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 Nelin. Dinam., 2019, Volume 15, Number 4, Pages 457–475 (Mi nd673)

Two Integrable Cases of a Ball Rolling over a Sphere in $\mathbb{R}^n$

B. Gajić, B. Jovanović

Mathematical Institute SANU, Kneza Mihaila 36, 11000, Belgrade, Serbia

Abstract: We consider the nonholonomic problem of rolling without slipping and twisting of a balanced ball over a fixed sphere in $\mathbb{R}^n$. By relating the system to a modified LR system, we prove that the problem always has an invariant measure. Moreover, this is a $SO(n)$-Chaplygin system that reduces to the cotangent bundle $T^*S^{n-1}$. We present two integrable cases. The first one is obtained for a special inertia operator that allows the Chaplygin Hamiltonization of the reduced system. In the second case, we consider the rigid body inertia operator $\mathbb I\omega=I\omega+\omega I$, ${I=diag(I_1,\ldots,I_n)}$ with a symmetry $I_1=I_2=\ldots=I_{r} \ne I_{r+1}=I_{r+2}=\ldots=I_n$. It is shown that general trajectories are quasi-periodic, while for $r\ne 1$, $n-1$ the Chaplygin reducing multiplier method does not apply.

Keywords: nonholonomic Chaplygin systems, invariant measure, integrability

 Funding Agency Grant Number Serbian Ministry of Science and Technological Development 174020 This research was supported by the Serbian Ministry of Science Project 174020, Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems.

DOI: https://doi.org/10.20537/nd190405

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MSC: 37J60, 37J15, 70E18
Accepted:28.08.2019

Citation: B. Gajić, B. Jovanović, “Two Integrable Cases of a Ball Rolling over a Sphere in $\mathbb{R}^n$”, Nelin. Dinam., 15:4 (2019), 457–475

Citation in format AMSBIB
\Bibitem{GajJov19} \by B. Gaji\'c, B. Jovanovi\'c \paper Two Integrable Cases of a Ball Rolling over a Sphere in $\mathbb{R}^n$ \jour Nelin. Dinam. \yr 2019 \vol 15 \issue 4 \pages 457--475 \mathnet{http://mi.mathnet.ru/nd673} \crossref{https://doi.org/10.20537/nd190405}