Integrable billiards on surfaces of constant curvature
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.
scientific article; zbMATH DE number 484234