Regular and Chaotic Dynamics

, Volume 14, Issue 4–5, pp 455–465 | Cite as

Isomorphisms of geodesic flows on quadrics

  • A. V. BorisovEmail author
  • I. S. Mamaev


We consider several well-known isomorphisms between Jacobi’s geodesic problem and some integrable cases from rigid body dynamics (the cases of Clebsch and Brun). A relationship between these isomorphisms is indicated. The problem of compactification for geodesic flows on noncompact surfaces is stated. This problem is hypothesized to be intimately connected with the property of integrability.

Key words

quadric geodesic flows integrability compactification regularization isomorphism 

MSC2000 numbers

53C22 37Kxx 


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  1. 1.
    Knörrer, H., Geodesics on Quadrics and a Mechanical Problem of C. Neumann, J. Reine Angew. Math., 1982, vol. 334, pp. 69–78.zbMATHMathSciNetGoogle Scholar
  2. 2.
    Minkowski, H., Über die Bewegung eines festen Körpers in einer Flüssigkeit, Sitzungsber. König. Preuss. Akad. Wiss. Berlin, 1888, vol. 30, pp. 1095–1110.Google Scholar
  3. 3.
    Moser, J., Various Aspects of Integrable Hamiltonian Systems, Dynamical Systems (Bressanone, 1978), Naples: Liguori, 1980, pp. 137–195.Google Scholar
  4. 4.
    Kozlov, V.V., Some Integrable Extensions of the Jacobi’s Problem of Geodesics on an Ellipsoid, Prikl. Mat. Mekh., 1995, vol. 59, no. 1, pp. 3–9 [J. Appl. Math. Mech., 1995, vol. 59, no. 1, pp. 1–7].MathSciNetGoogle Scholar
  5. 5.
    Kozlov, V.V. and Onishchenko, D. A., Nonintegrability of Kirchhoff’s Equations, Dokl. Akad. Nauk SSSR, 1982, vol. 266, no. 6, pp. 1298–1300 [Soviet Math. Dokl., 1982, vol. 26, no. 2, pp. 495–498].MathSciNetGoogle Scholar
  6. 6.
    Kozlov, V.V., Two Integrable Problems of Classical Dynamics, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1981, no. 4, pp. 80–83.Google Scholar
  7. 7.
    Kolosov, G.V., On Certain Modifications of Hamilton’s Principle in its Application to the Solution of Problems of the Rigid-Body Mechanics, St. Petersburg, 1908.Google Scholar
  8. 8.
    Borisov, A.V. and Mamaev, I. S., Rigid Body Dynamics — Hamiltonian Methods, Integrability, Chaos, Moscow-Izhevsk: Institute of Computer Science, 2005.Google Scholar
  9. 9.
    Novikov, S.P., The Hamiltonian Formalism and a Many-Valued Analogue of Morse Theory, Uspekhi Mat. Nauk, 1982, vol. 37, no. 5, pp. 3–49 [Russian Math. Surveys, 1982, vol. 37, no. 5, pp. 1–56].MathSciNetGoogle Scholar
  10. 10.
    Borisov, A. V., Necessary and Sufficient Conditions of Kirchhoff Equation Integrability, Regul. Chaotic Dyn., 1996, vol. 1, no. 2, pp. 61–76.zbMATHGoogle Scholar
  11. 11.
    Stekloff, V. A., Remarque sur un problème de Clebsch sur le mouvement d’un corps solide dans un liqiude indéfini en sur le problème de M. de Brun // C. R. Acad. Sci. Paris, 1902, vol. 135, pp. 526–528.zbMATHGoogle Scholar
  12. 12.
    Mumford, D., Tata Lectures on Theta I, Boston: Birkhäuser, 1983.zbMATHGoogle Scholar
  13. 13.
    Veselov, A.P., Two Remarks about the Connection of Jacobi and Neumann Integrable Systems, Math. Z., 1994, vol. 216, no. 3, pp. 337–345.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Bolsinov, A.V. and Dullin, H.R., On Euler Case in Rigid Body Dynamics and Jacobi Problem, Regul. Chaotic Dyn., 1997, vol. 2, no. 1, pp. 13–25.zbMATHMathSciNetGoogle Scholar
  15. 15.
    Braden, H.W., A Completely Integrable Mechanical System, Lett. Math. Phys., 1982, vol. 6, pp. 449–452.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Borisov, A.V. and Mamaev, I. S., Modern Methods of the Theory of Integrable Systems, Moscow-Izhevsk: Institute of Computer Science, 2003 (in Russian).zbMATHGoogle Scholar
  17. 17.
    Borisov, A.V. and Mamaev, I. S., Conservation Laws, Hierarchy of Dynamics and Explisit Integration of Nonholonomic System, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 443–490.CrossRefMathSciNetGoogle Scholar
  18. 18.
    Fedorov, Yu.N., Geometrical Properties of Finite-Dimensional Integrable Systems and Their Descritezation, Dissertation, Moscow State University, 2001.Google Scholar
  19. 19.
    Jung, C., Poincaré Map for Scaftaring States, J. Phys. A: Math Gen., 1997, vol. 19, pp. 1345–1353.CrossRefGoogle Scholar
  20. 20.
    Veselov, A.P., A Few Things I Learnt from Jürgen Moser, Regul. Chaotic Dyn., 2008, vol. 13, no. 6, pp. 515–524.CrossRefMathSciNetGoogle Scholar
  21. 21.
    Knauf, A. and Taimanov, I. A., On Integrability of the n-Centre Problem, Math. Ann., 2005, vol. 331, no. 3, pp. 631–649.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Multiparticle Systems: The Algebra of Integrals and Integrable Cases, Rus. J. Nonlin. Dyn., 2009, vol. 5, no. 1, pp. 53–82 [Regul. Chaotic Dyn., 2009, vol. 14, no. 1, pp. 18–41].MathSciNetGoogle Scholar
  23. 23.
    Borisov, A. V., Kilin, A.V., and Mamaev, I. S., The Hamiltonian Dynamics of Self-Gravitating Liquid and Gas Ellipsoids, Regul. Chaotic Dyn., 2009, vol. 19, no. 2, pp. 179–217.CrossRefMathSciNetGoogle Scholar
  24. 24.
    Ziglin, S. L., Dokl. Akad. Nauk, 2009 (in press).Google Scholar
  25. 25.
    Borisov, A.V. and Mamaev, I. S., Some Comments to the Paper by A.M.Perelomov ¡¡A Note on Geodesics on Ellipsoid¿¿, Regul. Chaotic Dyn., 2000, vol. 5, pp. 92–94.CrossRefMathSciNetGoogle Scholar
  26. 26.
    Brun, F., Rotation kring fix punkt, Öfvers. Kongl. Svenska Vetenskaps-Akad. Förhandl., 1893, vol. 7, pp. 455–468.Google Scholar
  27. 27.
    Jacobi, C.G. J., Vorlesungen über Dynamik, vol. 8 (Supplementband) of Gesammelte Werke, Berlin: Reimer, 1884.Google Scholar
  28. 28.
    Neumann, C., De problemate quodam mechanico, quod ad primam integralium ultraellipticorum classem revocatur, J. Rein. Angew. Math., 1859, vol. 56, pp. 46–63.zbMATHGoogle Scholar
  29. 29.
    Bogoyavlensky, O. I., Reversing Solitons: Nonlinear Integrable Equations, Moscow: Nauka, 1991 (in Russian).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Institute of Computer ScienceUdmurt State UniversityIzhevskRussia

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