Billiards - An introduction to Dynamical Systems with Impact: Elements of impact theory, genetic method, integrable and non integrable billiards, KAM theory

An impact is a short term interaction of bodies. It is assumed that the impact occurs almost instantaneously. The positions of the colliding bodies do not change at the moment of impact while their velocities acquire finite increments. A central feature of the impact theory is therefore, investigation of the relationship between the velocities before and after impact. The transformation law of the velocities in an impact interaction can be represented in purely geometric form and therefore in the simplest cases while describing the motion of systems with impacts, it is possible to get by with entirely elementary tools. The first detailed investigation of an impact phenomena was undertaken in 1668 at the suggestion of Royal Society of London. Three outstanding mathematicians Wallis, Wren and Huygens presented their works in which they expounded the laws of motion of colliding bodies.Later in 1687, Newton referred to these experiments in Mathematical Foundations of Natural Philosophy. Here, we elaborate in greater detail, what was done 350 years ago.