AI 中文总结
该研究针对均匀磁场中球面上带电测试粒子的运动，通过推广的诺特定理推导轨迹闭合判据，计算旋转数并确定其导数符号，还得到了恰当磁流的Mañé严格临界值。
AI 中文摘要
我们研究了被限制在单位球面上的带电测试粒子在均匀环境磁场中的运动。我们运用带磁力系统的诺特定理的推广形式，将该问题约化为求积问题。轨迹的旋转数（即其在纬度振荡一周所获得的方位角）是勒让德形式的第三类完全椭圆积分。轨迹闭合当且仅当该方位角增量为整周的有理倍数，该判据在两个首次积分的每个水平集上均成立。随后，我们对旋转数关于半回旋频率求导，并在水平集划分出的两个分支上确定该导数的符号。旋转数通过一个无量纲比率来体现电荷、质量、磁场和速度的影响，该比率为半回旋频率与速度之比。因此，闭合轨迹确定了该比率的一个取值。我们说明了对于给定的有理缠绕数，存在多少个这样的取值。极点可在单个水平集上达到，此时运动可约化为单摆运动。由于磁场穿过球面的通量为零，故其整体是恰当的。我们计算了由此产生的恰当磁流的Mañé严格临界值，当取值高于该临界值时，闭合轨迹是显式接触形式的闭合Reeb轨道。
英文摘要
We consider the motion of a charged test particle confined to the round unit sphere in a uniform ambient magnetic field. We use an extension of Noether's theorem for systems with magnetic forces to reduce the problem to a quadrature. The rotation number of a trajectory, the azimuth it gains over one oscillation in latitude, is a complete elliptic integral of the third kind in Legendre form. A trajectory closes if and only if that advance is a rational multiple of a full turn. The criterion holds on every level set of the two first integrals. Then, we differentiate the rotation number with respect to the half-cyclotron frequency and sign that derivative on each of the two branches into which a level set divides. The rotation number sees the charge, the mass, the field and the speed through a single dimensionless ratio, that of the half-cyclotron frequency to the speed. A closed trajectory therefore fixes a value of that ratio. We say how many values carry a given rational winding. The poles are attainable on a single level set, where the motion reduces to a pendulum. The field has no flux through the sphere, so it is globally exact. We compute the Mañé strict critical value of the resulting exact magnetic flow. Above that value the closed trajectories are closed Reeb orbits of an explicit contact form.
