The EOMs under the linearization approximation (Eq. 98) are shown to have three equivalent forms, documented in Sec. VI with (1) the conjugated action-angle type variables (Eq. 102), (2) the generalized Euler angles on multi-dimensional Bloch sphere (Eq. 113), and (3) the MMST Cartesian phase space variables (Eq. 117). The TCF and the EOMs under the linearized approximation are used as a trajectory-based non-adiabatic method in the mixed Wigner/S-W formalism. We perform numerical simulations to demonstrate the accuracy of this method by simulating non-adiabatic population dynamics of challenging model systems. We emphasize that despite that our work is inspired by the recent spin mapping formalism, we have made several unique theoretical contributions in the current work, which are detailed in the Conclusion section.