论文标题
磁性台球：强磁场的非整合性； Gutkin型示例
Magnetic billiards: Non-integrability for strong magnetic field; Gutkin type examples
论文作者
论文摘要
我们考虑在强恒定磁场下进行磁性台球。本文的目的是双重的。我们研究了台球磁流的多项式积分的存在问题。我们成功地将这个问题减少到了关于多项式积分存在的代数几何测试，该测试显示了所有幅度的多项式非共同性。在本文的第二部分中，我们构建了具有所谓的$δ$ - 古特金属性的磁性台球的示例，这意味着以相同的角度$δ$相同的角度进入域的任何Larmor圆圈进入域。对于飞机上的普通伯克霍夫（Birkhoff）台球，该例子是由E. gutkin引入的，非常明确。我们对Gutkin磁性台球的构造依赖于所谓的Zindler曲线的F.Wegner的精美例子，该曲线与平衡中的漂浮物体问题有关，后者可以追溯到S.Ulam。我们证明可以作为韦格纳曲线的平行曲线获得古特金磁台球。韦格纳曲线可以通过极性坐标中的椭圆函数写成，因此磁性古特金台球的构建相当明确，但要复杂得多。
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. 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. 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.
