Abstract:
We consider the possibility of using Dirac’s ideas of the deformation of Poisson brackets in nonholonomic mechanics. As an example, we analyze the composition of external forces that do no work and reaction forces of nonintegrable constraints in the model of a nonholonomic Chaplygin sphere on a plane. We prove that, when a solenoidal field is applied, the general mechanical energy, the invariant measure and the conformally Hamiltonian representation of the equations of motion are preserved. In addition, we consider the case of motion of the nonholonomic Chaplygin sphere in a constant magnetic field taking dielectric and ferromagnetic (superconducting) properties of the sphere into account. As a by-product we also obtain two new integrable cases of the Hamiltonian rigid body dynamics in a constant magnetic field taking the magnetization by rotation effect into account.
Keywords:
nonholonomic mechanics, magnetic field, deformation of Poisson brackets, Grioli problem, Barnett – London moment.
The work of A.V.Tsiganov (Sections 1, 2) is supported by the Russian Science Foundation
(project no. 19-71-30012) and performed in Steklov Mathematical Institute of Russian Academy
of Sciences. The work of A. V. Borisov (Sections 3, 4) was carried out within the framework of the
state assignment of the Ministry of Education and Science of Russia (1.2404.2017/4.6).
\Bibitem{BorTsi19}
\by Alexey V. Borisov, Andrey V. Tsiganov
\paper On the Chaplygin Sphere in a Magnetic Field
\jour Regul. Chaotic Dyn.
\yr 2019
\vol 24
\issue 6
\pages 739--754
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\crossref{https://doi.org/10.1134/S156035471906011X}
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https://www.mathnet.ru/eng/rcd1037
https://www.mathnet.ru/eng/rcd/v24/i6/p739
This publication is cited in the following 2 articles:
Alexey V. Borisov, Andrey V. Tsiganov, “On the Nonholonomic Routh Sphere in a Magnetic Field”, Regul. Chaotic Dyn., 25:1 (2020), 18–32
A. V. Borisov, A. V. Tsyganov, “Vliyanie effektov Barnetta-Londona i Einshteina-de Gaaza na dvizhenie negolonomnoi sfery Rausa”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 29:4 (2019), 583–598
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