References

The elliptical stadium is a plane region bounded by a curve \Gamma, constructed by joining two half-ellipses, with major axes a? 1 and minor axes b = 1, by two straight segments of equal length 2h. The billiard on the elliptical stadium consists in the study of the free motion of a point particle inside the stadium, being reflected elastically at the impacts with \Gamma

The presence and lack of Fermi acceleration in nonintegrable billiards

The unlimited energy growth ( Fermi acceleration) of a classical particle moving in a billiard with a parameter-dependent boundary oscillating in time is numerically studied. The shape of the boundary is controlled by a parameter and the billiard can change from a focusing one to a billiard with dispersing pieces of the boundary. The complete and simplified versions of the model are considered in the investigation of the conjecture that Fermi acceleration will appear in the time-dependent case when the dynamics is chaotic for the static boundary. Although this conjecture holds for the simplified version, we have not found evidence of Fermi acceleration for the complete model with a breathing boundary. When the breathing symmetry is broken, Fermi acceleration appears in the complete model

Fermi acceleration in non-autonomous billiards

Fermi acceleration can be modelled by a classical particle moving inside a time-dependent domain and elastically reflecting from its boundary. In this paper, we describe how the results from the dynamical system theory can be used to explain the existence of trajectories with unbounded energy. In particular, we show for slowly oscillating boundaries that the energy of the particle may increase exponentially fast in time