Toroidal regularization of the guiding center Lagrangian

TL;DR

This paper introduces a coordinate transformation that regularizes the guiding center Lagrangian, removing singularities and simplifying the formulation of symplectic integrators for guiding center motion in magnetic fields.

Contribution

A novel coordinate transformation is proposed that eliminates singularities and simplifies canonical variable identification in guiding center dynamics.

Findings

Eliminates large-velocity singularity in guiding center equations.

Simplifies the transition to canonical coordinates, even without flux surfaces.

Provides a stable variational integrator for time-dependent electromagnetic fields.

Abstract

In the Lagrangian theory of guiding center motion, an effective magnetic field $\mathbf{B}^* = \mathbf{B}+(m/e)v_\parallel\nabla \times {\mathbf{b}}$ appears prominently in the equations of motion. Because the parallel component of this field can vanish, there is a range of parallel velocities where the Lagrangian guiding center equations of motion are either ill-defined or very badly behaved. Moreover, the velocity dependence of $\mathbf{B}^*$ greatly complicates the identification of canonical variables, and therefore the formulation of symplectic integrators for guiding center dynamics. This Letter introduces a simple coordinate transformation that alleviates both of these problems simultaneously. In the new coordinates, the Liouville volume element is equal to the toroidal cotravariant component of the magnetic field. Consequently, the large-velocity singularity is completely eliminated. Moreover, passing from the new coordinate system to canonical coordinates is extremely simple, even if the magnetic field is devoid of flux surfaces. We demonstrate the utility of this approach to regularizing the guiding center Lagrangian by presenting a new and stable one-step variational integrator for guiding centers moving in arbitrary time-dependent electromagnetic fields.