Abstract
In this paper, we prove the explicit integrability of certain classes of dynamical systems on the tangent bundles of 2- and 3-dimensional spheres in the case where the forces are fields with so-called variable dissipation.
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V. I. Arnold, V. V. Kozlov, and A. I. Neustadt, Mathematical Aspects of Classical and Celestial Mechanics [in Russian], VINITI, Moscow (1985).
N. Bourbaki, Groupes et Algèbres de Lie, Hermann, Paris (1968).
S. A. Chaplygin, “On the motion of heavy bodies in an incompressible fluid,” in: Complete Collecton of Works [in Russian], Vol. 1, Leningrad (1933), pp. 133–135.
S. A. Chaplygin, Selected Works [in Russian], Nauka, Moscow (1976).
D. V. Georgievsky and M. V. Shamolin, “First integrals of the equations of motion of a generalized gyroscope in ℝn,” Vestn. Mosk. Univ. Ser. 1, Mat. Mekh., 5, 37–41 (2003).
V. V. Kozlov, “Integrability and nonintegrability in Hamiltonian mechanics,” Usp. Mat. Nauk, 38, No. 1, 3–67 (1983).
M. V. Shamolin, “On an integrable case in the spatial dynamics of a rigid body interacting with a medium,” Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 2, 65–68 (1997). 505
M. V. Shamolin, “On the integrability in transcendental functions,” Usp. Mat. Nauk, 53, No. 3, 209–210 (1998).
M. V. Shamolin, “New Jacobi-integrable cases in the dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 364, No. 5, 627–629 (1999).
M. V. Shamolin, “Jacobi integrability in the problem of the motion of a four-dimensional rigid body in a resisting medium,” Dokl. Ross. Akad. Nauk, 375, No. 3, 343–346 (2000).
M. V. Shamolin, “On an integrable case of equations of dynamics on so(4)×ℝ4,” Usp. Mat. Nauk, 60, No. 6, 233–234 (2005).
M. V. Shamolin, “The case of complete integrability in the spatial dynamics of a rigid body interacting with the medium, taking into account the rotational derivative of the angular velocity,” Dokl. Ross. Akad. Nauk, 403, No. 4, 482–485 (2005).
M. V. Shamolin, Methods of Analysis of Dynamical Systems with Variable Dissipation in Rigid- Body Dynamics [in Russian], Ekzamen, Moscow (2007).
M. V. Shamolin, “A case of complete integrability in dynamics on the tangent bundle of a twodimensional sphere,” Usp. Mat. Nauk, 62, No. 5, 169–170 (2007).
M. V. Shamolin, “Dynamic systems with variable dissipation: approaches, methods, applications,” Fundam. Prikl. Mat., 14, No. 3, 3–237 (2008).
M. V. Shamolin, “New integrable cases in the dynamics of a body interacting with the medium, taking into account the dependence of the moment of the resistance force on the angular velocity,” Prikl. Mat. Mekh., 72, No. 2, 273–287 (2008).
M. V. Shamolin, “New cases of complete integrability in the dynamics of a dynamically symmetric four-dimensional rigid body in a nonconservative field,” Dokl. Ross. Akad. Nauk, 425, No. 3, 338– 342 (2009).
M. V. Shamolin, “Classification of cases of complete integrability in the dynamics of a symmetric four-dimensional rigid body in non-conservative field,” J. Math. Sci., 165, No. 6, 743–754 (2010).
M. V. Shamolin, “New cases of integrability in the spatial dynamics of a rigid body,” Dokl. Ross. Akad. Nauk, 431, No. 3, 339–343 (2010).
M. V. Shamolin, “The case of complete integrability in the dynamics of a four-dimensional rigid body in a nonconservative field,” Usp. Mat. Nauk, 65, No. 1, 189–190 (2010).
M. V. Shamolin, “A new case of integrability in the dynamics of a four-dimensional rigid body in a nonconservative field,” Dokl. Ross. Akad. Nauk, 437, No. 2, 190–193 (2011).
M. V. Shamolin, “A complete list of first integrals in the problem of the motion of a fourdimensional rigid body in a nonconservative field in the presence of linear damping,” Dokl. Ross. Akad. Nauk, 440, No. 2, 187–190 (2011).
M. V. Shamolin, “A new case of integrability in the dynamics of a four-dimensional rigid body in a nonconservative field in the presence of linear damping,” Dokl. Ross. Akad. Nauk, 444, No. 5, 506–509 (2012).
M. V. Shamolin, “A new case of integrability in the spatial dynamics of a rigid body interacting with a medium, taking into account linear damping,” Dokl. Ross. Akad. Nauk, 442, No. 4, 479–481 (2012).
M. V. Shamolin, “Comparison of complete integrability cases in dynamics of a two-, three-, and four-dimensional rigid body in a nonconservative field,” J. Math. Sci., 187, No. 3, 346–359 (2012).
M. V. Shamolin, “Variety of integrable cases in dynamics of low- and multi-dimensional rigid bodies in nonconservative force fields,” J. Math. Sci., 204, No. 4, 379–530 (2015).
M. V. Shamolin, “Integrable systems with variable dissipation on the tangent bundle to a multidimensional sphere and applications,” Fundam. Prikl. Mat., 20, No. 4, 3–231 (2015).
M. V. Shamolin, “Low- and multidimensional pendulums in nonconservative field. Part 1,” J. Math. Sci., 233, No. 2, 173–299 (2018).
M. V. Shamolin, “Low- and multidimensional pendulums in nonconservative field. Part 1,” J. Math. Sci., 233, No. 3, 301–397 (2018).
V. V. Trofimov and M. V. Shamolin, “Geometric and dynamical invariants of integrable Hamiltonian and dissipative systems,” Fundam. Prikl. Mat., 16, No. 4, 3–229 (2010).
Acknowledgment
This work was partially supported by the Russian Foundation for Basic Research (project No. 15-01-00848-a).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 145, Geometry and Mechanics, 2018.
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Shamolin, M.V. Integrable Systems with Dissipation on the Tangent Bundles of 2- and 3-Dimensional Spheres. J Math Sci 245, 498–507 (2020). https://doi.org/10.1007/s10958-020-04706-3
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DOI: https://doi.org/10.1007/s10958-020-04706-3