发表机构
Universidad Autónoma Metropolitana – Azcapotzalco（墨西哥自治大学阿兹卡波察尔科分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究求解均匀磁场中球面带电粒子的利萨茹型轨道，发现其对应Randers型芬斯勒度量的测地线，为非黎曼几何的本科教学提供了实例。
AI 中文摘要
一个带电检验粒子被限制在均匀环境磁场中的单位球面上。我们通过初等方法将其运动约化为积分形式，并显式求解出到达两极的轨道族。在该轨道族上，方位角以恒定速率推进，纬度满足单摆方程。每振荡获得的绕转量是第一类完全椭圆积分，该积分是到正数集的严格递增双射；因此，每个正有理绕转量对应恰好一个半回旋频率值。随后我们探究这些轨道所属的几何：没有仿射联络能将它们包含在其测地线中，而Randers型芬斯勒度量可包含所有这些轨道，划分两种动力学 regime 的阈值恰好是该度量存在的条件，闭合轨道即为其闭合测地线。该例子使非黎曼几何可被高年级本科生或低年级研究生在经典力学课程中理解，我们提供推导过程、数值方法和习题。
英文摘要
A charged test particle is confined to the round sphere in a uniform ambient magnetic field. We reduce its motion to quadrature by elementary means and solve explicitly the family of trajectories which reaches the poles. On that family the azimuth advances at a constant rate and the latitude obeys a pendulum equation. The winding gained per oscillation is a complete elliptic integral of the first kind. That integral is a strictly increasing bijection onto the positive numbers. Every positive rational winding is therefore carried by exactly one value of the half-cyclotron frequency. Then we ask which geometry has these trajectories. No affine connection has them among its geodesics. A Finsler metric of Randers type has all of them. The threshold which separates the two dynamical regimes is precisely the condition for that metric to exist. The closed trajectories are its closed geodesics. The example places a non-Riemannian geometry within reach of an upper-level undergraduate or beginning graduate course in classical mechanics. We supply the derivations, the numerical recipes and the exercises.
