Spherical and Planar Ball Bearings -- a Study of Integrable Cases
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| Format: | Preprint |
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2022
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| author | Dragović, Vladimir Gajić, Borislav Jovanović, Božidar |
| author_facet | Dragović, Vladimir Gajić, Borislav Jovanović, Božidar |
| contents | 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 to the moving balls $\mathbf B_1,\dots,\mathbf B_n$. The problem is considered in four different configurations. We derive the equations of motion and prove that these systems possess an invariant measure. As the main result, for $n=1$ we found two cases that are integrable in 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, that roll without slipping over a fixed plane $Σ_0$ with a plane $Σ$ that moves without slipping over these balls. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_11586 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Spherical and Planar Ball Bearings -- a Study of Integrable Cases Dragović, Vladimir Gajić, Borislav Jovanović, Božidar Mathematical Physics Dynamical Systems Exactly Solvable and Integrable Systems 37J60, 37J35, 70E40, 70F25 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 to the moving balls $\mathbf B_1,\dots,\mathbf B_n$. The problem is considered in four different configurations. We derive the equations of motion and prove that these systems possess an invariant measure. As the main result, for $n=1$ we found two cases that are integrable in 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, that roll without slipping over a fixed plane $Σ_0$ with a plane $Σ$ that moves without slipping over these balls. |
| title | Spherical and Planar Ball Bearings -- a Study of Integrable Cases |
| topic | Mathematical Physics Dynamical Systems Exactly Solvable and Integrable Systems 37J60, 37J35, 70E40, 70F25 |
| url | https://arxiv.org/abs/2210.11586 |