Аннотация:
The problem of Hamiltonization of nonholonomic systems, both integrable and non-integrable, is considered. This question is important in the qualitative analysis of such systems and it enables one to determine possible dynamical effects. The first part of the paper is devoted to representing integrable systems in a conformally Hamiltonian form. In the second part, the existence of a conformally Hamiltonian representation in a neighborhood of a periodic solution is proved for an arbitrary (including integrable) system preserving an invariant measure. Throughout the paper, general constructions are illustrated by examples in nonholonomic mechanics.
Образец цитирования:
A.V. Bolsinov, A.V. Borisov, I. S. Mamaev, “Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds”, Regul. Chaotic Dyn., 16:5 (2011), 443–464
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\by A.V. Bolsinov, A.V. Borisov, I. S. Mamaev
\paper Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds
\jour Regul. Chaotic Dyn.
\yr 2011
\vol 16
\issue 5
\pages 443--464
\mathnet{http://mi.mathnet.ru/rcd446}
\crossref{https://doi.org/10.1134/S1560354711050030}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2844857}
\zmath{https://zbmath.org/?q=an:1309.37049}
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https://www.mathnet.ru/rus/rcd446
https://www.mathnet.ru/rus/rcd/v16/i5/p443
Эта публикация цитируется в следующих 57 статьяx:
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А. В. Цыганов, “О тензорных инвариантах для интегрируемых случаев движения твердого тела Эйлера, Лагранжа и Ковалевской”, Изв. РАН. Сер. матем., 89:2 (2025), 161–188
Elizaveta Artemova, Evgeny Vetchanin, “The motion of a circular foil in the field of a fixed point singularity: Integrability and asymptotic behavior”, Physics of Fluids, 36:2 (2024)
Paula Balseiro, Maria Eugenia Garcia, Cora Inés Tori, Marcela Zuccalli, “Momentum map reduction for nonholonomic systems”, Nonlinearity, 36:10 (2023), 5401
Vladimir Dragović, Borislav Gajić, Božidar Jovanović, “Gyroscopic Chaplygin Systems and Integrable Magnetic Flows on Spheres”, J Nonlinear Sci, 33:3 (2023)
Misael Avendaño-Camacho, Claudio César García-Mendoza, José Crispín Ruiz-Pantaleón, Eduardo Velasco-Barreras, “Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems”, SIGMA, 18 (2022), 038, 29 pp.
Ivan A. Bizyaev, Ivan S. Mamaev, “Permanent Rotations in Nonholonomic Mechanics.
Omnirotational Ellipsoid”, Regul. Chaotic Dyn., 27:6 (2022), 587–612
Balseiro P., Yapu L.P., “Conserved Quantities and Hamiltonization of Nonholonomic Systems”, Ann. Inst. Henri Poincare-Anal. Non Lineaire, 38:1 (2021), 23–60
Vladimir Dragović, Borislav Gajić, Božidar Jovanović, “Demchenko's nonholonomic case of a gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis”, Theor. Appl. Mech., 47:2 (2020), 257–287
Ivan Yu. Polekhin, “Precession of the Kovalevskaya and Goryachev – Chaplygin Tops”, Regul. Chaotic Dyn., 24:3 (2019), 281–297
Vyacheslav P. Kruglov, Sergey P. Kuznetsov, “Topaj – Pikovsky Involution in the Hamiltonian Lattice of Locally Coupled Oscillators”, Regul. Chaotic Dyn., 24:6 (2019), 725–738
Kurt M. Ehlers, Jair Koiller, “Cartan meets Chaplygin”, Theor. Appl. Mech., 46:1 (2019), 15–46
Borisov A.V., Ivanova T.B., Kilin A.A., Mamaev I.S., “Nonholonomic Rolling of a Ball on the Surface of a Rotating Cone”, Nonlinear Dyn., 97:2 (2019), 1635–1648
Garcia-Naranjo L.C., “Generalisation of Chaplygin'S Reducing Multiplier Theorem With An Application to Multi-Dimensional Nonholonomic Dynamics”, J. Phys. A-Math. Theor., 52:20 (2019), 205203
Jovanovic B., “Rolling Balls Over Spheres in R-N”, Nonlinearity, 31:9 (2018), 4006–4030
Andrey V. Tsiganov, “Bäcklund Transformations for the Nonholonomic Veselova System”, Regul. Chaotic Dyn., 22:2 (2017), 163–179
А. В. Борисов, И. С. Мамаев, И. А. Бизяев, “Динамические системы с неинтегрируемыми связями: вакономная механика, субриманова геометрия и неголономная механика”, УМН, 72:5(437) (2017), 3–62; A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics”, Russian Math. Surveys, 72:5 (2017), 783–840
Isaac A García, Benito Hernández-Bermejo, “Inverse Jacobi multiplier as a link between conservative systems and Poisson structures”, J. Phys. A: Math. Theor., 50:32 (2017), 325204
Alexey V. Borisov, Ivan S. Mamaev, “Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics”, Regul. Chaotic Dyn., 21:2 (2016), 232–248
Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Dynamics of Vortex Sources in a Deformation Flow”, Regul. Chaotic Dyn., 21:3 (2016), 367–376
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