×

zbMATH — the first resource for mathematics

The behaviour of cyclic variables in integrable systems. (English. Russian original) Zbl 1282.37031
J. Appl. Math. Mech. 77, No. 2, 128-136 (2013); translation from Prikl. Mat. Mekh. 77, No. 2, 179-190 (2013).
Summary: A general theorem on the behaviour of the angular variables of integrable dynamical systems as functions of time is established. Problems on the motion of the nodal line of a Kovalevskaya top and of a three- dimensional rigid body in a fluid are considered in integrable cases as examples. This range of topics is discussed for systems of a more general form obtained from completely integrable systems after changing the time.

MSC:
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arnold, B. I.; Kozlov, V. V.; Neishtadt, A. I., Mathematical aspects of classical and celestial mechanics, (1997), Springer Berlin · Zbl 0885.70001
[2] Levitan, B. M., Almost periodic functions, (1953), Gostekhizdat Moscow · Zbl 1222.42002
[3] Kozlov, V. V., Methods of qualitative analysis in the body dynamics rigid, (2000), NITs “RKhD” Moscow-Izhevsk · Zbl 1101.70003
[4] Kharlamov, M. P., Topological analysis of integrable problems in the body dynamics rigid, (1988), Izd LGU Leningrad
[5] Whittaker, E. T., Analytische dynamik, (1924), Springer Berlin, 1937.500c
[6] Kozlov, V. V., Some properties of the particular integrals of canonical equations, Vestnik MGU Ser 1 Matematika Mekhanika, 1, 81-84, (1973)
[7] Kornfeld, I. P.; Sinai YaG; Fomin, S. V., Ergodic theory, (1980), Nauka Moscow
[8] Borisov, A. V.; Mamayev, I. S., The body dynamics rigid, (2005), RKhD Moscow-Izhevsk · Zbl 1066.70504
[9] Chaplygin, S. A., On some cases of the motion of a rigid body in a fluid. article 1, Tr Otd Fiz Nauk Imp Obshchestva Lyubitelei yestestvozn, 6, 2, 20-42, (1894)
[10] Chaplygin, S. A., A case of the motion of a rigid body in a fluid. article 2, Mat Sbornik, 20, 1, 115-170, (1897)
[11] Chaplygin, S. A., The theory of the motion of nonholonomic systems. theorem on a reducing factor, Mat Sbornik, 28, 2, 303-314, (1912)
[12] Bolsinov, A. V.; Borisov, A. V.; Mamaev, I. S., Hamiltonization of nonholonomic systems in the neighbor-Hood of invariant manifolds, Regular and Chaotic Dynamics, 16, 5, 443-464, (2011) · Zbl 1309.37049
[13] Chaplygin, S. A., The rolling of a sphere along a horizontal plane, Mat Sbornik, 24, 1, 139-168, (1903)
[14] Duistermaat J.J. Chaplygin’s Sphere//a rXiv:math/0409019vl.
[15] Kilin, A. A., The dynamics of Chaplygin ball: the qualitative and computer analysis, Regular and Chaotic Dynamics, 6, 3, 291-306, (2001) · Zbl 1074.70513
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.