RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Nelin. Dinam.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

Full text: PDF file (361 kB)
References: PDF file   HTML file

MSC: 37J60, 37J15, 70E18
Received: 26.06.2019
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}


Linking options:
  • http://mi.mathnet.ru/eng/nd673
  • http://mi.mathnet.ru/eng/nd/v15/i4/p457

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Нелинейная динамика
    Number of views:
    This page:14
    Full text:4
    References:1

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020