Topological analysis of an integrable system related to the rolling of a ball on a sphere
- 64 Downloads
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.
Keywordsintegrable system bifurcation diagram conformally Hamiltonian system bifurcation Liouville foliation critical periodic solution
MSC2010 numbers37J60 37J35 70H45
Unable to display preview. Download preview PDF.
- 5.Duistermaat, J. J., Chaplygin’s Sphere, arXiv:math/0409019v1 [math.DS] 1 Sep 2004.Google Scholar
- 8.Bolsinov, A.V. and Fomenko, A. T., Integrable Hamiltonian Systems: Geometry, Topology and Classification, Boca Raton, FL: CRC Press, 2004.Google Scholar
- 12.The Clebsch System: Separation of Variables and Explicit Integration?: Collected Papers, A. V. Borisov, A.V. Tsiganov (Eds.), Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2009, pp. 7–20 (Russian).Google Scholar
- 13.Fundamental and Applied Problems in the Theory of Vortices, A. V. Borisov, I. S. Mamaev, M. A. Sokolovskiy (Eds.), Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2003 (Russian).Google Scholar
- 14.Kharlamov, M.P., Topological Analysis of Integrable Problems of Rigid Body Dynamics, Leningrad: Leningr. Gos. Univ., 1988.Google Scholar
- 16.Chaplygin, S. A., On a Ball’s Rolling on a Horizontal Plane, Math. Sb., 1903, vol. 24, no. 1, pp. 139–168 [Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 131–148].Google Scholar