  The relativistic spin dynamics defined by the pseudoscalar field of dark
matter axions is rigorously determined. The relativistic Hamiltonian in the
Foldy-Wouthuysen representation is derived. It agrees with the previously
obtained nonrelativistic Hamiltonians and the relativistic classical estimation
of the axion wind effect. In the relativistic Hamiltonian, the biggest term
describes the extraordinary (three orders of magnitude) enhancement of the
axion wind effect in storage ring experiments as compared with experiments with
immobile particles. This term defines the spin rotation about the longitudinal
axis. The effects caused by the axion-induced oscillating EDM and the axion
wind consist in the spin rotations about the different horizontal axes and
phases of stimulating oscillations differ by $\pi/2$. Experiments in a search
for dark matter axions has been discussed. The two experimental designs for
axion search experiments in storage rings have been elaborated.
