Integrable billiards on surfaces of constant curvature
type: Item
instance of: scholarly article
title: Integrable billiards on surfaces of constant curvature
zbMATH Open document ID: 0795.58028
DOI: 10.1007/BF02102114
published in: Mathematical Notes
publication date: 1994-02-08T00:00:00Z
review text: The work is devoted to searching for conditions under which a billiard on a surface of constant curvature is integrable. This implies to find necessary and sufficient conditions for the structure of the billiard's boundary which provide for the integrability. The integrability is realized according to Birkhoff as existence of higher integrals of motion. It is proved that, if the billiard's boundary is built of pieces of curves, each of which is a set of points whose radius-vectors belong to the kernel of the matrix \((B-\lambda I)\), \(B\) being a symmetric matrix and \(\lambda\) being its eigenvalue, the billiard is integrable. Actually, these curves must be algebraic curves of the first or second degree. For instance, on the surface of the zeroth curvature, the curves must be cofocal conics. It is also proved that, under some additional technical assumptions, the boundary of the integrable billiard must consist of pieces of the algebraic curves of the second degree, or pieces of geodesics on the surfaces of constant curvature.
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 58E17
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 34C40
zbMATH DE Number: 484234
zbMATH Keywords: Birkhoff integrability
zbMATH Keywords: algebraic curves
zbMATH Keywords: geodesics
author: S. V. Bolotin
reviewed by: B. A. Malomed
MaRDI profile type: Publication
cites work: Q5665930
cites work: Q3935173
cites work: Poisson manifolds and the Schouten bracket
cites work: First integrals of systems with elastic reflections
cites work: Q3312348
cites work: Q3667659
description: scientific article; zbMATH DE number 484234