磁台球：强磁场的不可积性; Gutkin型例子
Magnetic billiards: Non-integrability for strong magnetic field; Gutkin type examples
摘要：
我们考虑一个强恒定磁场下的磁性台球。本文的目的是双重的。研究了台球磁场中多项式积分的存在性问题。我们成功地把这个问题归结为多项式积分存在性的代数几何检验，证明了多项式积分的不可积性。在本文的第二部分中，我们构造了具有所谓的$δ$ -Gutkin性质的磁性台球的例子，这意味着任何以角度$δ$进入域的Larmor圆以相同的角度$δ$离开域。对于平面上的普通Birkhoff台球，E. Gutkin和是非常明确的。我们的Gutkin磁台球的结构依赖于F.Wegner所谓的Zindler曲线的美丽例子，这与平衡中的浮动物体的问题有关，这可以追溯到S. Ulam。我们证明了Gutkin磁台球可以作为一个平行的曲线韦格纳曲线。韦格纳曲线在极坐标下可用椭圆函数表示，因而磁性Gutkin台球的构造比较明确，但要复杂得多。
英文摘要：
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.
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