Approximate symmetries of guiding-centre motion

TL;DR

This paper investigates approximate symmetries in guiding-centre motion, establishing conditions for quasisymmetry, including phase-space symmetries, and demonstrating that magnetohydrostatics enforces quasisymmetry at leading order.

Contribution

It extends the understanding of quasisymmetry by analyzing approximate symmetries and deriving weaker conditions, revealing broader classes of conserved quantities and the influence of magnetohydrostatics.

Findings

Leading-order conditions match exact quasisymmetry for purely spatial symmetries.

Allowing phase-space symmetries introduces weaker, broader conditions.

Magnetohydrostatics enforces quasisymmetry at leading order.

Abstract

Quasisymmetry builds a third invariant for charged-particle motion besides energy and magnetic moment. We address quasisymmetry at the level of approximate symmetries of first-order guiding-centre motion. We find that the conditions to leading order are the same as for exact quasisymmetry if one insists that the symmetry is purely spatial. We also generalise to allow for approximate phase-space symmetries, and derive weaker conditions. The latter recover "weak quasisymmetry" as a subcase, thus we prove it is spatial only to leading order, but also that it implies the existence of a wider class of independent approximate conserved quantities. Finally, we demonstrate that magnetohydrostatics imposes quasisymmetry to leading order.