A qualitative analysis of the motion of a heavy solid of revolution on an absolutely rough plane

• Published in: Journal of applied mathematics and mechanics 1988, Vol.52 (2), p.159-165
• Main Author: Moshchuk, N.K.
• Format: Article
• Language: English
• ISSN: 0021-8928; 0021-8928
• DOI: 10.1016/0021-8928(88)90128-1

A strictly convex heavy solid of revolution, moving without slipping on a horizontal plane in a uniform gravitational field, is considered. Thanks to the existence of three first integrals (the explicit form of two of which is not known), the motion is amenable to qualitative analysis. The variation of the angle of nutation is studied and the motion of the point of contact on the surface of the body and on the supporting plane is analysed. It is shown that in phase space there are three-dimensional tori with conditionally-periodic motions. The problem of the motion of a body similar in shape and mass distribution to a dynamically and geometriçally symmetric body is considered. Generalizations of KAM-theory to reversible systems are used to establish the conservation of the majority of invariant tori. The problem of the rolling of a heavy body of revolution was first studied by Chaplygin /1/. The investigation was carried further in /2/. Up to now, fairly detailed attention has been devoted to the existence and stability of steady motions of a solid of revolution on an absolutely rough plane /3–8/. A qualitative analysis has been carried out /9/ of the motion without slipping of a heavy homogeneous tri-axial ellipsoid on a horizontal plane, on the assumption that it is similar to a sphere, and the periodic motions of the ellipsoid have been studied /10/.