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Symmetric and asymmetric separation of variables for an integrable case of the complex Kirchhoff's problem
Journal of Geometry and Physics ( IF 1.6 ) Pub Date : 2021-11-22 , DOI: 10.1016/j.geomphys.2021.104418
T. Skrypnyk 1
Affiliation  

In the present paper we develop a method of the vector fields Zi [9] in the theory of separation of variables. For an integrable case of the complex Kirchhoff's problem on e(3), which has been never considered before, we construct—with the help of this method—two types of separation of variables (SoV): symmetric and asymmetric ones. Our asymmetric SoV is unusual: it is characterized by the quadratures containing differentials defined on two different curves of separation. It is a direct analogue of asymmetric SoV for the Clebsch model [17]. In the case of symmetric SoV both curves of separation are the same. This case has an additional bonus: on zero level set of one of the Casimir functions it yields a direct analogue of the famous Weber-Neumann separated coordinates. They are also considered in the present paper in some details.



中文翻译:

复杂基尔霍夫问题的可积情况下变量的对称和非对称分离

在本文中,我们开发了一种矢量场的方法 Z一世[9] 在变量分离理论中。对于复基尔霍夫问题的可积情况电子(3),这是以前从未考虑过的,我们在这种方法的帮助下构建了两种类型的变量分离 (SoV):对称和非对称。我们的非对称 SoV 是不寻常的:它的特点是包含定义在两条不同分离曲线上的差分的正交。它是 Clebsch 模型 [17] 的非对称 SoV 的直接模拟。在对称 SoV 的情况下,两条分离曲线是相同的。这种情况有一个额外的好处:在一个 Casimir 函数的零水平集上,它产生了著名的 Weber-Neumann 分离坐标的直接模拟。在本文中也详细讨论了它们。

更新日期:2021-11-25
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