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Integrable Dynamical Systems with Dissipation on Tangent Bundles of 2D and 3D Manifolds

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Abstract

In many problems of dynamics, one has to deal with mechanical systems whose configurational spaces are two- or three-dimensional manifolds. For such a system, the phase space naturally coincides with the tangent bundle of the corresponding manifold. Thus, the problem of a flow past a (four-dimensional) pendulum on a (generalized) spherical hinge leads to a system on the tangent bundle of a two- or threedimensional sphere whose metric has a particular structure induced by an additional symmetry group. In such cases, dynamical systems have variable dissipation, and their complete list of first integrals consists of transcendental functions in the form of finite combinations of elementary functions. Another class of problems pertains to a point moving on a two- or three-dimensional surface with the metric induced by the encompassing Euclidean space. In this paper, we establish the integrability of some classes of dynamical systems on tangent bundles of two- and three-dimensional manifolds, in particular, systems involving fields of forces with variable dissipation and of a more general type than those considered previously.

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References

  1. V. I. Arnold, V. V. Kozlov, and A. I. Neustadt, Mathematical Aspects of Classical and Celestial Mechanics [in Russian], VINITI, Moscow (1985).

    Google Scholar 

  2. G. S. Byushgens and R. V. Studnev, Airplane Dynamics. Spatial Motion [in Russian], Mashinostroenie, Moscow (1988).

    Google Scholar 

  3. D. V. Georgievskii and M. V. Shamolin, “Trofimov Valerii Vladimirovich,” Sovrem. Mat. Fund. Napr., 23, 5–15 (2007).

    MATH  Google Scholar 

  4. D. V. Georgievskii and M. V. Shamolin, “Proceedings of V. V. Trofimov Seminar “Current Problems in Geometry and Mechanics” supervised by D. V. Georgievskii, M. V. Shamolin, and S. A. Agafonov,” Sovrem. Mat. Fund. Napr., 23, 16–45 (2007).

    Google Scholar 

  5. D. V. Georgievskii and M. V. Shamolin, “Proceedings of V. V. Trofimov Seminar “Current Problems in Geometry and Mechanics” supervised by D. V. Georgievskii, M. V. Shamolin, and S. A. Agafonov,” Sovrem. Mat. Prilozh., 62, 3–15 (2009).

    Google Scholar 

  6. D. V. Georgievskii and M. V. Shamolin, “Proceedings of V. V. Trofimov Seminar “Current Problems in Geometry and Mechanics” supervised by D. V. Georgievskii, M. V. Shamolin, and S. A. Agafonov,” Sovrem. Mat. Prilozh., 65, 3–10 (2009).

    Google Scholar 

  7. D. V. Georgievskii and M. V. Shamolin, “Proceedings of V. V. Trofimov Seminar “Current Problems in Geometry and Mechanics” supervised by D. V. Georgievskii, M. V. Shamolin, and S. A. Agafonov,” Sovrem. Mat. Prilozh., 76, 3–10 (2012).

    Google Scholar 

  8. D. V. Georgievskii and M. V. Shamolin, “Proceedings of V. V. Trofimov Seminar “Current Problems in Geometry and Mechanics” supervised by D. V. Georgievskii, M. V. Shamolin, and S. A. Agafonov,” Sovrem. Mat. Prilozh., 88 (2015)

  9. D. V. Georgievskii and M. V. Shamolin, “Proceedings of V. V. Trofimov Seminar “Current Problems in Geometry and Mechanics” supervised by D. V. Georgievskii, M. V. Shamolin, and S. A. Agafonov,” Sovrem. Mat. Prilozh., 98, 3–8 (2015).

    Google Scholar 

  10. D. V. Georgievskii and M. V. Shamolin, “Proceedings of V. V. Trofimov Seminar “Current Problems in Geometry and Mechanics” supervised by D. V. Georgievskii, M. V. Shamolin, and S. A. Agafonov,” Sovrem. Mat. Prilozh., 100, 3–11 (2016).

    Google Scholar 

  11. V. V. Kozlov, “Integrability and non-integrability in Hamiltonian Mechanics,” Usp. Mat. Nauk, 38, No. 1, 3–67 (1983).

    MATH  Google Scholar 

  12. B. Ya. Lokshin, V. A. Samsonov, and M. V. Shamolin, “Pendulum systems with a dynamic symmetry,” Sovrem. Mat. Prilozh., 100, 76–133 (2016).

    Google Scholar 

  13. H. Poincaré, New Methods in Clestial Mechanics, Springer (1992).

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

    Google Scholar 

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

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

    Google Scholar 

  17. M. V. Shamolin, “An integrable case in spatial dynamics of a rigid body interacting with a medium,” Izv. RAN Mekh. Tverd. Tela, No. 2, 65–68 (1997).

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

    Article  Google Scholar 

  19. M. V. Shamolin, “New cases integrable according to Jacobi in the dynamics of a rigid body interacting with a medium,” Dokl. RAN, 364, No. 5, 627–629 (1999).

    Google Scholar 

  20. M. V. Shamolin, “Complete integrability of the equations of motion of a spatial pendulum in incoming flow,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 5, 22–28 (2001).

  21. M. V. Shamolin, “On integrability of some classes of nonconservative systems,” Usp. Mat. Nauk, 57, No. 1, 169–170 (2002).

    Article  Google Scholar 

  22. M. V. Shamolin, “A case of complete integrability in spatial dynamics of a rigid body interacting with a medium, with rotational derivatives of the torque with respect to the angular velocity,” Dokl. RAN, 403, No. 4, 482–485 (2005).

    MathSciNet  Google Scholar 

  23. M. V. Shamolin, Methods for the Analysis of Dynamical Systems with Variable Dissipation in the Dynamics of Solids [in Russian], Examen, Moscow (2007).

    Google Scholar 

  24. M. V. Shamolin, “Complete integrability of the equations of motion of a spatial pendulum in a medium flow with rotational derivatives of the torque produced by the medium taken into account,” Izv. RAN. Mekh. Tverd. Tela, No. 3, 187–192 (2007).

  25. M. V. Shamolin, “A case of complete integrability in dynamics on the tangent bundle of a two-dimensional sphere,” Usp. Mat. Nauk, 62, No. 5, 169–170 (2007).

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  27. M. V. Shamolin, “On integrability in elementary functions of some classes of dynamical systems,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 3, 43–49 (2008).

  28. M. V. Shamolin, “On integrability in elementary functions of some classes of nonconservative dynamical systems,” Sovrem. Mat. Prilozh., 62, 131–171 (2009).

    Google Scholar 

  29. M. V. Shamolin, “New cases of integrability in spatial dynamics of rigids,” Dokl. RAN, 431, No. 3, 339–343 (2010).

    Google Scholar 

  30. M. V. Shamolin, “A new case of integrability in the dynamics of a four-dimensional rigid in a nonconservative field,” Dokl. RAN, 437, No. 2, 190–193 (2011).

    MathSciNet  Google Scholar 

  31. M. V. Shamolin, “A new case of integrability in the dynamics of a four-dimensional rigid in a nonconservative field with linear damping,” Dokl. RAN, 444, No. 5, 506–509 (2012).

    Google Scholar 

  32. M. V. Shamolin, “A new case of integrability in spatial dynamics of a rigid body interacting with a medium, with linear damping taken into account,” Dokl. RAN, 442, No. 4, 479–481 (2012).

    Google Scholar 

  33. M. V. Shamolin, “A comparison of cases of complete integrability in dynamics of three- and four-dimensional bodies in a nonconservative field,” Sovrem. Mat. Prilozh., 76, 84–99 (2012).

    MathSciNet  Google Scholar 

  34. M. V. Shamolin, “Some classes of integrable problems in spatial dynamics of a rigid body in a nonconservative force field,” Tr. Semin. Petrovskogo, 30, 287–350 (2014).

    MathSciNet  Google Scholar 

  35. M. V. Shamolin, “A new case of complete integrability of dynamic equations on the tangent bundle of a three-dimensional sphere,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 3, 11–14 (2015).

  36. M. V. Shamolin, “Integrable systems in dynamics on the tangent bundle of a sphere,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 25–30 (2016).

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

    MATH  Google Scholar 

  38. M. V. Shamolin, “Transcendental first integrals of dynamical systems on the tangent bundle of a sphere,” Sovrem. Mat. Prilozh., 100, 58–75 (2016).

    Google Scholar 

  39. M. V. Shamolin, “New cases of integrability of systems with dissipation on tangent bundles of twoand three-dimensional spheres,” Dokl. RAN, 471, No. 5, 547–551 (2016).

    Google Scholar 

  40. M. V. Shamolin, “Low-dimensional and multi-dimensional pendulums in nonconservative fields. Pt. 1,” Itogi Nauki Tekh. Ser. Sovrem. Mat. Pril. Temat. Obz., 134, 6–128 (2017).

    Google Scholar 

  41. M. V. Shamolin, “Low-dimensional and multi-dimensional pendulums in nonconservative fields. Pt. 2,” Itogi Nauki Tekh. Ser. Sovrem. Mat. Pril. Temat. Obz., 135, 3–93 (2017).

    Google Scholar 

  42. C. Jacobi, Lectures on Dynamics [Russian translation], ONTI, Moscow (1936).

    Google Scholar 

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Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    M. V. Shamolin

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  1. M. V. Shamolin
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Correspondence to M. V. Shamolin.

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Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 32, pp. 349–382, 2019.

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Shamolin, M.V. Integrable Dynamical Systems with Dissipation on Tangent Bundles of 2D and 3D Manifolds. J Math Sci 244, 335–355 (2020). https://doi.org/10.1007/s10958-019-04622-1

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