AI 中文总结
该研究拓展经典相对论性Størmer问题，纳入朗道-栗弗席兹辐射反作用，推导相关方程，给出解析解与数值结果，为强非均匀磁场中辐射驱动的相空间输运研究提供框架。
AI 中文摘要
我们通过朗道-栗弗席兹（Landau-Lifshitz）形式纳入辐射反作用，拓展了经典相对论性Størmer问题，从而为静态偶极磁场中耗散带电粒子运动提供自洽描述。我们推导了完整的无量纲方程，并将其与仅保留精确能量损失规律、但忽略与磁场梯度相关方向效应的简化阻力模型区分开。对于平面运动，简化系统给出粒子能量和正则角动量的精确瞬时演化规律，以及规则束缚天平动的平均输运方程。当运动被约束在瞬时圆分支时，长期演化可闭式积分并趋近于简单的大半径幂律。该解析解提供了有用的基准，不过圆分支在径向是不稳定的，因此不描述一般邻近轨迹。数值积分进一步展示了平面玫瑰形轨道的非均匀耗散变形。在三维情形中，完整的朗道-栗弗席兹力对小垂直扰动产生局域指数阻尼，主导贡献来自简化模型中缺失的场梯度项。这些精确、平均、条件及局域结果为研究强非均匀磁场中辐射驱动的相空间输运建立了可控解析框架，并为未来的全局模拟、动力学描述及相关电磁辐射计算奠定了基础。
英文摘要
We extend the classical relativistic Størmer problem by incorporating radiation reaction through the Landau--Lifshitz formulation, thereby providing a self-consistent description of dissipative charged-particle motion in a static dipole magnetic field. We derive the complete dimensionless equations and distinguish them from a reduced drag-only model that preserves the exact energy-loss law while omitting directional effects associated with magnetic field gradients. For planar motion, the reduced system yields exact instantaneous evolution laws for the particle energy and canonical angular momentum, together with averaged transport equations for regular bound librations. When the motion is constrained to the instantaneous circular branch, the secular evolution can be integrated in closed form and approaches a simple large-radius power law. This analytical solution provides a useful benchmark, although the circular branch is radially unstable and therefore does not describe generic nearby trajectories. Numerical integrations further illustrate the nonuniform dissipative deformation of planar rosette-like orbits. In three dimensions, the complete Landau--Lifshitz force produces local exponential damping of small vertical perturbations, with the leading contribution arising from the field-gradient term absent from the reduced model. These exact, averaged, conditional, and local results establish a controlled analytical framework for studying radiation-driven phase-space transport in strongly inhomogeneous magnetic fields and provide a foundation for future global simulations, kinetic descriptions, and calculations of the associated electromagnetic emission.
Comments16 pages, 2 figures
