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Spherical and Planar Ball Bearings — Nonholonomic Systems with Invariant Measures


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.

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We are very grateful to the referees for valuable remarks that helped us to improve the exposition. The research which led to this paper was initiated during the GDIS conference in the summer of 2018, when all three authors visited the Moscow Institute of Physics and Technology, kindly invited and hosted by Professor Alexey V. Borisov and his team.


This research has been supported by project no. 7744592 MEGIC “Integrability and Extremal Problems in Mechanics, Geometry and Combinatorics” of the Science Fund of Serbia, Mathematical Institute of the Serbian Academy of Sciences and Arts and the Ministry for Education, Science and Technological Development of Serbia, and the Simons Foundation grant no. 854861.

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Correspondence to Vladimir Dragović, Borislav Gajić or Božidar Jovanović.

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Dedicated to the memory of Professor Alexey Vladimirovich Borisov


37J60, 37J35, 70E40, 70F25

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Dragović, V., Gajić, B. & Jovanović, B. Spherical and Planar Ball Bearings — Nonholonomic Systems with Invariant Measures. Regul. Chaot. Dyn. 27, 424–442 (2022).

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  • nonholonimic dynamics
  • rolling without slipping
  • invariant measure
  • integrability