When the cE x BB2 drift velocity is large enough to exceed the particle gyro velocity, adiabatic theory becomes more complicated and less useful: more complicated because there are five drifts in addition to the E¿B, gradient, and line curvature drifts and three more terms in the parallel equation of motion; less useful because the second and third invariants J and &PHgr; are no longer valid in mirror geometry due to the rapid drift across field lines that destroys any semblance of periodicity in the bounce motion. (Approximate periodicity is necessary to have an adiabatic invariant.) But the special case of a rapidly rotating rigid magnetic field, with E‖‖ = 0, is an exception. If the field at any time is merely a rotation of that at an earlier time, the drift and parallel equations simplify, and even better, the second invariant is again valid. The drift equation now has two rather than five additional terms-a Coriolis drift and a centrifugal drift. Just as in a slowly or nonrotating mirror system, the second invariant now provides a (rapidly rotating) dirft shell to which the guiding center is confined. The particle kinetic energy in the rotating frame, minus the centrifugal potential, is an exact constant of motion in a rigid rotator. This constant is well known but does not by itself put limits on radial motion toward or away from the rotation axis, hence it does not limit particle energy changes. But the drift shell does not limit and make periodic any radial excursions, and this is why previous studies of particle motion, in particular examples of rigid rotators, have shown that particles experience no steady energy gain or loss.