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 Fundam. Prikl. Mat., 2015, Volume 20, Issue 4, Pages 3–231 (Mi fpm1663)

Integrable variable dissipation systems on the tangent bundle of a multi-dimensional sphere and some applications

M. V. Shamolin

Lomonosov Moscow State University

Abstract: This paper is a survey of integrable cases in dynamics of a multi-dimensional rigid body under the action of a nonconservative force field. We review both new results and results obtained earlier. Problems examined are described by dynamical systems with so-called variable dissipation with zero mean. The problem of the search for complete sets of transcendental first integrals of systems with dissipation is quite topical; a large number of works are devoted to it. We introduce a new class of dynamical systems that have a periodic coordinate. Due to the existence of nontrivial symmetry groups of such systems, we can prove that these systems possess variable dissipation with zero mean, which means that on the average for a period with respect to the periodic coordinate, the dissipation in the system is equal to zero, although in various domains of the phase space, either the energy pumping or dissipation can occur. Based on the facts obtained, we analyze dynamical systems that appear in dynamics of a multi-dimensional rigid body and obtain a series of new cases of complete integrability of the equations of motion in transcendental functions, which can be expressed through a finite combination of elementary functions. As applications, we study dynamical equations of motion arising in the study of the plane and spatial dynamics of a rigid body interacting with a medium and also a possible generalization of the obtained methods for the study of general systems arising in the qualitative theory of ordinary differential equations, in the theory of dynamical systems, and also in oscillation theory.

 Funding Agency Grant Number Russian Foundation for Basic Research 12-01-00020-à

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English version:
Journal of Mathematical Sciences (New York), 2018, 230:2, 185–353

Document Type: Article
UDC: 517+531.01

Citation: M. V. Shamolin, “Integrable variable dissipation systems on the tangent bundle of a multi-dimensional sphere and some applications”, Fundam. Prikl. Mat., 20:4 (2015), 3–231; J. Math. Sci., 230:2 (2018), 185–353

Citation in format AMSBIB
\Bibitem{Sha15} \by M.~V.~Shamolin \paper Integrable variable dissipation systems on the tangent bundle of a~multi-dimensional sphere and some applications \jour Fundam. Prikl. Mat. \yr 2015 \vol 20 \issue 4 \pages 3--231 \mathnet{http://mi.mathnet.ru/fpm1663} \elib{http://elibrary.ru/item.asp?id=29494664} \transl \jour J. Math. Sci. \yr 2018 \vol 230 \issue 2 \pages 185--353 \crossref{https://doi.org/10.1007/s10958-018-3738-8} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. M. V. Shamolin, “New cases of integrable systems with dissipation on the tangent bundle of a three-dimensional manifold”, Dokl. Phys., 62:11 (2017), 517–521
2. M. V. Shamolin, “New cases of integrable systems with dissipation on a tangent bundle of a two-dimensional manifold”, Dokl. Phys., 62:8 (2017), 392–396
3. M. V. Shamolin, “New cases of integrable systems with dissipation on a tangent bundle of a multidimensional sphere”, Dokl. Phys., 62:5 (2017), 262–265
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