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

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