In this work, we provide rigorous theoretical derivations of non-adiabatic mapping dynamics in the phase space of the SU (N ) Lie group. Thus, the current work can be viewed as a rigorous theoretical justification of the generalized spin mapping formalism by Rune-son and Richardson. We will not distinguish between the nomenclature of generalized spin mapping and the SU (N ) mapping in this paper when we mention this mapping formalism. In Sec. II, we briefly review the generators of the su(N ) Lie algebra which are used as the operator basis to represent the Hamiltonian and any operators in the electronic Hilbert space, and derive the analytic expression of the structure constants of the su(N ) Lie algebra (presented in Appendix A). In Sec. III we briefly review some basic properties of the S-W transform which effectively constructs a mapping relation between an operator described by the generators in the Hilbert space to a continuous function on the Lie group/manifold. Note that Sec. II and Sec. III are used to set up the theoretical background for our work presented in this paper, where similar expressions/discussions can be found in part in the previous works. Sec. IV to Sec. VI present our main theoretical contributions in this work. In Sec. IV, we present a mixed Wigner/S-W representation that performs Wigner transform on the nuclear DOFs and S-W transform on the electronic DOFs, and derive the exact expressions of the time-correlation function (TCF). In Sec. V, we derive the exact quantum Liouvillian expression (Eq. 85-89). under the mixed Wigner/S-W representation. Using this exact expression, we can explore approximate forms of the Liouvillian under the linearization approximation, which gives the EOMs with the variables of the spin coherent state expectation values (Eq. 98). These EOMs are identical to those proposed in the GDTWA approach, which can also be viewed as a generalization of the Hioe-Eberly theory of the N -dimensional spin precession with the explicit presence of the nuclear DOFs.