A survey on Polynomial in momenta integrals for billiard problems

TL;DR

This paper surveys recent algebraic approaches to integrable billiards on constant curvature surfaces, discusses magnetic billiards, and explores polynomial integrals for two-sided magnetic billiards, highlighting new applications of algebraic techniques.

Contribution

It provides a comprehensive survey of algebraic methods in billiard integrability and introduces new applications to magnetic billiards and polynomial integrals.

Findings

Recent results on algebraic Birkhoff conjecture for billiards

Discussion of integrable magnetic billiards

Existence of polynomial integrals for two-sided magnetic billiards

Abstract

In this paper we give a short survey of recent results on algebraic version of the Birkhoff conjecture for integrable billiards on surfaces of constant curvature. We also discuss integrable magnetic billiards. As a new application of the algebraic technique we study the existence of polynomial integrals for the two-sided magnetic billiards introduced by Kozlov and Polikarpov.