Evolution of Motion of a Solid Suspended on a Thread in a Homogeneous Gravity Field

The plane motion of a solid in a uniform gravity field is studied. The solid is suspended on a weightless inextensible thread, which remains stretched during the motion. It is assumed that the length of the thread is large (similar to epsilon(-1/2)) and the distance from the point of solid suspension to its center of gravity is small (similar to epsilon). The equations of motion are presented as equations of a system with one rapidly rotating phase. This system is analyzed using classical perturbation theory and KAM theory. It is shown that for all values of time, the movement differs little (by similar to epsilon) from the slow oscillations of the thread in the vicinity of the descending vertical and the solid rotation relative to the suspension point with an almost constant angular velocity. The measure of the set of motions, different from the above motions, is estimated from above by the value of the order of exp (-c/epsilon)(c > 0 = const).