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Fundamentalnaya i Prikladnaya Matematika, 2008, Volume 14, Issue 3, Pages 3–237 (Mi fpm1121)  

This article is cited in 73 scientific papers (total in 73 papers)

Dynamical systems with variable dissipation: Approaches, methods, and applications

M. V. Shamolin

M. V. Lomonosov Moscow State University
References:
Abstract: This work is devoted to the development of qualitative methods in the theory of nonconservative systems that arise, e.g., in such fields of science as the dynamics of a rigid body interacting with a resisting medium, oscillation theory, etc. This material can call the interest of specialists in the qualitative theory of ordinary differential equations, in rigid body dynamics, as well as in fluid and gas dynamics since the work uses the properties of motion of a rigid body in a medium under the streamline flow around conditions.
The author obtains a full spectrum of complete integrability cases for nonconservative dynamical systems having nontrivial symmetries. Moreover, in almost all cases of integrability, each of the first integrals is expressed through a finite combination of elementary functions and is a transcendental function of its variables, simultaneously. In this case, the transcendence is meant in the complex analysis sense, i.e., after the continuation of the functions considered to the complex domain, they have essentially singular points. The latter fact is stipulated by the existence of attracting and repelling limit sets in the system considered (for example, attracting and repelling foci).
The author obtains new families of phase portraits of systems with variable dissipation on lower- and higher-dimensional manifolds. He discusses the problems of their absolute or relative roughness. He discovers new integrable cases of the rigid body motion, including those in the classical problem of motion of a spherical pendulum placed in the over-running medium flow.
English version:
Journal of Mathematical Sciences (New York), 2009, Volume 162, Issue 6, Pages 741–908
DOI: https://doi.org/10.1007/s10958-009-9657-y
Bibliographic databases:
UDC: 517.925+531.01+531.552
Language: Russian
Citation: M. V. Shamolin, “Dynamical systems with variable dissipation: Approaches, methods, and applications”, Fundam. Prikl. Mat., 14:3 (2008), 3–237; J. Math. Sci., 162:6 (2009), 741–908
Citation in format AMSBIB
\Bibitem{Sha08}
\by M.~V.~Shamolin
\paper Dynamical systems with variable dissipation: Approaches, methods, and applications
\jour Fundam. Prikl. Mat.
\yr 2008
\vol 14
\issue 3
\pages 3--237
\mathnet{http://mi.mathnet.ru/fpm1121}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2482029}
\zmath{https://zbmath.org/?q=an:1189.37022}
\elib{https://elibrary.ru/item.asp?id=12174967}
\transl
\jour J. Math. Sci.
\yr 2009
\vol 162
\issue 6
\pages 741--908
\crossref{https://doi.org/10.1007/s10958-009-9657-y}
\elib{https://elibrary.ru/item.asp?id=15307901}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-70350662331}
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  • https://www.mathnet.ru/eng/fpm/v14/i3/p3
  • This publication is cited in the following 73 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Фундаментальная и прикладная математика
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    References:115
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