We present the rigorous analytical derivation of the non-adiabatic dynamics using the generators (generalized spin matrices) of the su(N ) Lie algebra. Applying the S-W transform on the SU (N )-based mapping Hamiltonian provides the continuous variables (generalized spin coherence states) that can be viewed as the angle variables on a multi-dimensional Bloch sphere (or so-called general Euler angles ), hence establishing a mapping between discrete electronic states and continuous variables. The main advantage of the SU (N ) representation is that the corresponding S-W transform exactly preserves the identity operator in the N dimensional Hilbert space, as opposed to the MMST formalism where the identity is not preserved through the mapping and there is an ambiguity of how to evaluate it. This is because the MMST representation has a larger size of Hilbert space (that contains other states outside the single excitation manifold of the mapping oscillators) compared to the original electronic subspace, which then requires a projection back to the single excitation subspace of the mapping oscillators to obtain accurate results. The SU (N ) representation, on the other hand, completely alleviates these problems and is the most natural way to map a N -level system into a classical phase space. More discussions can be found in Appendix F.