Abstract
We consider the integrable system with three degrees of freedom for which V. V. Sokolov and A. V. Tsiganov specified the Lax pair. The Lax representation generalizes the L-A pair found by A. G. Reyman and M. A. Semenov-Tian-Shansky for the Kovalevskaya gyrostat in a double field. We give explicit formulas for the additional first integrals K and G (independent almost everywhere), which are functionally related to the coefficients of the spectral curve for the Sokolov-Tsiganov L-A pair. Using this form of the additional integrals K and G and the Kharlamov parametric reduction, we analytically present two invariant four-dimensional submanifolds where the induced dynamical system is Hamiltonian (almost everywhere) with two degrees of freedom. The system of equations specifying one of the invariant submanifolds is a generalization of the invariant relations for the integrable Bogoyavlensky case (rotation of a magnetized rigid body in homogeneous gravitational and magnetic fields). We use the method of critical subsystems to describe the phase topology of the whole system. For each subsystem, we construct the bifurcation diagrams and specify the bifurcations of the Liouville tori both inside the subsystems and in the whole system.
Similar content being viewed by others
References
V. V. Sokolov and A. V. Tsiganov, Theor. Math. Phys., 131, 543–549 (2002).
A. G. Reyman and M. A. Semenov-Tian-Shansky, Lett. Math. Phys., 14, 55–61 (1987).
A. V. Borisov and I. S. Mamaev, Modern Methods in the Theory of Integrable Systems: Bi-Hamiltonian Description, Lax Representation, Separation of Variables [in Russian], IKI, Moscow (2003).
A. V. Borisov and I. S. Mamaev, Dynamics of a Rigid Body: Hamiltonian Methods, Integrability, Chaos [in Russian], IKI, Moscow (2005).
V. V. Sokolov, “A generalized Kowalevski Hamiltonian and new integrable cases on e(3) and so(4),” in: The Kowalevski Property (CRM Proc. Lect. Notes, Vol. 32, V. B. Kuznetsov, ed.), Amer. Math. Soc., Providence, R. I. (2002), pp. 307–315; arXiv:nlin/0110022v1 (2001).
P. E. Ryabov, Mekh. Tverd. Tela, 37, 97–111 (2007).
A. I. Bobenko, A. G. Reyman, and M. A. Semenov-Tian-Shansky, Commun. Math. Phys., 122, 321–354 (1989).
M. P. Kharlamov, Mekh. Tverd. Tela, 34, 47–58 (2004).
M. P. Kharlamov, Rus. J. Nonlin. Dynamics, 3, 331–348 (2007).
M. P. Kharlamov, Hiroshima Math. J., 39, 327–350 (2009).
O. I. Bogoyavlensky, Math. USSR-Izv., 25, 207–257 (1985).
D. B. Zotev, Regul. Chaotic Dyn., 5, 437–457 (2000).
A. V. Bolsinov, “Methods of calculation of the Fomenko-Zieschang invariant,” in: Topological Classification of Integrable Systems (Adv. Sov. Math., Vol. 6, A. T. Fomenko, ed.), Amer. Math. Soc., Providence, R. I. (1991), pp. 147–183.
A. T. Fomenko, “The theory of invariants of multidimensional integrable Hamiltonian systems (with arbitrary many degrees of freedom): Molecular table of all integrable systems with two degrees of freedom,” in: Topological Classification of Integrable Systems (Adv. Sov. Math., Vol. 6, A. T. Fomenko, ed.), Amer. Math. Soc., Providence, R. I. (1991), pp. 1–35.
M. P. Kharlamov and A. Yu. Savushkin, Ukrainian Math. Bull., 1, 569–586 (2004).
M. P. Kharlamov, Regul. Chaotic Dyn., 10, 381–398 (2005).
M. P. Kharlamov, Mekh. Tverd. Tela, 37, 97–111 (2007).
M. P. Kharlamov, Regul. Chaotic Dyn., 12, 267–280 (2007); arXiv:0803.1024v1 [nlin.SI] (2008).
M. P. Kharlamov, Regul. Chaotic Dyn., 14, 621–634 (2009).
M. P. Kharlamov, Rus. J. Nonlin. Dynamics, 7, 25–51 (2011).
P. E. Ryabov and M. P. Kharlamov, Sb. Math., 203, 257–287 (2012).
A. T. Fomenko, Math. USSR-Izv., 39, 731–759 (1992).
M. P. Kharlamov and P. E. Ryabov, Dokl. Math., 86, 839–842 (2012).
M. P. Kharlamov, Mekh. Tverd. Tela, 32, 32–38 (2002).
A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems: Geometry, Topology, Classification [in Russian], Vol. 1, 2, RKdD, Izhevsk (1999).
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 176, No. 2, pp. 205–221, August 2013.
Rights and permissions
About this article
Cite this article
Ryabov, P.E. Phase topology of one irreducible integrable problem in the dynamics of a rigid body. Theor Math Phys 176, 1000–1015 (2013). https://doi.org/10.1007/s11232-013-0087-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11232-013-0087-0