Dynamics of rigid spheres in complex motion

This paper presents solutions to selected challenging problems concerning the dynamics of two rigid spheres in complex motion. These unusual problems are formulated, without provided solutions, in old, hard-to-access literature [1-4] and were the subject of examinations at English universities at the end of the 19th century.1. INTRODUCTION
Problems in analytical dynamics concerning a system of rigid spheres in complex motion cause many difficulties in their solution. The presented tasks are contained in very old, hard-to-access literature [1-4] and originate from examination topics from the end of the 19th century at English universities. Due to the limited length of the paper, this study will provide the full solution to two problems, assuming that others can be solved using the ideas contained in these examples.
2. MATERIAL POINT AND RIGID SPHERE INSIDE A SPHERICAL VOID
CONTAINED IN A GLASS CUBE
In accordance with the notation adopted in Figure 1, we consider a glass cube with a centrally hollowed spherical void of a given radius R. Let the total mass of the cube together with the void be M. At the lowest position of the hollowed spherical void, there is initially a material point of mass m. The cube rests on an ideally smooth, horizontal, frictionless surface of the void. (…)

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