Perturbation analysis of trapped-particle dynamics in axisymmetric dipole geometry

The perturbation analysis of the bounce action-angle coordinates $(J,ζ)$ for charged particles trapped in an axisymmetric dipole magnetic field is presented. First, the lowest-order bounce action-angle coordinates are derived for deeply-trapped particles in the harmonic-oscillator approximation. Next, the Lie-transform perturbation method is used to derive higher-order anharmonic action-angle corrections. Explicit expressions (with anharmonic corrections) for the canonical parallel coordinates $s(J,ζ)$ and $p_{\|}(J,ζ)$ are presented, which satisfy the canonical identity $\{s,\; p_{\|}\}(J,ζ) \equiv 1$. Lastly, analytical expressions for the bounce and drift frequencies (which include anharmonic corrections) yield excellent agreement with exact numerical results.

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