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A survey on polynomial in momenta integrals for billiard problems.

Misha Bialy1, Andrey E Mironov2

  • 1School of Mathematical Sciences, Tel Aviv University, Israel bialy@post.tau.ac.il.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|September 19, 2018
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This paper surveys algebraic Birkhoff conjecture results for integrable billiards. It also explores polynomial integrals for magnetic billiards, applying new algebraic techniques.

Keywords:
Birkhoff conjecturemagnetic billiardspolynomial integralstwo-sided magneticbilliards

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Area of Science:

  • Integrable Systems
  • Dynamical Billiards
  • Differential Geometry

Background:

  • The Birkhoff conjecture is a fundamental problem in the study of dynamical systems, particularly for billiard systems.
  • Integrable systems, characterized by a sufficient number of conserved quantities (integrals), exhibit predictable and often structured behavior.
  • Surfaces of constant curvature provide a rich geometric setting for studying billiard dynamics.

Purpose of the Study:

  • To provide a concise overview of recent advancements in the algebraic approach to the Birkhoff conjecture for integrable billiards on surfaces of constant curvature.
  • To discuss the theory and applications of integrable magnetic billiards.
  • To investigate the existence of polynomial integrals for a specific class of magnetic billiards (two-sided) using novel algebraic methods.

Main Methods:

  • Survey of existing literature on algebraic methods applied to the Birkhoff conjecture.
  • Discussion of theoretical frameworks for integrable magnetic billiards.
  • Application of algebraic techniques to analyze polynomial integrals in two-sided magnetic billiards.

Main Results:

  • Recent findings on the algebraic Birkhoff conjecture for integrable billiards on surfaces of constant curvature are presented.
  • The properties and integrability of magnetic billiards are discussed.
  • The existence of polynomial integrals for the Kozlov-Polikarpov two-sided magnetic billiards is established through algebraic analysis.

Conclusions:

  • The algebraic approach offers a powerful tool for studying integrable billiard systems and related conjectures.
  • Magnetic fields introduce new dynamics and integrability properties to billiard systems.
  • The study demonstrates the utility of algebraic techniques in uncovering new results for complex integrable systems.