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We apply the semi-classical limit of the generalized $SO(3)$ map forrepresentation of variable-spin systems in a four-dimensional symplecticmanifold and approximate their evolution terms of effective classical dynamicson $T^{\ast }\mathcal{S}_{2}$. Using the asymptotic form of the star-product,we manage to "quantize" one of the classical dynamic variables and introduce adiscretized version of the Truncated Wigner Approximation (TWA). Two emblematicexamples of quantum dynamics (rotor in an external field and two coupled spins)are analyzed, and the results of exact, continuous, and discretized versions ofTWA are compared.

Semi-Classical Discretization and Long-Time Evolution of Variable Spin Systems