A relativistic bouncer on a vibrating surface

We consider a special relativity scenario in order to study the dynamical behavior of a particle flying under the influence of a gravitational field, colliding successive times against a rigid vibrating surface, via a restitution coefficient. We define two re-scaled dimensionless dynamical variables, namely: the relative particle velocity W with respect to the surface's velocity, and the real parameter τ accounting for the temporal evolution of the system. In order to analyze the system's nonlinear dynamical behavior, we construct the mapping at the particle-surface contact point, described for the k'th collision, by the couple of variables (τ k Wk). Then, from the dynamical mapping, we compute the fixed point trajectory and analyze its stability. We find the dynamical behavior of the fixed point trajectory to be stable or unstable, according to the values of the re-scaled vibrating surface amplitude, the restitution coefficient and the auxiliary variable β G (0,1), sweeping from the non relativistic to the ultra relativistic regime. Other important dynamical aspects such as the phase space volume and the one cycle vibrating surface (decomposed into absorbing and transmitting regions) are also discussed. Furthermore, the model rescues well known results in the non relativistic limit.