×

zbMATH — the first resource for mathematics

Explicit determination of certain periodic motions of a generalized two-field gyrostat. (English) Zbl 1412.70002
Summary: The case of motion of a generalized two-field gyrostat found by V. V. Sokolov and A. V. Tsyganov is known as a Liouville integrable Hamiltonian system with three degrees of freedom. For this system, we find some special periodic motions at which the momentum mapping has rank 1. For such motions, all phase variables can be expressed in terms of algebraic functions of a single auxiliary variable and a set of constants. This auxiliary variable satisfies a differential equation which can be integrated in elliptic functions of time. As an application, the explicit formulas of characteristic exponents for determining the Williamson type of the special periodic motions are obtained.
MSC:
70E05 Motion of the gyroscope
37N05 Dynamical systems in classical and celestial mechanics
70H12 Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bolsinov, A. V.; Borisov, A. V.; Mamaev, I. S., Topology and stability of integrable systems, Russian Math. Surveys, 65, 259-318, (2010) · Zbl 1202.37077
[2] Bolsinov, A. V.; Borisov, A. V.; Mamaev, I. S., Bifurcation analysis and the Conley index in mechanics, Regul. Chaotic Dyn., 17, 451-478, (2012) · Zbl 1252.76055
[3] Kharlamov, M. P., Regions of existence of critical motions of the generalized kowalewski top and bifurcation diagrams, Mekh. Tverd. Tela, 36, 13-22, (2006)
[4] Kharlamov, M. P., Critical set and bifurcation diagram on the problem of motion of the kowalewski top in two fields, Mekh. Tverd. Tela, 34, 47-58, (2004)
[5] Bogoyavlensky, O. I., Two integrable cases of a rigid body dynamics in the field of force, Dokl. Akad. Nauk USSR, 275, 1359-1363, (1984)
[6] Zotev, D. B., Fomenko-zieschang invariant in the bogoyavlenskyi case, Regul. Chaotic Dyn., 5, 437-458, (2000) · Zbl 1005.70007
[7] Kharlamov, M. P., Special periodic solutions of the generalized Delone case, Mekh. Tverd. Tela, 36, 23-33, (2006)
[8] Ryabov, P. E., Explicit integration and the topology of the D.N.Goryachev case, Dokl. Math., 84, 502-505, (2011) · Zbl 1243.37061
[9] Bogoyavlensky, O. I., Euler equations on finite-dimension Lie algebras arising in physical problems, Comm. Math. Phys., 95, 307-315, (1984) · Zbl 0581.58022
[10] Sokolov, V. V.; Tsiganov, A. V., Lax pairs for the deformed Kowalevski and Goryachev-Chaplygin tops, Theoret. and Math. Phys., 131, 543-549, (2002) · Zbl 1051.70002
[11] Kharlamov, M. P., Bifurcation diagrams of the Kowalevski top in two constant fields, Regul. Chaotic Dyn., 10, 381-398, (2005) · Zbl 1133.70306
[12] Reyman, A. G.; Semenov-Tian-Shansky, M. A., Lax representation with a spectral parameter for the kowalewski top and its generalizations, Lett. Math. Phys., 14, 55-61, (1987) · Zbl 0627.58027
[13] Ryabov, P. E., Phase topology of one irreducible integrable problem in the dynamics of a rigid body, Theoret. and Math. Phys., 176, 1000-1015, (2013) · Zbl 1286.70023
[14] Bobenko, A. I.; Reyman, A. G.; Semenov-Tian-Shansky, M. A., The kowalewski top 99 years later: a Lax pair, generalizations and explicit solutions, Comm. Math. Phys., 122, 321-354, (1989) · Zbl 0819.58013
[15] Kharlamov, M. P., Periodic motions of the Kowalevski gyrostat in two constant fields, J. Phys. A: Math. Theoret., 41, 275-307, (2008) · Zbl 1195.70010
[16] Yehia, H. M., On certain integrable motions of a rigid body acted upon by gravity and magnetic fields, Int. J. Nonlinear Mech., 36, 1173-1175, (2001) · Zbl 1345.70008
[17] Kharlamov, P. V., One case of integrability of the equations of the motion of a rigid body having a fixed point, Mekh. Tverd. Tela, 3, 57-64, (1971)
[18] Kharlamova, E. I.; Kharlamov, P. V., New solution of the differential equations of the motion of a body having a fixed point under the conditions of S. V. Kovalevskaya, Mekh. Tverd. Tela, 189, 967-968, (1969)
[19] Sokolov, S. V.; Ramodanov, S. M., Falling motion of a circular cylinder interacting dynamically with a point vortex, Regul. Chaotic Dyn., 18, 184-193, (2013) · Zbl 1273.70022
[20] Bezglasnyi, S. P., Stabilization of stationary motions of a gyrostat with a cavity filled with viscous fluid, Russ. Aeronaut., 57, 333-338, (2014)
[21] Ryabov, P. E.; Oshemkov, A. A.; Sokolov, S. V., The integrable case of Adler-Van moerbeke. discriminant set and bifurcation diagram, Regul. Chaotic Dyn., 21, 581-592, (2016) · Zbl 1368.70029
[22] Akbarzadeh, R.; Haghighatdoost, G., The topology of Liouville foliation for the borisov-mamaev-Sokolov integrable case on the Lie algebra so(4), Regul. Chaotic Dyn., 20, 317-344, (2015) · Zbl 1345.37056
[23] Akbarzadeh, R., Topological analysis corresponding to the borisov-mamaev-Sokolov integrable system on the Lie algebra so(4), Regul. Chaotic Dyn., 21, 1-17, (2016) · Zbl 1337.37039
[24] Bizyaev, I. A.; Borisov, A. V.; Mamaev, I. S., Dynamics of the Chaplygin sleigh on a cylinder, Regul. Chaotic Dyn., 21, 136-146, (2016) · Zbl 1346.70006
[25] Borisov, A. V.; Mamaev, I. S.; Bizyaev, I. A., The spatial problem of 2 bodies on a sphere. reduction and stochasticity, Regul. Chaotic Dyn., 21, 556-580, (2016) · Zbl 1402.70015
[26] Borisov, A. V.; Lebedev, V. G., Dynamics of three vortices on a plane and a sphere-II. general compact case, Regul. Chaotic Dyn., 3, 99-114, (1998) · Zbl 0933.76016
[27] Ryabov, P. E., New invariant relations for the generalized two-field gyrostat, J. Geom. Phys., 87, 415-421, (2015) · Zbl 1302.70012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.