On the number of independent adiabatic invariants for gyrating particles
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.