First integrals in the problem of the motion of a heavy rigid body suspended on a string

Abstract

The problem of the motion of a heavy rigid body suspended on an inextensible string is considered. The conditions ensuring the existence of first integrals, which hold both when the string is stretched and during free flight, are formulated. The possibility of extending the results to the case where the motion is executed by a chain of bodies is discussed. It is known that three additional integrals are lacking for the integrability of the equations of the motion of a heavy body on a stretched string in the general case. However, this problem is completely integrable when the body is suspended at the center of mass. In this case, the integration is performed by separation of variables [1]. It is also known that the problem has one additional integral when the body is suspended at a point in the axis of dynamic symmetry [1]. The existence of particular integrals such as the Hess integral [2] for this problem was studied in [3]. The problem was generalized to the case of chains of rigid bodies in [4]. The general issue of the existence of first integrals in the problems of rigid-body dynamics in the presence of unilateral constraints was discussed in [5, 6].