Anisotropic Kepler problem in a non–rotationally-symmetric Eaton lens

Abstract

In this paper, we prove that the anisotropic Kepler problem in two-body motion can be analogous to optical non--rotationally-symmetric Eaton lens. According to the numerical calculation and simulation results, when the mass ratio of anisotropic Kepler problem $\ensuremath{\mu}>1$, the corresponding non--rotationally-symmetric Eaton lens can simulate and reproduce the periodic orbits and chaotic characteristics of anisotropic Kepler problem in geometric optics and wave optics, respectively. In the extended study, when a beam is incident to the non--rotationally-symmetric Eaton lens from air, different from the light properties in traditional Eaton lens, the deflected beam is no longer along 180\ifmmode^\circ\else\textdegree\fi{}, and may have scatterings in other directions, which are influenced by the incident direction and mass ratio \ensuremath{\mu}. The results provide a successful reference for the analogy between optical and mechanical phenomena.