Newtonian reciprocality between spinmotive forces (SMF) and spin-torque-transfer (STT)

An SMF and the STT effect are reflected in definitions $$ \vec p = m\vec v + e\vec A_s \ \ \ \ \ {\rm and } \ \ \ \ \ H = H_0+e\Phi_s + eA_{0s}, $$ valid beyond the adiabatic approximation, where the momentum $m\vec v$ is mechanical, while $\vec p$ is that conjugate to the position $\vec r$, $H$ is the full Hamiltonian while $H_0$ is that for uniform magnet. Here $(\vec A_s, A_{0s})$ is the four vector which reflects the {\it linear\/} momentum of the magnetic system. The spin magnetic and electric fields $$ B_i = - \frac{im^2}{\hbar}\epsilon_{ijk}[v_j,v_k]\ \ \ \ {\rm and} \ \ \ \ E_i = - \frac{im^2}{\hbar} [v_i,v_0]; \ \ \ \ i = \{x,y,z\} $$ where $i\partial_t = mv_0 + e A_0$, and involve commutators. The Landau-Lfttshitz equations are an {\it emergent}. They correspond to $$ [S_z, (H_0+ e\Phi_s + eA_{0s})] = 0 $$ as required to separate the {\it slow\/} magnetic and {\it fast\/} electronic degrees of freedom. E.g. $ L = (enA \dot z - \dot q)\frac{\hbar}{e} \phi + g \mu_B B nAx - q{\mathcal E} $ is the Lagrangian for simple domain wall connected to a battery emf ${\mathcal E}$. SMF-SST reciprocality reflects Newtonian third law and {\it not\/} an Onsager relationship between transport coefficients. Experiment for spin-valves and MTJs will be reviewed.