Radiation reaction in the classical relativistic Størmer problem

Abstract

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.