Thomas-BMT equation generalized to electric dipole moments and field gradients

An expression is presented for the relativistic equations of motion, including field gradients, of a particle and its spin with electric and magnetic dipole moments aligned along the spin axis. An electromagnetic duality transformation is used to generalize a Thomas-BMT equation with gradient terms. Corrections to particle dynamics in storage rings for precision $(g-2)$ and electric dipole moment measurements are calculated, and applications to precision particle tracking programs are considered.