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The stability in a strict non-linear sense of a trivial relative equilibrium position in the classical and generalized versions of Sitnikov’s problem. (English. Russian original) Zbl 1282.70019
J. Appl. Math. Mech. 77, No. 2, 172-180 (2013); translation from Prikl. Mat. Mekh. 77, No. 2, 239-250 (2013).
Summary: Stability, in a strict non-linear sense, of a trivial relative equilibrium position is investigated in the classical and generalized versions of Sitnikov’s problem in the case of small eccentricities of the orbits of bodies of finite dimensions. In the classical version \((n = 2)\) of the problem, it is proved that there are no second-, third- and fourth-order resonances and a degenerate case. In the generalized version \((2 < n \leq 5 {\cdot} 10^5)\), it is proved that there are no second- and third-order resonances and a degenerate case. A fourth-order resonance occurs in versions of the problem in which the number of finite size bodies satisfies the inequality \(45000 \leq n \leq 62597\) and the orbital eccentricities \(e < 0.25\). Use of the Arnold-Moser and Markeyev theorems enables one to establish the Lyapunov stability of the trivial positions of relative equilibrium in the above-mentioned versions of Sitnikov’s problem.
70F07 Three-body problems
70K20 Stability for nonlinear problems in mechanics
Full Text: DOI
[1] Sitnikov, K. A., The existence of oscillatory motions in the three-body problem, Dokl Akad Nauk SSSR, 133, 2, 303-306, (1960)
[2] Kalas VO, Krasilnikov PS. Equilibrium stability in Sitnikov’s problem. In: Collection of Papers of the II Interindustry Youth Scientific-Technical Forum “Young People and Future Aviation and Cosmonautics-2010. Moscow:Mosk. Aviats Inst; 2010; 214-9.
[3] Prokopenya, A. N., Investigation of the stability of equilibrium solutions of an elliptic bounded many-body problem by methods of computer algebra, Mat Modelirovaniye, 18, 10, 102-112, (2006) · Zbl 1104.70009
[4] Arnold, V. I., The stability of the equilibrium position of a Hamiltonian system of ordinary differential equations in the general elliptic case, Dokl Akad Nauk SSSR, 137, 2, 255-257, (1961)
[5] Arnold, V. I.; Kozlov, V. V.; Neishtadt, A. I., Mathematical aspects of classical and celestial mechanics, (1997), Spinger Berlin · Zbl 0885.70001
[6] Markeyev, A. P., Libration points in celestial mechanics and space dynamics, (1978), Nauka Moscow
[7] Grebenikov Ye, A., The existence of exact symmetric solutions in the plane Newtonian many- body problem, Mat Modelirovaniye, 10, 8, 74-80, (1998) · Zbl 1189.70026
[8] Grebenikov Ye, A.; Prokopenya, A. N., Determination of the boundaries between the domains of stability and instability of the Hill’s equation, Nonlinear Oscillations, 6, 1, 42-51, (2003) · Zbl 1086.34530
[9] Tkhai, V. N., Periodic motions of a reversible second-order mechanical system. application to Sitnikov’s problem, J Appl Mat Mech, 60, 5, 734-753, (2006)
[10] Tkhai, V. N., Rectilinear motions of a particle in the field of a double star, Problems of Research on the Stability and Stabilization of Motion, Part1, 30-36, (2001), VTs Ross Akad Nauk Moscow
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