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2013
Impact Factor

Borislav Gajić

Publications:

Dragović V., Gajić B., Jovanović B.
Spherical and Planar Ball Bearings — Nonholonomic Systems with Invariant Measures
2022, vol. 27, no. 4, pp.  424-442
Abstract
We first construct nonholonomic systems of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with centers $O_1,\ldots,O_n$ and with the same radius $r$ that are rolling without slipping around a fixed sphere $\mathbf S_0$ with center $O$ and radius $R$. In addition, it is assumed that a dynamically nonsymmetric sphere $\mathbf S$ of radius $R+2r$ and the center that coincides with the center $O$ of the fixed sphere $\mathbf S_0$ rolls without slipping over the moving balls $\mathbf B_1,\dots,\mathbf B_n$. We prove that these systems possess an invariant measure. As the second task, we consider the limit, when the radius $R$ tends to infinity. We obtain a corresponding planar problem consisting of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with centers $O_1,\ldots,O_n$ and the same radius $r$ that are rolling without slipping over a fixed plane $\Sigma_0$, and a moving plane $\Sigma$ that moves without slipping over the homogeneous balls. We prove that this system possesses an invariant measure and that it is integrable in quadratures according to the Euler – Jacobi theorem.
Keywords: nonholonimic dynamics, rolling without slipping, invariant measure, integrability
Citation: Dragović V., Gajić B., Jovanović B.,  Spherical and Planar Ball Bearings — Nonholonomic Systems with Invariant Measures, Regular and Chaotic Dynamics, 2022, vol. 27, no. 4, pp. 424-442
DOI:10.1134/S1560354722040037

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