Realization of focal singularities of integrable systems using billiard books with a Hooke potential field

TL;DR

This paper demonstrates how to realize specific focal singularities of integrable systems with two degrees of freedom using billiard systems in a Hooke potential field, providing a geometric construction for these singularities.

Contribution

It introduces a method to realize nondegenerate focal singularities of integrable systems through billiard models in a Hooke potential field, linking geometric billiard dynamics with singularity theory.

Findings

Complete realization of semi-local focal singularities in billiard systems

Construction of billiard systems with singularity fibers homeomorphic to given ones

Extension of billiard models to integrable systems with Hooke potential

Abstract

Systems of particle motion in the Hooke central potential field on a billiard book glued from flat circular billiard domains are considered. An important class of nondegenerate focal singularities of the rank 0 of integrable systems with 2 degrees of freedom is completely realized by this class of billiards. Namely, for each semi-local focal singularity the constructed billiard system has a singularity fiberwise homeomorphic to the given one.