  We treat the guiding-center dynamics in a non-uniform external Maxwell field
using a manifestly Lorentz covariant action principle which easily reproduces
the known Vandervoort equations of motion. We derive the corresponding kinetic
theory and ideal hydrodynamic theory. In contrast to conventional five-equation
hydrodynamics, the guiding-center hydrodynamics needs only three equations due
to a constraint on the motion across magnetic field. We argue that this
hydrodynamics applies more generally than the kinetic theory, e.g., for
strongly-coupled quark-gluon plasma.
