Some results on ideal impacts of billiard balls

TL;DR

This paper investigates the impacts of billiard balls under idealized conditions using differential geometric methods, providing insights consistent with real-world behaviors through symbolic computation analysis.

Contribution

It applies recent theoretical results on ideal impacts to analyze billiard ball collisions in various ideal scenarios, utilizing symbolic computation for complex systems.

Findings

Impacts align with physical intuition and real system behaviors.

Different impact scenarios show consistent theoretical predictions.

Symbolic computation effectively handles high degrees of freedom.

Abstract

We analyze the impact of two equal billiard balls in three ideal situations: when the balls freely slide on the plane of the billiard, when they roll without sliding and when one of them freely slides and the other rolls. In all the cases we suppose that the contact between the balls is smooth. We base our analysis on some recent general theoretical results on ideal impacts obtained by means of Differential Geometric Impulsive Mechanics. We use symbolic computation software to solve the computational difficulties arising by the high number of degrees of freedom of the system. Some particular but significative impacts, with opportunely assigned left velocities and positions of the balls, are analyzed in details. The results admit easy interpretations that turn out to be in good agreement with the reasonable forecasts and the behaviours of real systems.