Abstract

Orbital invariants of integrable billiards on two-dimensional book tables are studied at constant energy values. These invariants are calculated from rotation functions defined on one-parameter families of Liouville 2-tori. For two-dimensional billiard books, a complete analogue of Liouville’s theorem is proved, action–angle variables are introduced, and rotation functions are defined. A general formula for the rotation functions of such systems is obtained. For a number of examples, the monotonicity of these functions is studied, and edge orbital invariants (rotation vectors) are calculated. It turned out that not all billiards have monotonic rotation functions, as was originally assumed by A. Fomenko’s hypothesis. However, for some series of billiards, this hypothesis is true.

中文翻译：

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作为轨道不变量的可积台球的旋转函数

摘要

研究了二维书桌上可积台球在恒定能量值下的轨道不变量。这些不变量是根据在 Liouville 2-tori 的单参数族上定义的旋转函数计算的。对于二维台球书，证明了刘维尔定理的完整模拟，引入了作用角度变量，并定义了旋转函数。获得了此类系统的旋转函数的通用公式。通过一些例子，研究了这些函数的单调性，并计算了边缘轨道不变量（旋转向量）。事实证明，并非所有台球都具有单调旋转函数，正如 A. Fomenko 的假设最初所假设的那样。然而，对于某些系列的台球来说，这个假设是正确的。