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Precession and recession of the rock'n'roller
Author(s)
Date Issued
30 September 2009
Date Available
08T11:10:13Z April 2011
Abstract
We study the dynamics of a spherical rigid body that rocks and rolls on a plane
under the effect of gravity. The distribution of mass is non-uniform and the
centre of mass does not coincide with the geometric centre. The symmetric case,
with moments of inertia I1 = I2 < I3, is integrable and themotion is completely
regular. Three known conservation laws are the total energy E, Jellett’s quantity
QJ and Routh’s quantity QR. When the inertial symmetry I1 = I2 is broken,
even slightly, the character of the solutions is profoundly changed and new
types of motion become possible. We derive the equations governing the
general motion and present analytical and numerical evidence of the recession,
or reversal of precession, that has been observed in physical experiments. We
present an analysis of recession in terms of critical lines dividing the (QR,QJ )
plane into four dynamically disjoint zones. We prove that recession implies
the lack of conservation of Jellett’s and Routh’s quantities, by identifying
individual reversals as crossings of the orbit (QR(t ),QJ (t)) through the critical
lines. Consequently, a method is found to produce a large number of initial
conditions so that the system will exhibit recession.
Sponsorship
Not applicable
Type of Material
Journal Article
Publisher
IOP Publishing
Journal
Journal of Physics A: Mathematical and Theoretical
Volume
42
Issue
42
Start Page
425203 (25pp)
Copyright (Published Version)
2009 IOP Publishing Ltd
Keywords
Subject – LCSH
Rotational motion (Rigid dynamics)
Hamiltonian systems
Precession
Web versions
Language
English
Status of Item
Peer reviewed
ISSN
1751-8121 (Online)
This item is made available under a Creative Commons License
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