Non-uniform drag force on the Fermi accelerator model

Some dynamical properties of a particle suffering the action of a generic drag force are obtained for a dissipative Fermi Acceleration model. The dissipation is introduced via a viscous drag force, like a gas, and is assumed to be proportional to a power of the velocity: F∝−v^γ$F\propto -v^{\gamma }$. The dynamics is described by a two-dimensional nonlinear area-contracting mapping obtained via the solution of Newton’s second law of motion. We prove analytically that the decay of high energy is given by a continued fraction which recovers the following expressions: (i) linear for γ=1$\gamma =1$; (ii) exponential for γ=2$\gamma =2$; and (iii) second-degree polynomial type for γ=1.5$\gamma =1.5$. Our results are discussed for both the complete version and the simplified version. The procedure used in the present paper can be extended to many different kinds of system, including a class of billiards problems.