Skip to main content
SpringerLink
Log in

Examples of Integrable Systems with Dissipation on the Tangent Bundles of Multidimensional Spheres

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

In this paper, we prove the integrability of certain classes of dynamical systems that appear in the dynamics of multidimensional rigid bodies and the dynamics of a particle moving on a multidimensional sphere. The force field considered has the so-called variable dissipation with zero mean; they are generalizations of fields studied earlier. We present examples of the application of the method for integrating dissipative systems on the tangent bundles of two-dimensional surfaces of revolution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. O. I. Bogoyavlenskii, “Some integrable cases of Euler equation,” Dokl. Akad. Nauk SSSR, 287, No. 5, 1105–1108 (1986).

    MathSciNet  Google Scholar 

  2. S. A. Chaplygin, “On motion of heavy bodies in an incompressible fluid,” in: Complete Collection of Works [in Russian], Vol. 1, Izd. Akad. Nauk SSSR, Leningrad (1933), pp. 133–135.

    Google Scholar 

  3. S. A. Chaplygin, Selected Works [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  4. B. A. Dubrovin and S. P. Novikov, “On Poisson brackets of hydrodynamic type,” Dokl. Akad. Nauk SSSR, 279, No. 2, 294–297 (1984).

    MathSciNet  MATH  Google Scholar 

  5. B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry. Theory and Applications [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  6. D. V. Georgievskii and M. V. Shamolin, “Sessions of the Workshop of the Mathematics and Mechanics Department of Lomonosov Moscow State University, “Urgent Problems of Geometry and Mechanics” Named after V. V. Trofimov,” J. Math. Sci., 227, No. 4, 387–394 (2017).

    Article  Google Scholar 

  7. E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen. I. Gewöhnliche Differentialgleichungen, Teubner, Stuttgart (1977).

    Book  Google Scholar 

  8. B. Ya. Lokshin, V. A. Samsonov, and M. V. Shamolin, “Pendulum systems with dynamical symmetry,” J. Math. Sci., 227, No. 4, 461–519 (2017).

    Article  MathSciNet  Google Scholar 

  9. S. V. Manakov, “Note on the integration of Euler’s equations of the dynamics of an n-dimensional rigid body,” Funkts. Anal. Prilozh., 10, No. 4, 93–94 (1976).

    MATH  Google Scholar 

  10. B. V. Shabat, Introduction to Complex Analysis [in Russian], Nauka, Moscow (1987).

    Google Scholar 

  11. M. V. Shamolin, “On integrability in transcendental functions,” Usp. Mat. Nauk, 53, No. 3, 209–210 (1998).

    Article  MathSciNet  Google Scholar 

  12. M. V. Shamolin, “Dynamical systems with variable dissipation: Approaches, methods, and applications,” Fundam. Prikl. Mat., 14, No. 3, 3–237 (2008).

    MathSciNet  Google Scholar 

  13. M. V. Shamolin, “A new case of integrability in dynamics of a four-dimensional rigid body in a nonconservative field,” Dokl. Ross. Akad. Nauk, 437, No. 2, 190–193 (2011).

    MathSciNet  Google Scholar 

  14. M. V. Shamolin, “A new case of integrability in spatial dynamics of a rigid solid interacting with a medium under assumption of linear damping,” Dokl. Ross. Akad. Nauk, 442, No. 4, 479–481 (2012).

    Google Scholar 

  15. M. V. Shamolin, “Variety of integrable cases in dynamics of low- and multi-dimensional rigid bodies in nonconservative force fields,” in: Itogi Nauki i Tekh. Ser. Sovr. Mat. Prilozh. Temat. Obzory [in Russian], 125, All-Russian Institute for Scientific and Technical Information, Russian Academy of Sciences, Moscow (2013), pp. 5–254.

    Google Scholar 

  16. M. V. Shamolin, “New case of integrability in the dynamics of a multidimensional solid in a nonconservative field,” Dokl. Ross. Akad. Nauk, 453, No. 1, 46–49 (2013).

    MathSciNet  Google Scholar 

  17. M. V. Shamolin, “Integrable systems with variable dissipation on the tangent bundle of a multidimensional sphere and their applications,” Fundam. Prikl. Mat., 20, No. 4, 3–231 (2015).

    MathSciNet  Google Scholar 

  18. M. V. Shamolin, “Complete list of first integrals of equations of motion of a multidimensional rigid body in a nonconservative field under assumption of linear damping,” Dokl. Ross. Akad. Nauk, 464, No. 6, 688–692 (2015).

    MathSciNet  Google Scholar 

  19. M. V. Shamolin, “New examples of integrable systems with dissipation on the tangent bundles of the two-dimensional and three-dimensional spheres,” Dokl. Ross. Akad. Nauk, 471, No. 5, 547–551 (2016).

    Google Scholar 

  20. M. V. Shamolin, “Four-dimensional rigid body (pendulum) in a nonconservative field,” in: Proc. Int. Conf. “Voronezh Winter Mathematical School of S. G. Krein–2016,” Voronezh (2016), pp. 433–436.

  21. M. V. Shamolin, “Integrable systems in dynamics on the tangent bundles of spheres,” Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 2, 25–30 (2016).

    Google Scholar 

  22. M. V. Shamolin, “Integrable nonconservative dynamical systems on the tangent bundles of multidimensional spheres,” Differ. Uravn., 52, No. 6, 743–759 (2016).

    MathSciNet  Google Scholar 

  23. M. V. Shamolin, “First integrals of dynamical systems with variable dissipation in dynamics of rigid bodies,” in: Proc. XIII Int. Conf. “Stability and Oscillations of Nonlinear Control Systems,” Moscow, June 1–3, 2016, Moscow (2016), pp. 421–423.

  24. M. V. Shamolin, “First integrals of dynamical systems with dissipation on the tangent bundle of a finite-dimensional sphere,” in: Proc. III Int. School-Conf. “Geometric Analysis and Its Applications,” Volgograd, May 30–June 3, 2016, Volgograd (2016), pp. 217–222.

  25. M. V. Shamolin, “Multidimensional pendulum in nonconservative field under linear damping,” Dokl. Ross. Akad. Nauk, 470, No. 3, 288–292 (2016).

    Google Scholar 

  26. M. V. Shamolin, “Integrable systems with variable dissipation on the tangent bundle of the sphere,” Probl. Mat. Anal., 86, 139–151 (2016).

    MATH  Google Scholar 

  27. M. V. Shamolin, “Integrable motions of a pendulum in a two-dimensional plane,” J. Math. Sci., 227, No. 4, 419–441 (2017).

    Article  MathSciNet  Google Scholar 

  28. M. V. Shamolin, “Transcendental first integrals of dynamical systems on the tangent bundle to the sphere,” J. Math. Sci., 227, No. 4, 442–460 (2017).

    Article  MathSciNet  Google Scholar 

  29. M. V. Shamolin, “Examples of integrable systems corrersponding to the motion of a pendulum in the three-dimensional space,” Vestn. Samar. Univ. Estestvennonauch. Ser., Nos. 3-4, 75–97 (2016).

  30. M. V. Shamolin, “New examples of integrable systems with dissipation on the tangent bundle of the multidimensional sphere,” Dokl. Ross. Akad. Nauk, 474, No. 2, 177–181 (2017).

    Google Scholar 

  31. M. V. Shamolin, “Low-dimensional and multi-dimensional pendulums in nonconservative fields. Part 1,” J. Math. Sci., 233, No. 2, 173–299 (2018).

    Article  MathSciNet  Google Scholar 

  32. M. V. Shamolin, “Low-dimensional and multi-dimensional pendulums in nonconservative fields. Part 1,” J. Math. Sci., 233, No. 2, 300–397 (2018).

    Article  MathSciNet  Google Scholar 

  33. M. V. Shamolin, “New examples of integrable systems with dissipation on the tangent bundles of multidimensional spheres,” J. Math. Sci., xxx, No. y, zzz–zzz (20xx).

  34. M. V. Shamolin, “Integrable systems with variable dissipation on the tangent bundle of a twodimensional manifold,” in: Proc. Int. Conf. “Mathematical Theory of Optimal Control” Dedicated to the 90th Anniversary of Academician R. V. Gamkrelidze, Moscow, June 1-2, 2017, Mat. Inst. Steklova, Moscow (2017), pp. 124–127.

    Google Scholar 

  35. M. V. Shamolin, “New examples of integrable systems with dissipation on the tangent bundles of two-dimensional manifolds,” Dokl. Ross. Akad. Nauk, 475, No. 5, 519–523 (2017).

    MathSciNet  Google Scholar 

  36. M. V. Shamolin, “Integrability in elementary functions of certain classes of nonconservative systems,” in: Proc. 7th Eur. Conf. on Applied Mathematics and Informatics (AMATHI’16), Venice, Italy, January 29–31, 2016, Math. Comput. Sci. Engin. Ser., 57, WSEAS Press, Seoul (2016), pp. 50–58.

    Google Scholar 

  37. M. V. Shamolin, “Cases of integrability corresponding to the motion of a pendulum in the threedimensional space,” in: Proc. XLIV Summer School-Conference “Advanced Problems in Mechanics” Dedicated to the 30th Anniversary of IPME RAS, St. Petersburg, June 27–July 2, 2016, Saint Petersburg (2016), pp. 375–387.

  38. M. V. Shamolin, “Cases of integrability corresponding to the motion of a pendulum in the threedimensional space,” in: Proc. Global Conf. on Applied Physics and Mathematics, Rome, July 25–27, 2016, Rome (2016).

  39. M. V. Shamolin, “First integrals of variable dissipation dynamical systems in rigid body dynamics,” in: Proc. Int. Conf. “Stability and Oscillations of Nonlinear Control Systems” (Pyatnitskiy’s Conference), Moscow, June 1–3, 2016, IEEE (2016), pp. 1–4.

  40. M. V. Shamolin, “Cases of integrability corresponding to the motion of a pendulum in the fourdimensional space,” in: Proc. XLV Summer School-Conf. “Advanced Problems in Mechanics,” St. Petersburg, June 22–27, 2017, Saint Petersburg (2017), pp. 401–413.

  41. V. V. Trofimov and M. V. Shamolin, “Geometrical and dynamical invariants of integrable Hamiltonian and dissipative systems,” Fundam. Prikl. Mat., 16, No. 4, 3–229 (2010).

    Google Scholar 

  42. A. P. Veselov, “On integrability conditions for the Euler equations on so(4),” Dokl. Akad. Nauk SSSR, 270, No. 6, 1298–1300 (1983).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mechanics of the M. V. Lomonosov Moscow State University, Moscow, Russia

    M. V. Shamolin

Authors
  1. M. V. Shamolin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. V. Shamolin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shamolin, M.V. Examples of Integrable Systems with Dissipation on the Tangent Bundles of Multidimensional Spheres. J Math Sci 250, 932–941 (2020). https://doi.org/10.1007/s10958-020-05054-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-020-05054-y

Keywords and phrases

AMS Subject Classification

Search

Navigation