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 Regul. Chaotic Dyn.: Year: Volume: Issue: Page: Find

 Regul. Chaotic Dyn., 2013, Volume 18, Issue 4, Pages 356–371 (Mi rcd117)

Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere

Alexey V. Borisovabc, Ivan S. Mamaevbca

a Institute of Mathematics and Mechanics of the Ural Branch of RAS, ul. S. Kovalevskoi 16, Yekaterinburg, 620990 Russia
b Institute of Computer Science; Laboratory of Nonlinear Analysis and the Design of New Types of Vehicles, Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
c A. A. Blagonravov Mechanical Engineering Research Institute of RAS, ul. Bardina 4, Moscow, 117334 Russia

Abstract: A new integrable system describing the rolling of a rigid body with a spherical cavity on a spherical base is considered. Previously the authors found the separation of variables for this system on the zero level set of a linear (in angular velocity) first integral, whereas in the general case it is not possible to separate the variables. In this paper we show that the foliation into invariant tori in this problem is equivalent to the corresponding foliation in the Clebsch integrable system in rigid body dynamics (for which no real separation of variables has been found either). In particular, a fixed point of focus type is possible for this system, which can serve as a topological obstacle to the real separation of variables.

Keywords: integrable system, bifurcation diagram, conformally Hamiltonian system, bifurcation, Liouville foliation, critical periodic solution

DOI: https://doi.org/10.1134/S1560354713040035

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Bibliographic databases:

Document Type: Article
MSC: 37J60, 37J35, 70E18, 70F25, 70H45
Language: English

Citation: Alexey V. Borisov, Ivan S. Mamaev, “Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere”, Regul. Chaotic Dyn., 18:4 (2013), 356–371

Citation in format AMSBIB
\Bibitem{BorMam13} \by Alexey V. Borisov, Ivan S. Mamaev \paper Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere \jour Regul. Chaotic Dyn. \yr 2013 \vol 18 \issue 4 \pages 356--371 \mathnet{http://mi.mathnet.ru/rcd117} \crossref{https://doi.org/10.1134/S1560354713040035} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3090206} \zmath{https://zbmath.org/?q=an:1334.37059} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000322878100003} 

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

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