Abstract

In the paper, a new class of integrable billiards, namely, billiards with slipping, is studied. At the reflection from the boundary, a billiard particle of such a system may not only change its velocity, but also move some distance along the border. Some laws of slipping preserve the integrability of flat confocal and circular billiards and billiard books glued from them, i.e., billiards on cell complexes. In the paper, the topology of the Liouville foliations for several integrable billiards with slipping, both flat and locally flat, is studied. Two such systems are Liouville equivalent to integrable geodesics flows of small degrees on nonorientable two-dimensional surfaces, namely, the projective plane and the Klein bottle. This shows that the nonorientability of a two-dimensional surface, in itself, is not an obstacle to its implementability by an appropriate integrable billiard.

中文翻译：

---

滑动的拓扑台球的Liouville叶面

摘要

在本文中，研究了一类新型的可整合台球，即打滑台球。在边界反射时，这种系统的台球粒子不仅会改变其速度，而且还会沿边界移动一些距离。一些滑移定律保留了平坦的共焦和圆形台球以及由它们胶合的台球书籍（即细胞复合体上的台球）的可集成性。在本文中，研究了几种具有滑动性的可整合台球（无论是平坦的还是局部平坦的）的Liouville叶面的拓扑。两个这样的系统是Liouville等效于不可定向二维表面（即投影平面和Klein瓶）上小程度的可积分测地线流。这表明二维表面本身的非定向性