Abstract
The bounce-action-angle coordinates (J,ζ) for charged particles trapped in an axisymmetric dipole magnetic field are constructed by perturbation analysis. First, the lowest-order bounce-action-angle coordinates (J0,ζ0) 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 (J=J0+ϵtJ1, ζ=ζ0+ϵtζ1), where the dimensionless parameter ϵt≡(sb/re)2⪡1 is defined as the ratio of the turning-point distance |sb| (measured from the equator) along a magnetic field line labeled by the equatorial distance re. 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∥}≡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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