New integrable problems of motion of a rigid body with a particle oscillating or bouncing in it

Some qualitative aspects of the problem of motion about a fixed point of a rigid body with a particle moving in it in a prescibed (sinusoidal) way was treated in [1–3]. The mechanical system comprised of a rigid body containing an internal mass that moves along a fixed line in the body was considered in several works [4–5]. Recently, an integrable case of this system was found, in which the body is dynamically axisymmetric and moves under no external forces while the particle moves on the axis of dynamical symmetry under the action of Hooke's force to the fixed point [5].In the present note we introduce a more general integrable case in which the particle moves on the axis of dynamical symmetry and is subject to an arbitary conservative force that depends only on the distance from the fixed point. Separation of variables is accomplished and the solution is reduced to quadratures. As a special version of this problem, the case when the particle bounces elastically between two points is briefly discussed.