# A Particular Integrable Case in the Nonautonomous Problem of a Chaplygin Sphere Rolling on a Vibrating Plane

A Particular Integrable Case in the Nonautonomous Problem of a Chaplygin Sphere Rolling on a... In this paper we investigate the motion of a Chaplygin sphere rolling without slipping on a plane performing horizontal periodic oscillations. We show that in the system under consideration the projections of the angular momentum onto the axes of the fixed coordinate system remain unchanged. The investigation of the reduced system on a fixed level set of first integrals reduces to analyzing a three-dimensional period advance map on \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$SO(3)$$\end{document}. The analysis of this map suggests that in the general case the problem considered is nonintegrable. We find partial solutions to the system which are a generalization of permanent rotations and correspond to nonuniform rotations about a body- and space-fixed axis. We also find a particular integrable case which, after time is rescaled, reduces to the classical Chaplygin sphere rolling problem on the zero level set of the area integral. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Regular and Chaotic Dynamics Springer Journals

# A Particular Integrable Case in the Nonautonomous Problem of a Chaplygin Sphere Rolling on a Vibrating Plane

, Volume 26 (6) – Nov 1, 2021
12 pages

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Publisher
Springer Journals
Copyright
Copyright © Pleiades Publishing, Ltd. 2021
ISSN
1560-3547
eISSN
1468-4845
DOI
10.1134/s1560354721060149
Publisher site
See Article on Publisher Site

### Abstract

In this paper we investigate the motion of a Chaplygin sphere rolling without slipping on a plane performing horizontal periodic oscillations. We show that in the system under consideration the projections of the angular momentum onto the axes of the fixed coordinate system remain unchanged. The investigation of the reduced system on a fixed level set of first integrals reduces to analyzing a three-dimensional period advance map on \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$SO(3)$$\end{document}. The analysis of this map suggests that in the general case the problem considered is nonintegrable. We find partial solutions to the system which are a generalization of permanent rotations and correspond to nonuniform rotations about a body- and space-fixed axis. We also find a particular integrable case which, after time is rescaled, reduces to the classical Chaplygin sphere rolling problem on the zero level set of the area integral.

### Journal

Regular and Chaotic DynamicsSpringer Journals

Published: Nov 1, 2021

Keywords: Chaplygin sphere; rolling motion; nonholonomic constraint; nonautonomous dynamical system; periodic oscillations; permanent rotations; integrable case; period advance map

### References

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