Selected problems in the dynamics of a system of two rigid spheres

This paper presents solutions to selected, more difficult problems concerning the dynamics of two rigid spheres. These unusual problems are formulated, without solutions, in old, inaccessible 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 regarding the dynamics of a system of two rigid spheres cause many difficulties in their solution. Such problems, without solutions, are contained in very old, hard-to-access literature [1-4]. They were the subject of examination papers from the end of the 19th century at English universities. Due to the limited length of the paper, we will provide the full solution to two problems in this study.
Other, similar problems can be solved using the ideas contained in these examples.
2. TWO SMOOTH SPHERES AGAINST A SMOOTH
AND SMOOTH HORIZONTAL FLOOR
VERTICAL WALL
According to the markings in Figure 1, we consider two smooth, identical spheres, each with mass M and radius R. The spheres rest one on top of the other. We assume that there are no friction forces between the spheres and between the spheres and the vertical wall and horizontal floor.
A negligibly small, horizontal impulse of force applied to the lower sphere moves both spheres from an unstable equilibrium position, causing them to move under the influence of gravitational forces. (…)

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