Fermi acceleration in time dependent billiards

Billiards with time dependent boundaries are a natural generalization of the one dimensional Fermi accelerator model and a simple example of time dependent hamiltonian systems. Besides their interest from the point of view of dynamical systems, the fundamental question is if a particle moving inside a bounded region and undergoing elastic collisions with a moving boundary can be given unlimited energy. We will address this question, presenting some exact and numerical results which indicate that the answer depends on the shape of the boundary and on the nature of the phase space of the static model. In particular Fermi acceleration seems to be related to chaotic dynamics, while integrability seems to imply the boundedness of energy.