当前位置: X-MOL 学术J. Geometr. Phys. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Magnetic billiards: Non-integrability for strong magnetic field; Gutkin type examples
Journal of Geometry and Physics ( IF 1.5 ) Pub Date : 2020-08-01 , DOI: 10.1016/j.geomphys.2020.103716
Misha Bialy , Andrey E. Mironov , Lior Shalom

Abstract We consider magnetic billiards under a strong constant magnetic field. The purpose of this paper is two-folded. We examine the question of existence of polynomial integral of billiard magnetic flow. As in our previous paper (Bialy and Mironov, 2016) we succeed to reduce this question to algebraic geometry test on existence of polynomial integral, which shows polynomial non-integrability for all but finitely many values of the magnitude. In the second part of the paper we construct examples of magnetic billiards which have the so called δ -Gutkin property, meaning that any Larmor circle entering the domain with angle δ exits the domain with the same angle δ . For ordinary Birkhoff billiard in the plane such examples were introduced by E. Gutkin and are very explicit. Our construction of Gutkin magnetic billiards relies on beautiful examples by F.Wegner of the so called Zindler curves, which are related to the problem of floating bodies in equilibrium, which goes back to S.Ulam (Problem 19 in The Scottish Book, 1981). We prove that Gutkin magnetic billiard can be obtained as a parallel curve to a Wegner curve. Wegner curves can be written by elliptic functions in polar coordinates so the construction of magnetic Gutkin billiard is rather explicit but much more complicated.

中文翻译:

磁力台球:强磁场不可积;Gutkin 类型示例

摘要 我们考虑强恒磁场下的磁性台球。本文的目的有两个。我们考察了台球磁流多项式积分的存在性问题。正如我们之前的论文(Bialy 和 Mironov,2016 年)一样,我们成功地将这个问题简化为关于多项式积分存在性的代数几何测试,这表明多项式不可积对于除有限多个大小值之外的所有值。在论文的第二部分,我们构建了具有所谓 δ -Gutkin 特性的磁性台球的例子,这意味着任何以角度 δ 进入域的拉莫尔圆以相同的角度 δ 离开域。对于飞机上的普通 Birkhoff 台球,E. Gutkin 介绍了这样的例子,并且非常明确。我们对 Gutkin 磁性台球的构建依赖于 F.Wegner 的所谓 Zindler 曲线的漂亮例子,这些例子与平衡中的浮体问题有关,这可以追溯到 S.Ulam(苏格兰书中的问题 19,1981) . 我们证明 Gutkin 磁性台球可以作为与 Wegner 曲线平行的曲线获得。Wegner 曲线可以用极坐标中的椭圆函数来写,因此磁性 Gutkin 台球的构造相当明确,但要复杂得多。
更新日期:2020-08-01
down
wechat
bug