Celestial mechanics

, Volume 24, Issue 4, pp 407–429 | Cite as

Gylden-Meščerskii problem

  • L. M. Berković
Article

Abstract

Classical non-stationary two-body problem, described by the equation of the form
$$\ddot r = - \mu (t)\frac{r}{{r^3 }}$$
is investigated using differential equation transformation methods developed by the author.

All laws of mass variation for which Gylden-Meščerskii problem is reduced to autonomous form are stated. The problem symmetry properties are investigated and reviews of integrable cases from the group point of view are made.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ambartsumjan, V. A.: 1952,Trans. I.A.U. 8, 665.Google Scholar
  2. Armellini, G.: 1916,Mem. Soc. ital. sci. (dei XL),19, 75–96.Google Scholar
  3. Berković, L. M.: 1971,Differencial'nye Uravnenija 7, No. 2, 353–356 (Russian).Google Scholar
  4. Berković, L. M.: 1978,The Transformation of Ordinary Linear Differential Equations, Kuybyshev State University, Kuybyshev, 92 p. (Russian).Google Scholar
  5. Berković, L. M.: 1979,Prikl. Mat. Meh. 43, No. 4, 629–638 (Russian).Google Scholar
  6. Berković, L. M.: 1980a,Dokl. Akad. Nauk SSSR 250, No. 5 (Russian).Google Scholar
  7. Berković, L. M.: 1980b,Prikl. Mat. Meh. 44, No. 2 (Russian).Google Scholar
  8. Berković, L. M. and Gelfgat, B. E.: 1972, inThe Second Chetayev Conference on Analytical Mechanics, Motion Stability and Optimal Control, Kazan, p. 4 (Russian).Google Scholar
  9. Berković, L. M. and Gelfgat, B. E.: 1975, InThe Problems of Analytical Mechanics, Stability Theory and Control, Nauka, Moscow, pp. 54–61 (Russian).Google Scholar
  10. Berković, L. M. and Netchayevskii, M. L.: 1976, inDifferential Equations, Kuybyshev State University, Kuybyshev, pp. 3–18 (Russian).Google Scholar
  11. Berković, L. M. and Netchayevskii, M. L.: 1979, inStability Theory and Its Application, Nauka, Novisibirsk, pp. 163–172.Google Scholar
  12. Besge, M.: 1844,J. de Math. pures et appl. 1 sér. No. 9.Google Scholar
  13. Dirac, P. A. M.: 1937,Nature, No. 3512, p. 323.Google Scholar
  14. Dirac, P. A. M.: 1938,Roy. Soc. Proc. Lond. 165A, No. 921, 199–208.Google Scholar
  15. Dommanget, J.: 1963,Ann. Observ. roy. Belg. 9, No. 5 (French).Google Scholar
  16. Duboshin, G. N.: 1925,Astron. J. 2, No. 4, 5–11, 138–151 (French).Google Scholar
  17. Duboshin, G. N.: 1928,Astron. J. 5, Nos. 2–3, (Russian, French summary)}.Google Scholar
  18. Duboshin, G. N.: 1930,Astron. J. 7, Nos. 3–4, 153–172 (Russian, French summary).Google Scholar
  19. Duboshin, G. N.: 1978,Celestial Mechanics. Analytical and Qualitative Methods. Nauka, Moscow (Russian).Google Scholar
  20. Ermakov, V. P.: 1880,Univ. Izv. Kiev 20, No. 9, 1–25 (Russian).Google Scholar
  21. Fessenkov, V. G.: 1952,Trans. I.A.U. 8, 702.Google Scholar
  22. Gelfgat, B. E.: 1959,Bull. Inst. Teor. Astr. AN SSSR, Leningrad,7, No. 5, 354–362 (Russian, German summary).Google Scholar
  23. Gelfgat, B. E.: 1968, InProc. Third K. E. Tsiolkovskii Reading, Sec.Space Flight Mechanics. Znanye, Moscow, pp 86–101 (Russian).Google Scholar
  24. Grossman, I. and Magnus, W.: 1964,Groups and their graphs, Random House, The L. W. Singer Company.Google Scholar
  25. Gylden, H.: 1889,Astron. Nachr. 109, No. 2593.Google Scholar
  26. Hadjidemetriou, J.: 1967, Advances in Astronomy and Astrophysics, No. 5, Academic Press, New York-London.Google Scholar
  27. Jeans, J. H.: 1920,Monthly Notices Roy. Astron. Soc. 85, No. 1.Google Scholar
  28. Kummer, E.: 1834, Abdruck sus dem Program des evangelischen Königl und Stadgymnasium in Liegnitz von Jahre, reprint:J. Reine Angew. Math. 1877100, 1–9.Google Scholar
  29. Lapin, A. S.: 1944,Uč. zap Leningrad Univ. ser. mat. nauk, No. 13, 3–55 (Russian).Google Scholar
  30. Lie, Sophus: 1893,Vorlesungen über continuerliche Gruppen, Teubner, Leipzig.Google Scholar
  31. Liouville, J.: 1837,J. de Math. Pures et appl.,2, 16–36.Google Scholar
  32. Lovett, E. O.: 1902,Astron. Nachr. 158, No. 3790.Google Scholar
  33. McMillan, W. D.: 1925,Monthly Notices Roy, Astron. Soc. 85, No. 9, 909–912.Google Scholar
  34. Meščerskii, I. V.: 1893,Astron. Nachr. 132, No. 3153.Google Scholar
  35. Meščerskii, I. V.: 1897Mass Variable Particle Dynamics, St. Petersburg University (Russian).Google Scholar
  36. Meščerskii, I. V.: 1902,Astron. Nachr. 159, No. 3807, 229–242.Google Scholar
  37. Meščerskii, I. V.: 1949,Works on Variable Mass Body Mechanics, GITTL, Moscow-Leningrad (Russian).Google Scholar
  38. Mikhailov, G. K.: 1974,On the History of Variable Composition Systems Dynamics and Reactive Motion Theory (till World War II), Inst. Mech. Problems, Acad. Sci USSR. preprint No. 49, Moscow (Russian).Google Scholar
  39. Mikhailov, G. K.: 1975,Trudy Mosk. Inst. Chim. Mashinostroyeniya, No. 65, 122–133 (Russian).Google Scholar
  40. Nechvile, V.: 1926,C. R. Achad. Sci. de Paris,v, 182.Google Scholar
  41. Niţa M. M.: 1958,Studii şi cercetärii mec. apl. Achad. RPR,9, No. 2.Google Scholar
  42. Niţâ, M. M.: 1973,Teoria Zborului sputiul, Bucareşti, Ed. Acad. Rep. Soc. Romania.Google Scholar
  43. Omarov, T. B.: 1975,Galaxy of Galaxies Gravitating Systems Dynamics, Nauka, Alma-Ata (Russian).Google Scholar
  44. Radzijevsky, V. V. and Gelfgat, B. E.: 1957,Astron. J. 34, No. 4, 581–587 (Russian, English summary).Google Scholar
  45. Stäckel, P.: 1893,J. Reine Angew. Math. 111, 290–302.Google Scholar
  46. Stepanov, W.: 1930,Astron. J. 7, 73–80 (French).Google Scholar
  47. Weyl, H.: 1952,Symmetry, Princeton University Press, Princeton, New Jersey.Google Scholar
  48. Wigner, E. P.: 1970,Symmetries and reflections, Indiana University Press, Bloomington-London.Google Scholar
  49. Wilczynski, E. J.: 1906,Projective Differential Geometry of Curves and ruled Surfaces, Teubner, Leipzig.Google Scholar

Copyright information

© D. Reidel Publishing Co. 1981

Authors and Affiliations

  • L. M. Berković
    • 1
  1. 1.Kuybyshev State University KuybyshevUSSR

Personalised recommendations