The second Lyapunov method is applied to the analysis of stability of triangular libration points in a three-dimensional restricted circular three-body problem. It is shown that the triangular libration points are unstable.
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M. Gascheau, “Exsamen d’une classe d’èquations differentielles et application à un cas particulier du probléme des trois corps,” Compt. Rend., 16, 393–394 (1843).
E. J. Routh, “On Laplace’s three particles, with a supplement on the stability of steady motion,” Proc. London Math. Soc., 6, 86–97 (1874).
V. I. Arnol’d, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of the Classical and Celestial Mechanics [in Russian], URSS, Moscow (2002).
A. N. Kolmogorov, “On the preservation of conditionally periodic motions for small variations of the Hamiltonian function,” Dokl. Akad. Nauk SSSR, 98, No. 4, 527–530 (1954).
V. I. Arnol’d, “Proof of the A. N. Kolmogorov theorem on the preservation of conditionally periodic motions for small variations of the Hamiltonian function,” Usp. Mat. Nauk, 18, Issue 5, 13–40 (1963).
J. Moser, “On invariant curves of area-preserving mappings of an annulus,” Nachr. Akad. Wiss. Göttingen. II Math-Phys. K1, 1, 1–20 (1962).
A. M. Leontovich, “On the stability of Lagrangian periodic solutions in a restricted three-body problem,” Dokl. Akad. Nauk SSSR, 143, No. 3, 525–528 (1962).
A. Deprit and A. Deprit-Bartholomé, “Stability of the triangular Lagrangian points,” Astron. J., 72, No. 2, 173–179 (1967).
A. P. Markeev, Libration Points in Celestial Mechanics and Cosmodynamics [in Russian], Nauka, Moscow (1978).
A. G. Sokol’skii, “Proof of the stability of Lagrangian solutions for the critical ratio of masses,” Pis’ma Astron. Zh., 4, No. 3, 148–152 (1978).
V. Szebehely, Theory of Orbits. The Restricted Problem of Three Bodies, Academic Press, New York (1967).
A. E. Roy, Orbital Motion, Wiley, New York (1979).
S. P. Sosnitskii, “On the stability of triangular Lagrangian points in the restricted three-body problem,” Astron. J., 135, No. 1, 187–195 (2008).
A. M. Lyapunov, “General problem of the stability of motion,” in: Collection of Works [in Russian], Vol. 2, Izd. Akad. Nauk SSSR, Moscow (1956), pp. 7–263.
V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations [in Russian], Gostekhteorizdat, Moscow–Leningrad (1949).
G. D. Birkhoff, Dynamical Systems, American Mathematical Society Colloquium Publ. (1926).
R. Douady and P. Le Galvez, “Example de point fixe elliptique non topologiquement stable en dimension 4,” C. R. Acad. Sci. A, 296, No. 21 895–898 (1983).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 11, pp. 1557–1563, November, 2009.
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Sosnyts’kyi, S.P. On one atypical scheme of application of the second Lyapunov method. Ukr Math J 61, 1830–1838 (2009). https://doi.org/10.1007/s11253-010-0315-9
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DOI: https://doi.org/10.1007/s11253-010-0315-9