The drifting of particles across flux surfaces is generally only a problem for trapped particles, which are trapped in a
magnetic mirror. Untrapped (or passing) particles, which can circulate freely around the flux surface, are automatically confined to stay on a flux surface. For trapped particles, omnigeneity relates closely to the
second adiabatic invariant \cal{J} (often called the parallel or longitudinal invariant). One can show that the radial drift a particle experiences after one full bounce motion is simply related to a derivative of \cal{J},\frac{\partial \cal{J}}{\partial \alpha} = q \Delta \psiwhere q is the charge of the particle, \alpha is the magnetic field line label, and \Delta \psi is the total radial drift expressed as a difference in toroidal flux. With this relation, omnigeneity can be expressed as the criterion that the second adiabatic invariant should be the same for all the magnetic field lines on a flux surface,\frac{\partial \cal{J}}{\partial \alpha} = 0This criterion is exactly met in axisymmetric systems, as the derivative with respect to \alpha can be expressed as a derivative with respect to the toroidal angle (under which the system is invariant). In
piecewise omnigenous fields, this criterion is met piecewisely on the flux surface. == References ==