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Topological analysis of an integrable system related to the rolling of a ball on a sphere

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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.

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Authors and Affiliations

  1. 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

    Alexey V. Borisov & Ivan S. Mamaev

  2. A.A. Blagonravov Mechanical Engineering Research Institute of RAS, ul. Bardina 4, Moscow, 117334, Russia

    Alexey V. Borisov & Ivan S. Mamaev

  3. Institute of Mathematics and Mechanics of the Ural Branch of RAS, ul. S. Kovalevskoi 16, Yekaterinburg, 620990, Russia

    Alexey V. Borisov & Ivan S. Mamaev

Authors
  1. Alexey V. Borisov
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  2. Ivan S. Mamaev
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Correspondence to Alexey V. Borisov.

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Borisov, A.V., Mamaev, I.S. Topological analysis of an integrable system related to the rolling of a ball on a sphere. Regul. Chaot. Dyn. 18, 356–371 (2013). https://doi.org/10.1134/S1560354713040035

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

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