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Rus. J. Nonlin. Dyn., 2019, Volume 15, Number 4, Pages 457–475 (Mi nd673)  

This article is cited in 1 scientific paper (total in 1 paper)

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

Full text: PDF file (361 kB)
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MSC: 37J60, 37J15, 70E18
Received: 26.06.2019
Accepted:28.08.2019
Language:

Citation: B. Gajić, B. Jovanović, “Two Integrable Cases of a Ball Rolling over a Sphere in $\mathbb{R}^n$”, Rus. J. Nonlin. Dyn., 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 Rus. J. Nonlin. Dyn.
\yr 2019
\vol 15
\issue 4
\pages 457--475
\mathnet{http://mi.mathnet.ru/nd673}
\crossref{https://doi.org/10.20537/nd190405}
\elib{https://elibrary.ru/item.asp?id=43620849}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85084294708}


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    This publication is cited in the following articles:
    1. V. Dragovic, B. Gajic, B. Jovanovic, “Demchenko's nonholonomic case of a gyroscopic ball rolling without sliding over a sphere after his 1923 belgrade doctoral thesis”, Theor. Appl. Mech., 47:2 (2020), 257–287  crossref  zmath  isi  scopus
  • Russian Journal of Nonlinear Dynamics
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