  We theoretically study the antiferromagnetic domain wall motion actuated by
an inhomogeneous external magnetic field. The Lagrangian and the equations of
motion of antiferromagnetic spins under an inhomogeneous magnetic field are
derived, first in terms of the N\'eel vector, and then using collective
coordinates of the domain wall. A solution is found that describes the
actuation of a domain wall by an inhomogeneous field, in which the motion is
initiated by a paramagnetic response of wall magnetization, which is then
driven by a Stern-Gerlach like force. The effects of pinning potential are also
investigated. These results are in good agreement with atomistic simulations.
While the present formulation contains the so-called intrinsic magnetization
associated with N\'eel texture, a supplementary discussion is given to
reformulate the theory in terms of physical magnetization without the intrinsic
magnetization.
