A computer-assisted geometric approach to the analysis of the impact of billiard balls. Part I: Ideal impacts

Abstract

It is shown how some recent general theoretical results on impacts of mechanical systems with unilateral constraints, obtained by means of Differential Geometric Impulsive Mechanics, can be operatively applied to the study of impacts between rigid bodies. The applicability of these geometric techniques is partly discussed for general impacts of rigid bodies. In particular, the general aspects of the algorithm are described and applied to analyze the impacts of two equal billiard balls moving on the plane in all possible ideal situations: when the balls can freely slide or roll on the plane of the billiard and/or between themselves. The use of symbolic-computation software is indispensable to solve the computational difficulties arising because of the high number of degrees of freedom of the system. It allows to obtain explicit expressions for the post-impact linear and angular velocities of the balls, and therefore a complete quantitative and qualitative analysis of any particular ideal impact with assigned pre-impact positions and velocities. The data regarding the usual billiard impact with the object ball at rest are explicitly listed and illustrated.