On the Dynamics of Inverse Magnetic Billiards

TL;DR

This paper studies the behavior of charged particles in a plane with a convex obstacle and a magnetic field that varies across the boundary, introducing a new type of billiard system called inverse magnetic billiards.

Contribution

It introduces and analyzes the dynamics of inverse magnetic billiards, a novel billiard system with a magnetic field discontinuity, comparing it to existing billiard models.

Findings

Some theorems align with standard billiards

Some results differ from magnetic billiards

Provides insights into particle trajectories in mixed magnetic environments

Abstract

Consider a strictly convex set $\Omega$ in the plane, and a homogeneous, stationary magnetic field orthogonal to the plane whose strength is $B$ on the complement of $\Omega$ and $0$ inside $\Omega$. The trajectories of a charged particle in this setting are straight lines concatenated with circular arcs of Larmor radius $\mu$. We examine the dynamics of such a particle and call this inverse magnetic billiards. Comparisons are made to standard Birkhoff billiards and magnetic billiards, as some theorems regarding inverse magnetic billiards are consistent with each of these billiard variants while others are not.