Abstract
Orbital stability of quasiperiodic motions in the many dimensional autonomic hamiltonian systems is considered. Studied motions are supposed to be not far from equilibrium, the number of their basic frequencies may be not equal to the number of degrees of freedom, and the procedure of their construction is supposed to be converged. The stability problem is solved in the strict nonlinear mode.
Obtained results are used in the stability investigation of small plane motions near the lagrangian solutions of the three-dimensional circular restricted three-body problem. The values of parameters for which the plane motions are unstable have been found.
Резюме
Рассматривается задача орбитальной устойчивости условноперио дических движений в многоменных автономных гамильтоновых системах. Исследуемые движения предполагаются близкими к положению навновесия, число их базисных частот может не совпадать с числом степеней свободы, а процедира их построения считается сходящейся. Задача об устоьчивости решается в строгой нелинейной постановке.
Полученные резильтаты применяются при исследовании устойчивости малых плоских движений, близких лагранжевым решениям пространственной круговой лграниченной задачи трех тел. Найдены значения параметрпв, при которых плоские движения неустойчивы.
Similar content being viewed by others
References
Alfrend, K. T.: 1972,Celest, Mech. 5, 502.
Arnold, V. I.: 1983,Uspeki Mat. Nauk USSR 18, 13.
Arnold, V. I.: 1963,Uspeki Mat. Nauk USSR 18, 91.
Arnold, V. I. and Avez, A.: 1968,Ergodic Problems of Classical Mechanics, Academic Press, New York.
Birkhoff, G. D.: 1927,Dynamical Systems, Am. Math. Soc., Providence, R.I.
Brjuno, A. D.: 1971,Trans. Moscow Math. Soc. 25, 131.
Brjuno, A. D.: 1970,Math. Collection (USSR) 83, 273.
Brjuno, A. D.: 1973,On the Local Problems of Mechanics, Preprint No. 96, Publ. Inst. of Appl. Math. Acad. Sci., U.S.S.R., Moscow.
Brjuno, A. D.: 1974,Set of Analyticity of Normalization Transformation, Preprint No. 98, Publ. Inst. of Appl. Math. Acad. Sci., U.S.S.R., Moscow.
Brjuno, A. D.: 1975,Analitic Integral Sets, Preprint No. 86, Publ. Inst. of Appl. Math. Acad. Sci., U.S.S.R., Moscow.
Brjuno, A. D.: 1976,Normal Form in Nonlinear Problems, Preprint No. 18, Publ. Inst. of Appl. Math. Acad. Sci., U.S.S.R., Moscow.
Brjuno, A. D.: 1976,Rep. Acad. Sci., USSR 230, 257.
Bulgakov, B. V.: 1946,Appl. Math. Mech. 10, 273.
Duboshin, G. N.: 1964,Celestial Mechanics, Analytical and Qualitative Methods, Publ. ‘Nauka’, Moscow.
Deprit, A. and Deprit-Bartholome, A.: 1967,Astron. J. 72, 173.
Deprit, A. and Henrard, J.: 1970, pp. in G. E. O. Giacaglia (ed.),Periodic Orbits, Stability and Resonances, D. Reidel Publ. Co., Dordrecht, Holland, pp. 1–18.
Henrard, J.: 1970,Celest. Mech,1, 437.
Leontovich, A. M.: 1962,Rep. Acad. Sci. USSR 143, 525.
Lyapunov, A. M.: 1966,Stability of Motion, Academic Press, New York.
Markeev, A. P.: 1968,Appl. Math. Mech. 32, 738.
Markeev, A. P.: 1969,Appl. Math. Mech. 33, 112.
Markeev, A. P.: 1970,Appl. Math. Mech. 34, 997.
Markeev, A. P.: 1971,Soviet Astron. J. 48, 862.
Markeev, A. P.: 1973,Appl. Math. Mech. 37, 753.
Markeev, A. P.: 1974, on ‘Arnold's Diffusion’ inMany Dimensional Problem of Stability of Triangular Libration Points, Preprint No. 109, Publ. Inst. of Appl. Math. Acad. Sci., U.S.S.R., 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., U.S.S.R., Moscow.
Markeev, A. P. and Sokolsky, A. G.: 1976,Some Computer Algorithms of Normalization of Hamiltonian Systems, Preprint No. 31, Publ. Inst. of Appl. Math. Acad. Sci., U.S.S.R., 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, 58.
Markeev, A. P. and Sokolsky, A. G.: 1978,Bull. Inst. Theor. Astron., in press.
Melnikov, V. K.: 1965,Rep. Acad. Sci., USSR 165, 1245.
Melnikov, V. K.: 1968,Rep. Acad. Sci. USSR 181, 546.
Moser, J.: 1958,Comm. Pure Appl. Math. 11, 81.
Moser, J.: 1967,Math. Ann. 169, 136.
Moser, J.: 1968,Mem. Am. Math. Soc. 81, 1.
Nehoroshev, N. N.: 1973, addition to book Moser, J.: 1973,Lectures on Hamiltonian Systems, (in Russian), Publ. ‘Mir’, Moscow.
Schmidt, D. S.: 1974,Celest. Mech. 9, 81.
Sokolsky, A. G.: 1974,Appl. Math. Mech. 38, 791.
Sokolsky, A. G.: 1975,Appl. Math. Mech. 39, 366.
Sokolsky, A. G.: 1977,Appl. Math. Mech. 41, 24.
Hazin, L. G.: 1977,Appl. Math. Mech. 35, 423.
Vries, J. P.: 1966, Am. Math. Soc., Providence, R.I.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sokolsky, A.G. On the stability of small quasiperiodic motions in the hamiltonian systems. Celestial Mechanics 17, 373–394 (1978). https://doi.org/10.1007/BF01228958
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01228958