Time Dependent Billiards

We discuss in this chapter some dynamical properties for time dependent billiards. We construct the equations of the mapping that describe the dynamics of the particles considering that the velocity of the particle is given by the application of the momentum conservation law at each impact with the moving boundary. After the collision, the velocity of the particle changes, consequently a new pair of variables is added to the usual pair of angular variables, namely the velocity of the particle after the collision and the instant of the collision. We discuss the Loskutov–Ryabov–Akinshin (LRA) conjecture that says the existence of chaos in the billiard with static boundary is sufficient condition for the Fermi acceleration (unlimited energy growth) when the particle is time dependent. The conjecture was tested in the oval billiard leading to the unlimited energy growth. In the elliptical billiard, which is integrable for the fixed boundary, the time dependency of the boundary transforms the separatrix curve into a distribution of points called as stochastic layer hence leading to the unlimited energy growth and producing the Fermi acceleration.