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This article is cited in 2 scientific papers (total in 2 papers)
Spherical and Planar Ball Bearings — a Study of Integrable Cases
Vladimir Dragovićab, Borislav Gajićb, Bozidar Jovanovićb a Department of Mathematical Sciences, The University of Texas at Dallas,
800 West Campbell Road, 75080 Richardson TX, USA
b Mathematical Institute, Serbian Academy of Sciences and Arts,
Kneza Mihaila 36, 11001 Belgrade, Serbia
Abstract:
We consider the nonholonomic systems of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with the same radius $r$ that are rolling without slipping about a fixed sphere $\mathbf S_0$ with center $O$ and radius $R$.
In addition, it is assumed that a dynamically nonsymmetric sphere $\mathbf S$ with the center that coincides with the center $O$ of the fixed sphere $\mathbf S_0$ rolls without
slipping in contact with the moving balls $\mathbf B_1,\dots,\mathbf B_n$. The problem is considered in four different configurations, three of which are new.
We derive the equations of motion and find an invariant measure for these systems.
As the main result, for $n=1$ we find two cases that are integrable by quadratures according to the Euler – Jacobi theorem.
The obtained integrable nonholonomic models are natural extensions of the well-known Chaplygin ball integrable problems.
Further, we explicitly integrate
the planar problem consisting of $n$ homogeneous balls of the same radius, but with different
masses, which roll without slipping
over a fixed plane $\Sigma_0$ with a plane $\Sigma$ that moves without slipping over these balls.
Keywords:
nonholonimic dynamics, rolling without slipping, invariant measure, integrability.
Received: 14.10.2022 Accepted: 04.01.2023
Citation:
Vladimir Dragović, Borislav Gajić, Bozidar Jovanović, “Spherical and Planar Ball Bearings — a Study of Integrable Cases”, Regul. Chaotic Dyn., 28:1 (2023), 62–77
Linking options:
https://www.mathnet.ru/eng/rcd1195 https://www.mathnet.ru/eng/rcd/v28/i1/p62
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Abstract page: | 71 | References: | 23 |
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