Abstract
The predictor-corrector method is described for numerically extending with respect to the parameters of the periodic solutions of a Lagrangian system, including recurrent solutions. The orbital stability in linear approximation is investigated simultaneously with its construction.
The method is applied to the investigation of periodic motions, generated from Lagrangian solutions of the circular restricted three body problem. Small short-period motions are extended in the plane problem with respect to the parameters h, µ (h = energy constant, µ = mass ratio of the two doninant gravitators); small vertical oscillations are extended in the three-dimensional problem with respect to the parameters h, µ. For both problems in parameter's plane h, µ domaines of existince and stability of derived periodic motions are constructed, resonance curves of third and fourth orders are distinguished.
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References
Birkhoff, G. D.: 1927, Dynamical Systems, Am. Math. Soc., Providence, R.I.
Brown, E. W.: 1911, Monthly Notices Roy. Astron. Soc. 71, 492.
Deprit, A.: 1966, Astron. J., 71.
Deprit, A. and Henrard, J.: 1968, Advances in Astronomy and Astrophysics, Academic Press, New-York; London, p. 3.
Deprit, A. and Henrard, J.: 1967, Astron. J., 72, 2.
Horn, J.: 1903, Z. Math. Phys., 48.
Karimov, S. R. and Sokolsky, A. G.: 1986, Letters to Soviet Astron. J., 12, 566.
Karimov, S. R. and Sokolsky, A. G.: 1987, The Method of Computer Extending with respect to the Parameters for Investigation of Periodic Motions of Many Dimensional Lagrangian Systems, Report of Soviet VINITI, No. 4487-1387.
Karimov, S. R. and Sokolsky, A. G.: 1987, The Periodic Motions in Three Body Problem, Generated from Space Librations in the Vicinity of Lagrangian Solutions, Report of Soviet VINITI, No. 6182-1387.
Lyapunov, A. M.: 1966, ‘About stability of motions in one participle case of three body problem’, in ‘Stability of Motion’, Academic Press, New York.
Lyapunov, A. M.: 1966, Stability of Motion, Academic Press, New York.
Markeev, A. P.: 1978, Libration Points in Celestial Mechanics and Cosmodynamics, Publ. ‘Nauka’, Moscow.
Markeev, A. P. and Sokolsky, A. G.: 1975, Investigation of Periodic Motions near the Lagrangian Solutions of Restricted Three Body Problem, Preprint No. 110, Publ. Inst. of Appl. Math. Acad. Sci., USSR, Moscow.
Markeev, A. P. and Sokolsky, A. G.: 1977, Soviet Astron. J. 54, 4.
Markeev, A. P. and Sokolsky, A. G.: 1978, Appl. Math. Mech. 42, 1.
Pedersen, P.: 1935, Monthly Notices Roy. Astron. Soc., 95.
Plummer, H. C.: 1901, Monthly Notices Roy. Astron. Soc., 62.
Puankare, A.: 1971, 1972, New Methods of Celestial Mechanics, Selected Works, vol. 1, 2, Publ. “Nauka”, Moscow.
Ryabov, U. A.: 1952, Soviet Astron. J. 29, 5.
Sharlier, C. L.: 1927, Die Mechanik des Himmels, Walter de Gryter and Co., Berlin and Leipzig.
Sokolsky, A. G.: 1978, Celest. Mech., 17, 4.
Sokolsky, A. G. and Khovansky, S. A.: 1983, Soviet Kozmic. Issled., 21, 6.
Sokolsky, A. G. and Khovansky, S. A.: 1986, Numerical Algorithm of Extending with Respect to the Parameters Periodic Motions of Hamiltonian System with Two Degrees of Freedom, Report of Soviet VINITI, No. 4042-1386.
Szebehely, V.: 1967, Theory of Orbits: The restricted Problem of Three Bodies, Academic Press, New York.
Wintner, A.: 1941, The Analytical Foundations of Celestial Mechanics, Princeton University Press, Princeton.
Yakubovich, V. A. and Starzhinskii, V. M.: 1975, Linear Differential Equations with Periodic Coefficients. v. I, II, J. Wiley, New York (in Russian, 1972).
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Karimov, S.R., Sokolsky, A.G. Periodic motions generated by Lagrangian solutions of the circular restricted three-body problem. Celestial Mech Dyn Astr 46, 335–381 (1989). https://doi.org/10.1007/BF00051487
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DOI: https://doi.org/10.1007/BF00051487