论文标题
磁场中的新类四二次集成系统：广义圆柱体和球形案例
New classes of quadratically integrable systems in magnetic fields: the generalized cylindrical and spherical cases
论文作者
论文摘要
我们研究具有三维欧几里得空间上具有二次运动积分的磁场的可整合和可促进的系统。与没有矢量电位的情况相反，相应的积分不再连接到汉密尔顿 - 雅各比方程中变量的分离，并且可以具有更一般的领先顺序项。我们专注于两种情况，以扩展物理相关的圆柱形和球形型积分。我们在广义圆柱情况下发现了三个新的可集成系统，但在球形上没有。我们猜想这与三维欧几里德代数的最大亚伯层亚代数的最大亚级代数相关，这是由一阶积分产生的。通过研究可累加性，我们在可集成的系统中仅发现一个（最少）可促进的系统。它在任何正交坐标系中均不分开。该系统提供了放置在无限螺线管中的螺旋波动器的数学模型。
We study integrable and superintegrable systems with magnetic field possessing quadratic integrals of motion on the three-dimensional Euclidean space. In contrast with the case without vector potential, the corresponding integrals may no longer be connected to separation of variables in the Hamilton-Jacobi equation and can have more general leading order terms. We focus on two cases extending the physically relevant cylindrical- and spherical-type integrals. We find three new integrable systems in the generalized cylindrical case but none in the spherical one. We conjecture that this is related to the presence, respectively absence, of maximal abelian Lie subalgebra of the three-dimensional Euclidean algebra generated by first order integrals in the limit of vanishing magnetic field. By investigating superintegrability, we find only one (minimally) superintegrable system among the integrable ones. It does not separate in any orthogonal coordinate system. This system provides a mathematical model of a helical undulator placed in an infinite solenoid.
