Abstract
With the aid of computer algebra methods, we have conducted qualitative analysis of the phase space for the classic and generalized Goryachev–Chaplygin problem. In particular, we have found a series of new invariant manifolds of various dimension which possess some extremal property. Motions on a one-dimensional invariant manifold have been investigated. It was shown that these motions are asymptotically stable on this manifold, and one of equilibrium points on the manifold is a limit point for these motions.
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References
Bolsinov, A.V., Kozlov, V.V., Fomenko, A.T.: The Maupertuis principle and geodesic flows on the sphere arising from integrable cases in the dynamics of a rigid body. Russian Math. Surveys. 50(3), 473–501 (1995)
Chaplygin, S.A.: A new particular solution of the problem of the rotation of a heavy rigid body about a fixed point. Trudy Otd. Fiz. Nauk Obshch. Lyubitelei Yestestvoznaniya 12(1), 1–4 (1904)
Godbillon, C.: Géometrie Différentielle et Mécanique Analytique, Collection Méthodes Hermann, Paris (1969)
Goryachev, D.N.: The motion of a heavy rigid body about a fixed point in the case when A=B=4C. Mat. Sbornik Kruzhka Lyubitelei Mat. Nauk. 21(3), 431–438 (1900)
Irtegov, V.D., Titorenko, T.N.: The invariant manifolds of systems with first integrals. J. Appl. Math. Mech. 73(4), 379–384 (2009)
Komarov, I.V., Kuznetsov, V.B.: The generalized Goryachev Chaplygin gyrostat in quantum mechanics. Differential Geometry, Lie Groups and Mechanics. Trans. of LOMI scientifc seminar USSR Acad. of Sciences 9, 134–141 (1987)
Lyapunov, A.M.: On Permanent Helical Motions of a Rigid Body in Fluid. Collected Works, vol. 1. USSR Acad. Sci., Moscow–Leningrad (1954)
Oden, M.: Rotating Tops: A Course of Integrable Systems. Udmurtiya univ., Izhevsk (1999)
Reyman, A.G., Semenov-Tian-Shansky, M.A.: Integrable Systems (Theoretic-group Approach). Institute of Computer Science, Izhevsk (2003)
Rumyantsev, V.V., Oziraner, A.S.: Motion Stability and Stabilization with Respect to Part of Variables. Nauka, Moscow (1987)
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Irtegov, V., Titorenko, T. (2014). Invariant Manifolds in the Classic and Generalized Goryachev–Chaplygin Problem. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2014. Lecture Notes in Computer Science, vol 8660. Springer, Cham. https://doi.org/10.1007/978-3-319-10515-4_16
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DOI: https://doi.org/10.1007/978-3-319-10515-4_16
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