Billiard systems with polynomial integrals of third and fourth degree (Q2716879)

A billiard dynamical system can be specified by a Lagrangian NEWLINE\[NEWLINEL= \tfrac 12 [\dot{\theta}^2 + f(\theta)\dot{\varphi}^2] - V(\theta, \varphi)NEWLINE\]NEWLINE in the polar coordinates \(\theta, \varphi\) mod \(2\pi\). The function \(f(\theta)\) determines the boundary of the billiard table and \(V\) stands for an external potential. In a previous paper, the author studied the existence of polynomial-in-momenta integrals of motion, independent of the total energy. She proved that a linear integral only exists if \(\varphi\) is a cyclic coordinate, and the existenceof a quadratic integral leads to the separation of variables \(V v(\varphi) / f(\theta)\), where \(v\) is a \(2\pi\)-periodic function. In this paper she constructs examples of billiards with irreducible polynomial integrals of third and fourth degrees.