Abstract
This paper discusses the calculation of the adiabatic invariant of a charged particle moving in an axisymmetric magnetic field with straight field lines. This calculation can be reduced to quadratures, and in several cases, the exact analytical results can be obtained. In particular, an exact expression is obtained for the charged particle motion in the equatorial field of a magnetic dipole, which can have important applications to the high-order guiding center theory for the particles in the terrestrial radiation belts. Another closed-form analytical result corresponds to magnetic field intensity which is inversely proportional to the radius. The results represent an extension of the classical series expansions of Kruskal for these specific magnetic fields.
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