An integrable system with a nonintegrable constraint

Abstract

INTRODUCTIONThepaper[1]studiesthemotion ofadynamicallysymmetricbalancedrounddiskonahorizontalsurface covered with smooth ice. Thus, the system is subjected to a nonholonomic constraint:the velocity of the disk’s point of tangency is parallel to its horizontal diameter. In [1], explicitquadratures were obtained and it was shown that the disk will almost never fall on the plane andthat the point of tangency performs a complicated motion in the general case; namely, it movesperiodically along a closed analytic curve that rotates as a rigid body about a fixed point at aconstant angular velocity.Stationary motions of the disk on horizontal ice and their stability were studied in [2]. In [3], thefall probability was studied and particular solutions were indicated for the already nonintegrableproblem on the motion of the disk on a tilted ice plane. The existence of an invariant measure wasused to establish, by the Schwarzschild–Littlewood theorem [4], that the set of fall trajectories hasmeasure zero (but is not empty).The paper [5] put forward the problem of findingthe probability of falling of a round disk in thepresence of dynamic asymmetry. In this case, attempts to find an invariant measure for the equa-tions have failed (apparently, there exists no invariant measure), and the Schwarzschild–Littlewoodtheorem does not apply. In the present paper, we show that there exists an invariant measure fora disk sliding on an icy sphere in the gravity field if the disk’s center of mass lies at an arbitrarypoint of the axis perpendicular to the plane of the disk and passing through the disk’s center.This results in the zero measure of the set of fall trajectories. For inertial motion, the problem isintegrable, and we manage to obtain a solution in quadratures.1. EQUATIONS OF MOTION AND AN INVARIANT MEASURELet a body with sharp circular edge move on the surface of a sphere covered with ice. Thismeans that the point of tangency of the circle with the sphere cannot move (slip) in the directionperpendicular to the plane in which the circle lies. We also assume that the center of mass of thesystem lies on the circle’s symmetry axis at a distance