On the number of independent adiabatic invariants for gyrating particles

TL;DR

This paper challenges the common belief that only three adiabatic invariants exist for gyrating particles, showing that additional invariants can be identified, which are useful for modeling plasma equilibria.

Contribution

It demonstrates the existence of more than three adiabatic invariants for gyrating particles, including a radial drift invariant, and explores their implications for plasma modeling.

Findings

Four useful invariants can exist for gyrating particles.

Radial drift invariant corresponds to the gyro center position.

Invariants enable MHD-type equilibrium modeling with local Maxwellian functions.

Abstract

It is pointed out that the three established adiabatic invariants are separating invariants in the sense of Liouville. It is widely claimed that no more than three adiabatic invariants can exist for the motion of a point charge. However, additional independent (not separating) adiabatic invariants do exist. For a force free motion, the components of angular momentum provide two additional constants of motion. This result can be generalized to the Hamilton Jacobi equation. The number of independent constants of motion is reduced if there is a global symmetry. For a gyrating particle, neglecting a gyro helix type of invariant, four 'useful' invariants could exist. A radial drift invariant, corresponding to the average of the radial coordinate of the particle, is a constant of motion for a confined gyrating particle. For the special case of a screw pinch where each gyro center moves on a magnetic flux surface without mirror trapping, the radial drift invariant is the radial coordinate of the gyro center. For a screw pinch, the set of constants of motion consising of the energy, parallel velocity and radial drift invariant is convenient to model the equilibrium. Local Maxwellian distribution functions expressed in this set of invariants are demonstrated to provide MHD-type of equilibria, for which it is straightforward to model the radial profiles of the particle and field components.