This article is cited in 8 papers

Nearly flat falling motions of the rolling disk

R. Cushman^ab, J. Duistermaat<sup>b

a</sup> University of Calgary, 2500 University Drive NW Calgary, Alberta, Canada T2N1N4
^b Mathematisch Instituut, Universiteit Utrecht, P.O. Box 80 010, 3508 TA Utrecht, The Netherlands

Abstract: We study the motion of a disk which rolls on a horizontal plane under the influence of gravity, without slipping or loss of energy due to friction. There is a codimension one semi-analytic subset $F$ of the phase space such that the disk falls flat in a finite time, if and only if its initial phase point belongs to $F$. We describe the motion of the disk when it starts at a point $p \notin F$ which is close to a point $f \in F$. It then almost falls flat, after which it rises up again. We prove that during the short time interval that the disk is almost flat, the point of contact races around the rim of the disk from a well defined position at the end of falling to a well defined position at the beginning of rising, where the increase of the angle only depends on the mass distribution of the disk and the radius of the rim. The sign of the increase of the angle depends on the side of $F$ from which $p$ approaches $f$.

Keywords: nonholonomic systems, rolling disk, nearly falling flat.

MSC: 70E18, 70E40, 70F25, 37J15, 37J60

Received: 17.07.2005
Accepted: 27.01.2006

Language: English

DOI: 10.1070/RD2006v011n01ABEH000333

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