The Jovian magnetosphere is a rapid rotator in the sense that the corotation velocity exceeds the gyrovelocity for most of the low-energy plasma particles. Ordinarily, an electric drift velocity greater than the gyrovelocity will destroy the near periodicity of the bounce motion, so the parallel invariant J will no longer be valid. However, this paper shows that J invariance actually remains valid in rapidly rotating magnetospheres so long as departures from rigid rotation are not too great. Northrop and Birmingham have already proved J invariance for the case of rapid but exactly rigid rotation, and this paper extends their results to nonrigid, rapid rotation with corotation lag and/or convection. These results allow the magnetic field to be arbitrarily asymmetric. The guiding-center equations in the rotating frame are also used to obtain conditions for plasma corotation in arbitrarily asymmetric but rigidly rotating magnetic fields. The plasma bulk velocity equals the corotation velocity for any trapped particle population that is steady state in the corotating frame provided E+(&OHgr;¿r)¿B/c=0 and all first-order drifts are negligible. These conditions are satisfied, and corotation is expected, for a steady-state cold plasma on equipotential field lines of any geometry threading a rigidly rotating central conductor.