Title | Integrable systems with dissipation on the tangent bundle of two-dimensional manyfold |
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Authors | M. V. Shamolin^{1}^{1}Lomonosov Moscow State University |

Annotation | The mechanical systems with position spaces as two-dimensional manifolds are arising in many problems of dynamics. The phase spaces of such systems naturally become the tangent bundles to them. For example, the study of three-dimensional pendulum on a spherical hinge in a medium flow leads to a dynamical system on the tangent bundle to two-dimensional sphere with a special metric on it. This metric is induced by an additional group of symmetries. In this case the dynamical systems have the variable dissipation, and the complete list of first integrals consists of the transcendental functions expressed in terms of a finite combination of elementary functions. There is also a class of problems on the motion of a point in two-dimensional surface with the metric which is induced by the Euclidean metric of a comprehensive space. The activity shows the integrability of certain classes of dynamical systems on the tangent bundles of two-dimensional manifolds. In this case, the force fields have the variable dissipation and generalize the previously considered. |

Keywords | dynamical system, variable dissipation, integrability, transcendental first integral |

Citation | Shamolin M. V. ''Integrable systems with dissipation on the tangent bundle of two-dimensional manyfold '' [Electronic resource]. Proceedings of the XIII International scientific conference ''Differential equations and their applications in mathematical modeling''. (Saransk, July 12-16, 2017). Saransk: SVMO Publ, 2017. - pp. 10-21. Available at: http://conf.svmo.ru/files/deamm2017/papers/paper02.pdf. - Date of access: 24.08.2019. |

**© SVMO, National Research Mordovia State University, 2019**

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