Relativistic spin dynamics conditioned by dark matter axions

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.