduced to the form H∗ = 3σ2 2 ( p∗21 + q∗21 ) − σ2 2 ( p∗22 + q∗22 ) + + 1 4 ( c20(p∗21 + q∗21 )2 + c11(p∗21 + q∗21 )(p∗22 + q∗22 ) + c02(p∗22 + q∗22 )2 ) + + B 4 ( p∗1p ∗3 2 − 3q∗22 p∗1p ∗ 2 − 3q∗1q ∗ 2p ∗2 2 + q∗1q ∗3 2 ) . (50) At last, making the transformation (38), we rewrite the Hamiltonian (50) as H = 3σ2τ1 − σ2τ2 + c20τ 2 1 + c11τ1τ2 + c02τ 2 2 + + Bτ2 √ τ1τ2 cos(ϕ1 + 3ϕ2) + H∗ 5 (ϕ1, ϕ2, τ1, τ2) + . . . . (51) According to Markeev’s theorem [2], stability of the equilibrium solutions under the fourth order resonance depends on the values of c20 + 3c11 + 9c02 and 3 √ 3B. Our calculations show that, for µ = µ3, c20 + 3c11 + 9c02 = 21, 4802, 3 √ 3B = 8, 99408 and, hence, c20 + 3c11 + 9c02 > 3 √ 3B. Thus, on the basis of Markeev’s theorem [2] we can make the following conclusion. Theorem 3. The bisector equilibrium solutions of the circular restricted problem of four bodies are stable in Liapunov’s sense for the parameter µ = µ3 from the interval (16) when the fourth order resonance takes place. Conclusion. In the present paper we have studied the stability of equilibrium solutions in the Newtonian circular restricted four-body problem in the strict nonlinear formulation. 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 belongs to the interval (16), except for two points µ = µj , j = 1, 2. We have showed that for µ = µ1 the bisector equilibri- um solutions become unstable because of the third order resonance. Nevertheless, for µ = µ3, when the fourth order resonance takes place, the bisector equilibrium solutions are stable in Liapunov’s sense. In the case µ = µ2, the parameter f becomes equal zero, which means that the third and the fourth order terms in expansion (40) are absent. Hence, in order to conclude on the stability of the bisector equilibrium solutions for µ = µ2, a further analysis of higher order terms in the Hamiltonian expansion is required. All symbolic and numerical calculations in the present paper are done with the computer algebra system Mathematica. ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1 82 E. A. GREBENIKOV, L. GADOMSKI, A. N. PROKOPENYA 1. Szebehely V. Theory of orbits. The restricted problem of three bodies (in Russian). — Moscow: Nauka, 1982. — 656 p. 2. Markeev A.P. The points of libration in celestial mechanics and cosmic dynamics (in Russian). — Moscow: Nauka, 1978. — 312 p. 3. Markeev A.P. Stability of the Hamiltonian systems // Nonlinear Mechanics / Eds V. M. Matrosov, V. V. Ru- myantsev, A. V. Karapetyan (in Russian). — Moscow: Fizmatlit, 2001. — P. 114 – 130. 4. Bogolyubov N. N., Mitropolskii Yu. A. Asymptotic methods in the theory of nonlinear oscillations (in Russi- an). — Moscow: Nauka, 1974. — 503 p. 5. Mitropolskii Yu. A. The averaging method in nonlinear mechanics (in Russian). — Kiev: Naukova Dumka, 1971. 6. Perko L. M., Walter E. L. Regular polygon solutions of the N -body problem // Proc. Amer. Math. Soc. — 1985. — 94. — P. 301 – 309. 7. Elmabsout B. Sur l’existence de certaines configurations d’equilibre relatif dans le probleme des N corps // Celest. Mech. and Dynam. Astron. — 1988. — 41. — P. 131 – 151. 8. Grebenikov E. A. New exact solutions in the plain symmetrical (n + 1)-body problem // Rom. Astron. J. — 1997. — 7. — P. 151 – 156. 9. Grebenikov E. A. Two new dynamical models in celestial mechanics // Ibid. — 1998. — 8. — P. 13 – 19. 10. Grebenikov E. A., Kozak-Skovorodkin D., Jakubiak M. The methods of computer algebra in the many-body problem (in Russian). — Moscow: RUDN, 2002. — 209 p. 11. Grebenikov E. A., Prokopenya A. N. Studying stability of the equilibrium solutions in the restricted Newton’s problem of four bodies // Bul. Acad. Sti. Republ. Moldova. Mat. — 2003. — № 2(42). — P. 28 – 36. 12. Kolmogorov A. N. On the conservation of quasi-periodic motions for a small change in the Hamiltonian function // Dokl. Akad. Nauk USSR. — 1954. — 98, № 4. — P. 527 – 530. 13. Arnold V. I. Small denominators and problems of stability of motion in classical and celestial mechanics // Uspekhi Math. Nauk. — 1963. — 18, № 6. — P. 91 – 192. 14. Moser J. Lectures on the Hamiltonian systems (in Russian). — Moscow: Mir, 1973. — 168 p. 15. Wolfram S. The mathematica book. — Cambridge: Univ. Press, 1996. — 1470 p. 16. Liapunov A. M. General problem about the stability of motion (in Russian). — Moscow: Gostekhizdat, 1950. — 471 p. 17. Birkhoff G. D. Dynamical systems (in Russian). — Moscow: Gostekhteorizdat, 1941. — 320 p. Received 30.10.2006 ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1

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