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Celestial mechanics

, Volume 17, Issue 4, pp 373–394 | Cite as

On the stability of small quasiperiodic motions in the hamiltonian systems

  • A. G. Sokolsky
Article

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.

Keywords

Hamiltonian System Plane Motion Stability Problem Basic Frequency Orbital Stability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

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References

  1. [1]
    Alfrend, K. T.: 1972,Celest, Mech. 5, 502.Google Scholar
  2. [2]
    Arnold, V. I.: 1983,Uspeki Mat. Nauk USSR 18, 13.Google Scholar
  3. [3]
    Arnold, V. I.: 1963,Uspeki Mat. Nauk USSR 18, 91.Google Scholar
  4. [4]
    Arnold, V. I. and Avez, A.: 1968,Ergodic Problems of Classical Mechanics, Academic Press, New York.Google Scholar
  5. [5]
    Birkhoff, G. D.: 1927,Dynamical Systems, Am. Math. Soc., Providence, R.I.Google Scholar
  6. [6]
    Brjuno, A. D.: 1971,Trans. Moscow Math. Soc. 25, 131.Google Scholar
  7. [7]
    Brjuno, A. D.: 1970,Math. Collection (USSR) 83, 273.Google Scholar
  8. [8]
    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.Google Scholar
  9. [9]
    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.Google Scholar
  10. [10]
    Brjuno, A. D.: 1975,Analitic Integral Sets, Preprint No. 86, Publ. Inst. of Appl. Math. Acad. Sci., U.S.S.R., Moscow.Google Scholar
  11. [11]
    Brjuno, A. D.: 1976,Normal Form in Nonlinear Problems, Preprint No. 18, Publ. Inst. of Appl. Math. Acad. Sci., U.S.S.R., Moscow.Google Scholar
  12. [12]
    Brjuno, A. D.: 1976,Rep. Acad. Sci., USSR 230, 257.Google Scholar
  13. [13]
    Bulgakov, B. V.: 1946,Appl. Math. Mech. 10, 273.Google Scholar
  14. [14]
    Duboshin, G. N.: 1964,Celestial Mechanics, Analytical and Qualitative Methods, Publ. ‘Nauka’, Moscow.Google Scholar
  15. [15]
    Deprit, A. and Deprit-Bartholome, A.: 1967,Astron. J. 72, 173.Google Scholar
  16. [16]
    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.Google Scholar
  17. [17]
    Henrard, J.: 1970,Celest. Mech,1, 437.Google Scholar
  18. [18]
    Leontovich, A. M.: 1962,Rep. Acad. Sci. USSR 143, 525.Google Scholar
  19. [19]
    Lyapunov, A. M.: 1966,Stability of Motion, Academic Press, New York.Google Scholar
  20. [20]
    Markeev, A. P.: 1968,Appl. Math. Mech. 32, 738.Google Scholar
  21. [21]
    Markeev, A. P.: 1969,Appl. Math. Mech. 33, 112.Google Scholar
  22. [22]
    Markeev, A. P.: 1970,Appl. Math. Mech. 34, 997.Google Scholar
  23. [23]
    Markeev, A. P.: 1971,Soviet Astron. J. 48, 862.Google Scholar
  24. [24]
    Markeev, A. P.: 1973,Appl. Math. Mech. 37, 753.Google Scholar
  25. [25]
    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.Google Scholar
  26. [26]
    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.Google Scholar
  27. [27]
    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.Google Scholar
  28. [28]
    Markeev, A. P. and Sokolsky, A. G.: 1977,Soviet Astron. J. 54, 4.Google Scholar
  29. [29]
    Markeev, A. P. and Sokolsky, A. G.: 1978,Appl. Math. Mech. 42, 58.Google Scholar
  30. [30]
    Markeev, A. P. and Sokolsky, A. G.: 1978,Bull. Inst. Theor. Astron., in press.Google Scholar
  31. [31]
    Melnikov, V. K.: 1965,Rep. Acad. Sci., USSR 165, 1245.Google Scholar
  32. [32]
    Melnikov, V. K.: 1968,Rep. Acad. Sci. USSR 181, 546.Google Scholar
  33. [33]
    Moser, J.: 1958,Comm. Pure Appl. Math. 11, 81.Google Scholar
  34. [34]
    Moser, J.: 1967,Math. Ann. 169, 136.Google Scholar
  35. [35]
    Moser, J.: 1968,Mem. Am. Math. Soc. 81, 1.Google Scholar
  36. [36]
    Nehoroshev, N. N.: 1973, addition to book Moser, J.: 1973,Lectures on Hamiltonian Systems, (in Russian), Publ. ‘Mir’, Moscow.Google Scholar
  37. [37]
    Schmidt, D. S.: 1974,Celest. Mech. 9, 81.Google Scholar
  38. [38]
    Sokolsky, A. G.: 1974,Appl. Math. Mech. 38, 791.Google Scholar
  39. [39]
    Sokolsky, A. G.: 1975,Appl. Math. Mech. 39, 366.Google Scholar
  40. [40]
    Sokolsky, A. G.: 1977,Appl. Math. Mech. 41, 24.Google Scholar
  41. [41]
    Hazin, L. G.: 1977,Appl. Math. Mech. 35, 423.Google Scholar
  42. [42]
    Vries, J. P.: 1966, Am. Math. Soc., Providence, R.I.Google Scholar

Copyright information

© D. Reidel Publishing Company 1978

Authors and Affiliations

  • A. G. Sokolsky
    • 1
  1. 1.Moscow Aircraft InstituteMoscowU.S.S.R.

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