Pensive billiards, point vortices, and the silver ratio

TL;DR

This paper introduces pensive billiards, a new class of plane billiards with boundary-dependent travel distances, explores their mathematical properties, and reveals connections to the golden and silver ratios in hydrodynamical vortex models.

Contribution

It defines pensive billiards, proves their variational and symplectic properties, and links these dynamics to the golden and silver ratios in vortex systems.

Findings

Pensive billiards generalize puck and vortex billiards.

Existence of periodic orbits and twist map conditions.

Appearance of golden and silver ratios in vortex dynamics.

Abstract

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', i.e. the motion of a point vortex dipole in 2D 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.