Trudy Seminara imeni I. G. Petrovskogo
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 Tr. Semim. im. I. G. Petrovskogo, 2019, Issue 32, Pages 349–382 (Mi tsp113)

Integrable dynamical systems with dissipation on tangent bundles of 2D and 3D manifolds

M. V. Shamolin

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.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-01-00848_à

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English version:
Journal of Mathematical Sciences (New York), 2020, 244:2, 335–355

UDC: 517+531.01

Citation: M. V. Shamolin, “Integrable dynamical systems with dissipation on tangent bundles of 2D and 3D manifolds”, Tr. Semim. im. I. G. Petrovskogo, 32, 2019, 349–382; J. Math. Sci. (N. Y.), 244:2 (2020), 335–355

Citation in format AMSBIB
\Bibitem{Sha19} \by M.~V.~Shamolin \paper Integrable dynamical systems with dissipation on tangent bundles of 2D and 3D manifolds \serial Tr. Semim. im. I. G. Petrovskogo \yr 2019 \vol 32 \pages 349--382 \mathnet{http://mi.mathnet.ru/tsp113} \elib{https://elibrary.ru/item.asp?id=43214525} \transl \jour J. Math. Sci. (N. Y.) \yr 2020 \vol 244 \issue 2 \pages 335--355 \crossref{https://doi.org/10.1007/s10958-019-04622-1} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85075861341}