Introduction to Billiard Dynamics - Leonel, Edson Denis [UNESP] (My University, 2021)
We discuss in this chapter the elementary concepts of billiards. In a billiard, a particle or in an equivalent way an ensemble of non-interacting particles move freely along a closed boundary to where they collide. The characterization of the time evolution of the particles is made by using a discrete mapping in the variables describing the position of the particle at the boundary given by the polar angle and the angle the trajectory the particle makes with the tangent line at the instant of the impact. We use three different types of boundary leading to different dynamics. One of them is the circular billiard, which is integrable. The other shape is the elliptical billiard, which is also integrable and finally an oval billiard, which shows mixed phase space being then non-integrable.