Pensive Billiards, Point Vortices, and the Silver Ratio

We define a new class of plane billiards - the pensive billiard - in which the billiard ball travels along the boundary for some distance depending on the incidence angle before reflecting, while preserving the billiard rule of equality of the angles of incidence and reflection. This generalizes so-called puck billiards proposed by M. Bialy, as well as a vortex billiard, that is, the motion of a point vortex dipole in two-dimensional hydrodynamics on domains with boundary. We prove the variational origin and invariance of a symplectic structure for pensive billiards, as well as study their properties including conditions for a twist map, the existence of periodic orbits, etc. We also demonstrate the appearance of both the golden and silver ratios in the corresponding hydrodynamical vortex setting. Finally, we introduce and describe basic properties of pensive outer billiards.