The generalized equation of motion for the orbital dynamics in the presence of current

The orbital dynamics induced by the charge current is studied theoretically. The equation of motion for the isospin vector $T^A$ in the SU(N) case is derived in the presence of the current, and is applied to the cases of $e_g$ (N=2) and $t_{2g}$ (N=3) orbitals. In spite of the anisotropic transfer integrals between orbitals, the dynamics is found to be isotropic for $e_g$-orbitals similarly to the spin case, while it is anisotropic for $t_{2g}$-orbitals. The implication of this result to the current driven orbital domain wall motion is discussed.