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 Nelin. Dinam., 2012, Volume 8, Number 5, Pages 957–975 (Mi nd381)

Topological analysis of one integrable system related to the rolling of a ball over a sphere

Alexey V. Borisovabc, Ivan S. Mamaevabc

a Institute of Computer Science; Laboratory of nonlinear analysis and the design of new types of vehicles, Udmurt State University, Izhevsk, Russia
b A. A. Blagonravov Mechanical Engineering Research Institute of RAS, Moscow, Russia
c Institute of Mathematics and Mechanics of the Ural Branch of RAS, Ekaterinburg, Russia

Abstract: A new integrable system describing the rolling of a rigid body with a spherical cavity over a spherical base is considered. Previously the authors found the separation of variables for this system at the zero level 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.

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Document Type: Article
UDC: 517.925+517.938.5
MSC: 37J60, 37J35, 70H45
Revised: 24.12.2012

Citation: Alexey V. Borisov, Ivan S. Mamaev, “Topological analysis of one integrable system related to the rolling of a ball over a sphere”, Nelin. Dinam., 8:5 (2012), 957–975

Citation in format AMSBIB
\Bibitem{BorMam12} \by Alexey~V.~Borisov, Ivan~S.~Mamaev \paper Topological analysis of one integrable system related to the rolling of a ball over a sphere \jour Nelin. Dinam. \yr 2012 \vol 8 \issue 5 \pages 957--975 \mathnet{http://mi.mathnet.ru/nd381} 

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This publication is cited in the following articles:
1. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “Dynamics and Control of an Omniwheel Vehicle”, Regul. Chaotic Dyn., 20:2 (2015), 153–172
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