Regular and Chaotic Dynamics

, Volume 18, Issue 4, pp 356–371 | Cite as

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

Article

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 

MSC2010 numbers

37J60 37J35 70H45 

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Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Institute of Computer Science; Laboratory of Nonlinear Analysis and the Design of New Types of VehiclesUdmurt State UniversityIzhevskRussia
  2. 2.A.A. Blagonravov Mechanical Engineering Research Institute of RASMoscowRussia
  3. 3.Institute of Mathematics and Mechanics of the Ural Branch of RASYekaterinburgRussia

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