Integrable topological billiards and equivalent dynamical systems
Integrable topological billiards and equivalent dynamical systems
复制标题
可积拓扑台球和等效动力系统
DOI：
10.1070/im8602
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
A. Fomenko
中科院分区：
文献类型：
--
作者：
V. V. Fokicheva;A. Fomenko
我们考虑几个拓扑可积台球，并证明他们是刘维尔等价于许多系统的刚体动力学。证明使用Fomenko- Zieschang理论的不变量的可积系统。我们研究了共焦二次曲面的弧和它们的推广，广义台球，其中的运动发生在一个局部平面上的表面上，通过粘合几个平面域等距沿着他们的边界，这是共焦二次曲面的弧。本文描述了两类新的以共焦二次曲面的弧为界的可积台球，即非紧台球和广义台球，它们是通过将平面台球沿着非凸部分的边界胶合而得到的。本文对以共焦二次曲面弧为边界的非紧台球进行了完全分类，并利用描述附加积分奇异叶分叉的Fomenko不变量研究了其拓扑结构。研究了一类非凸广义台球的等能曲面的拓扑。事实证明，他们拥有异国情调的刘维尔叶理：积分轨迹的台球，躺在一些奇异的叶子不承认连续的延伸。这样的台球似乎是leafwise相当于台球的共焦二次曲面的闵可夫斯基度量的弧。
We consider several topological integrable billiards and prove that they are Liouville equivalent to many systems of rigid body dynamics. The proof uses the Fomenko– Zieschang theory of invariants of integrable systems. We study billiards bounded by arcs of confocal quadrics and their generalizations, generalized billiards, where the motion occurs on a locally planar surface obtained by gluing several planar domains isometrically along their boundaries, which are arcs of confocal quadrics. We describe two new classes of integrable billiards bounded by arcs of confocal quadrics, namely, non- compact billiards and generalized billiards obtained by gluing planar billiards along non-convex parts of their boundaries. We completely classify non- compact billiards bounded by arcs of confocal quadrics and study their topology using the Fomenko invariants that describe the bifurcations of singular leaves of the additional integral. We study the topology of isoenergy surfaces for some non-convex generalized billiards. It turns out that they possess exotic Liouville foliations: the integral trajectories of the billiard that lie on some singular leaves admit no continuous extension. Such billiards appear to be leafwise equivalent to billiards bounded by arcs of confocal quadrics in the Minkowski metric.