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Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics. I

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Literature Cited

  1. G. Springer, Introduction to Riemann Surfaces, Addison-Wesley, Reading, Mass. (1957).

    Google Scholar 

  2. V. I. Arnol'd, Supplementary Chapters of the Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  3. V. I. Arnol'd, Ordinary Differential Equations [in Russian], Nauka, Moscow (1971).

    Google Scholar 

  4. V. I. Arnol'd and A. L. Krylov, "Uniform distribution of points on the sphere, and some ergodic properties of the solutions to ordinary differential equations in the complex domain," Dokl. Akad. Nauk SSSR,148, No. 1, 9–12 (1963).

    Google Scholar 

  5. A. Poincaré, Les Méthodes Nouvelles de la Mécanique Celeste, Vols. 1, 2, 3. Gauthier-Villars, Paris (1892); reprint, Dover, New York (1957).

    Google Scholar 

  6. S. Lang, Algebra, Addison-Wesley, Reading, Mass. (1965).

    Google Scholar 

  7. V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Graduate Texts in Math., No. 60, Springer-Verlag, New York (1978).

    Google Scholar 

  8. B. L. van der Waerden, Modern Algebra, Berlin (1937).

  9. M. F. Atiyah, "K-theory and reality," Q. J. Math.,17, 367–386 (1966).

    Google Scholar 

  10. V. V. Kozlov, "Nonexistence of uniform first integrals and branching of solutions in the dynamics of rigid body," Prikl. Mat. Mekh.,42, No. 3, 400–406 (1978).

    Google Scholar 

  11. V. V. Kozlov, Methods of Qualitative Analysis in the Dynamics of the Rigid Body [in Russian], Moscow State Univ. (1980).

  12. E. Husson, "Recherche des integrales algebraiques dans le mouvement d'un solide pesant autour d'un point fixé," Ann. Faculté Sci. Univ. Toulouse,8, Ser. 2, 73–152 (1906).

    Google Scholar 

  13. Yu. A. Arkhangel'skii, Analytical Dynamics of Rigid Body [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  14. S. L. Ziglin, "Branching of solutions, intersection of separatrices, and nonexistence of an integral in the dynamics of the rigid body," Tr. Mosk. Mat. Obshch.,41, 287–303 (1980).

    Google Scholar 

  15. V. V. Kozlov, "Nonexistence of an additional first integral in the problem of the motion of a nonsymmetric heavy body around a fixed point," Vestn. Mosk. Gos. Univ., Ser. Mat.-Mekh., No. 6, 99–104 (1976).

    Google Scholar 

  16. H. Bateman and A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, New York (1953–1955).

    Google Scholar 

  17. V. V. Golubev, Lectures on the Analytical Theory of Differential Equations [in Russian], Gostekhizdat, Moscow—Leningrad (1950).

    Google Scholar 

  18. V. V. Golubev, Lectures on the Integration of the Equations of Motion of a Heavy Rigid Body around a Fixed Point [in Russian], Gostekhizdat, Moscow (1953).

    Google Scholar 

  19. V. V. Kozlov, "On the qualitative analysis of the motion of a heavy rigid body in the Goryachev—Chaplygin case," Prikl. Mat. Mekh.,41, No. 2, 225–233 (1977).

    Google Scholar 

  20. O. Forster, Riemannsche Flächen, Springer-Verlag, Berlin—Heidelberg—New York (1977).

    Google Scholar 

  21. M. Henon and C. Heiles, "The applicability of the third integral of motion; some numerical experiments," Astron. J.,69, 73–79 (1964).

    Google Scholar 

  22. M. L. Berry, "Regular and irregular motions," in: Topics in Nonlinear Dynamics, S. Jorna (ed.), Am. Inst. Phys. (1978), pp. 16–120.

  23. N. N. Bogolyubov and D. V. Shirkov, Quantum Fields [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  24. V. E. Zakharov, M. F. Ivanov, and L. I. Shur, "On the abnormally slow stochastization in certain two-dimensional models of field theory," Pis'ma Zh. Eksp. Teor. Fiz.,30, No. 1, 39–44 (1979).

    Google Scholar 

  25. B. P. Demidovich, Lectures on the Mathematical Theory of Stability [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  26. S. V. Kovalevskaya, Scientific Works [in Russian], Izd. Akad. Nauk SSSR, Moscow (1948).

    Google Scholar 

  27. A. M. Lyapunov, "On one property of the differential equations of the motion of a heavy rigid body having a fixed point," Soobshch. Kharkovsk. Mat. Obshch., Ser. 2,4, 123–140 (1894).

    Google Scholar 

  28. S. L. Ziglin, "Self-intersection of complex separatrices, and nonexistence of integrals in Hamiltonian systems with one and a half degrees of freedom," Prikl. Mat. Mekh.,45, 564–566 (1981).

    Google Scholar 

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Radiotechnics and Electronics Institute, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 16, No. 3, pp. 30–41, July–September, 1982.

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Ziglin, S.L. Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics. I. Funct Anal Its Appl 16, 181–189 (1982). https://doi.org/10.1007/BF01081586

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  • DOI: https://doi.org/10.1007/BF01081586

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