Qualitative analysis of motion of a heavy solid body on a smooth horizontal plane

Abstract

The motion of a solid body on a stationary absolutely smooth horizontal surface in a gravitational field is considered. The surface that bounds the body is convex, and the body differs little from a dynamically and geometrically symmetric one. This difference is defined by the magnitude if the small parameters ε. The unperturbed problem (when ε = 0) is integrable /1/. The basic aim is the investigation of motion for 0 < ε ⪡ 1. The nondegeneracy of Hamiltonian function of the unperturbed motion is shown and on the basis of Kolmogorov theorem /2,3/ is established that the perpetual closeness of variables “action” to their initial values which correspond to conditional periodic motions in the unperturbed problem. By this is established the smallness of variation of basic geometrical characteristics of the unperturbed motion, when the solid body differs little from the geometrically and dynamically symmetric one.