We discuss an extension of the velocity Verlet method that accurately approximates the kinetic-energy-conserving charged particle motion that comes from magnetic forcing. For a uniform magnetic field, the method is shown to conserve both particle kinetic energy and magnetic dipole moment better than midpoint Runge–Kutta. We then use the magnetic velocity Verlet method to generate trapped particle trajectories, both in a cylindrical magnetic mirror machine setup and for dipolar fields like the earth's magnetic field. Finally, the method is used to compute an example of (single) mirror motion in the presence of a magnetic monopole field, where the trajectory can be described in closed form.
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We encourage curious readers to find the source current density that produces this field and compare that source with the one that produced the field in Eq. (28). In addition, one can compare the field in Eq. (29) with the full, off-axis, field produced by a pair of current loops placed at along the z axis.
There is also the possibility that the particle is not trapped at all. One can develop, from the expression for μ, constraints on the ratio of the perpendicular to longitudinal velocity components that prevent trapping. We leave this interesting opportunity, and its numerical verification, for the reader (Refs. 15 and 8).
