Differential Equations

, Volume 52, Issue 6, pp 722–738 | Cite as

Integrable nonconservative dynamical systems on the tangent bundle of the multidimensional sphere

Ordinary Differential Equations

Abstract

We construct a class of nonconservative systems of differential equations on the tangent bundle of the sphere of any finite dimension. This class has a complete set of first integrals, which can be expressed as finite combinations of elementary functions. Most of these first integrals consist of transcendental functions of their phase variables. Here the property of being transcendental is understood in the sense of the theory of functions of the complex variable in which transcendental functions are functions with essentially singular points.

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References

  1. 1.
    Samsonov, V.A. and Shamolin, M.V., Body Motion in a Resisting Medium, Vestnik Moskov. Univ. Ser. 1 Mat. Mekh., 1989, no. 3, pp. 51–54.MathSciNetMATHGoogle Scholar
  2. 2.
    Shamolin, M.V., On the Problem of the Motion of a Body in a Resistant Medium, Vestnik Moskov. Univ. Ser. 1 Mat. Mekh., 1992, no. 1, pp. 52–58.MathSciNetMATHGoogle Scholar
  3. 3.
    Shamolin, M.V., Metody analiza dinamicheskikh sistem s peremennoi dissipatsiei v dinamike tverdogo tela (Methods of Analysis of Variable Dissipation Dynamical Systems in Rigid Body Dynamics), Moscow, 2007.Google Scholar
  4. 4.
    Shamolin, M.V., On the Integrable Case in the 3D Dynamics of a Solid Interacting with a Medium, Izv. Ross. Akad. Nauk Mekh. Tverd. Tela, 1997, no. 2, pp. 65–68.Google Scholar
  5. 5.
    Shamolin, M.V., New Cases Integrable According to Jacobi in the Dynamics of a Solid Body Placed into Fluid Flow, Dokl. Akad. Nauk, 1999, vol. 364, no. 5, pp. 627–629.Google Scholar
  6. 6.
    Shamolin, M.V., The Case of Complete Integrability in Three-Dimensional Dynamics of a Rigid Body Interacting with a Medium with the Inclusion of Rotary Derivatives of the Force Moment with Respect to the Angular Velocity, Dokl. Akad. Nauk, 2005, vol. 403, no. 4, pp. 482–485.Google Scholar
  7. 7.
    Shamolin, M.V., A Case of Complete Integrability in the Dynamics on the Tangent Bundle of a Two-Dimensional Sphere, Uspekhi Mat. Nauk, 2007, vol. 62, no. 5, pp. 169–170.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Shamolin, M.V., Dynamical Systems with Variable Dissipation: Approaches, Methods, and Applications, Fund. Prikl. Mat., 2008, vol. 14, no. 3, pp. 3–237.Google Scholar
  9. 9.
    Chaplygin, S.A., On the Motion of Heavy Bodies in Incompressible Fluid, in Poln. sobr. soch. (Collection of Works), Leningrad, 1933, vol. 1.Google Scholar
  10. 10.
    Chaplygin, S.A., Izbrannye trudy (Selected Works), Moscow: Nauka, 1976.Google Scholar
  11. 11.
    Georgievskii, D.V. and Shamolin, M.V., Kinematics and Mass Geometry for a Solid Body with a Fixed Point in Rn, Dokl. Akad. Nauk, 2001, vol. 380, no. 1, pp. 47–50.MathSciNetGoogle Scholar
  12. 12.
    Georgievskii, D.V. and Shamolin, M.V., Generalized Euler’s Equations Describing the Motion of a Rigid Body with a Fixed Point in Rn, Dokl. Akad. Nauk, 2002, vol. 383, no. 5, pp. 635–637.MathSciNetGoogle Scholar
  13. 13.
    Georgievskii, D.V. and Shamolin, M.V., First Integrals of Motion Equations of a Generalized Gyroscope in Rn, Vestnik Moskov. Univ. Ser. 1 Mat. Mekh., 2003, no. 5, pp. 37–41.MathSciNetGoogle Scholar
  14. 14.
    Trofimov, V.V. and Shamolin, M.V., Geometric and Dynamical Invariants of Integrable Hamiltonian and Dissipative Systems, Fund. Prikl. Mat., 2010, vol. 16, no. 4, pp. 3–229.MathSciNetMATHGoogle Scholar
  15. 15.
    Shamolin, M.V., Variety of Integrable Cases in Dynamics of Low- and Multi-Dimensional Rigid Bodies in Nonconservative Force Fields, Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh., 2013, vol. 125, pp. 5–254.Google Scholar
  16. 16.
    Shamolin, M.V., Integrability According to Jacobi in the Problem of Motion of a Four-Dimensional Solid in a Resistant Medium, Dokl. Akad. Nauk, 2000, vol. 375, no. 3, pp. 343–346.Google Scholar
  17. 17.
    Shamolin, M.V., Comparison of Jacobi Integrable Cases of Plane and Spatial Motion of a Body in a Medium at Streamlining, Prikl. Mat. Mekh., 2005, vol. 69, no. 6, pp. 1003–1010.MathSciNetMATHGoogle Scholar
  18. 18.
    Shamolin, M.V., New Integrable Cases in Dynamics of a Medium-Interacting Body with Allowance for Dependence of Resistance Force Moment on Angular Velocity, Prikl. Mat. Mekh., 2008, vol. 72, no. 2, pp. 273–287.MathSciNetMATHGoogle Scholar
  19. 19.
    Shamolin, M.V., Classification of Complete Integrability Cases in Four-Dimensional Symmetric Rigid- Body Dynamics in a Nonconservative Field, Sovrem. Mat. Prilozh., 2009, vol. 65, pp. 132–142.Google Scholar
  20. 20.
    Shamolin, M.V., New Cases of Full Integrability in Dynamics of a Dynamically Symmetric Four-Dimensional Solid in a Nonconservative Field, Dokl. Akad. Nauk, 2009, vol. 425, no. 3, pp. 338–342.MathSciNetMATHGoogle Scholar
  21. 21.
    Shamolin, M.V., New Cases of Integrability in the Spatial Dynamics of a Rigid Body, Dokl. Akad. Nauk, 2010, vol. 431, no. 3, pp. 339–343.MathSciNetMATHGoogle Scholar
  22. 22.
    Shamolin, M.V., A Completely Integrable Case in the Dynamics of a Four-Dimensional Rigid Body in a Non-Conservative Field, Uspekhi Mat. Nauk, 2010, vol. 65, no. 1, pp. 189–190.MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Shamolin, M.V., A New Case of Integrability in Dynamics of a 4D-Solid in a Nonconservative Field, Dokl. Akad. Nauk, 2011, vol. 437, no. 2, pp. 190–193.MathSciNetGoogle Scholar
  24. 24.
    Arnol’d, V.I., Kozlov, V.V., and Neishtadt, A.I., Matematicheskie aspekty klassicheskoi i nebesnoi mekhaniki (Mathematical Aspects of Classical and Celestial Mechanics), Moscow: Akad. Nauk, 1985.Google Scholar
  25. 25.
    Bourbaki, N., Groupes et algébres de Lie, Paris: Hermann, 1971. Translated under the title Gruppy i algebry Li, Moscow: Mir, 1972.MATHGoogle Scholar
  26. 26.
    Dubrovin, B.A., Novikov, S.P., and Fomenko, A.T., Sovremennaya geometriya (Modern Geometry), Moscow: Nauka, 1979.MATHGoogle Scholar
  27. 27.
    Kozlov, V.V., Integrability and Nonintegrability in Hamiltonian Mechanics, Uspekhi Mat. Nauk, 1983, vol. 38, no. 1, pp. 3–67.MathSciNetGoogle Scholar
  28. 28.
    Poincaré, H., O krivykh, opredelyaemykh differentsial’nymi uravneniyami (On Curves Defined by Differential Equations), Moscow–Leningrad, 1947.Google Scholar
  29. 29.
    Shamolin, M.V., On Integrability in Transcendental Functions, Uspekhi Mat. Nauk, 1998, vol. 53, no. 3, pp. 209–210.MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Shamolin, M.V., Complete List of First Integrals in the Problem on the Motion of a 4D Solid in a Resisting Medium under Assumption of Linear Damping, Dokl. Akad. Nauk, 2011, vol. 440, no. 2, pp. 187–190.MathSciNetGoogle Scholar
  31. 31.
    Shamolin, M.V., A New Case of Integrability in the Dynamics of a 4D-Rigid Body in a Nonconservative Field under the Assumption of Linear Damping, Dokl. Akad. Nauk, 2012, vol. 444, no. 5, pp. 506–509.MathSciNetGoogle Scholar
  32. 32.
    Shamolin, M.V., A New Case of Integrability in Spatial Dynamics of a Rigid Solid Interacting with a Medium under Assumption of Linear Damping, Dokl. Akad. Nauk, 2012, vol. 442, no. 4, pp. 479–481.MathSciNetGoogle Scholar
  33. 33.
    Shamolin, M.V., Complete List of First Integrals for Dynamic Equations of Motion of a Solid Body in a Resisting Medium with Consideration of Linear Damping, Vestnik Moskov. Univ. Ser. 1 Mat. Mekh., 2012, no. 4, pp. 44–47.MathSciNetMATHGoogle Scholar
  34. 34.
    Shamolin, M.V., Comparison of Complete Integrability Cases in Dynamics of a Two-, Three-, and Four- Dimensional Rigid Body in a Nonconservative Field, Sovrem. Mat. Prilozh., 2012, vol. 76, pp. 84–99.MathSciNetMATHGoogle Scholar
  35. 35.
    Shamolin, M.V., An Integrable Case of Dynamical Equations on so(4)×R 4, Uspekhi Mat. Nauk, 2005, vol. 60, no. 6, pp. 233–234.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia

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