Composite pulses are a series of pulses with specifically chosen phases, that may compensate for the imperfections of a conventional single pulse; thus their performance in steering a two-level system to a final state is feasible, accurate and robust. Professor Malcolm H. Levitt, who is considered the father of the composite pulses scheme in NMR, has mentioned the scarcity of available composite pulses schemes for higher level systems. In this work, we derive a general analytical theory of composite pulses for higher order irreducible representation of SU(2) dynamics of N-level systems, allowing to tailor the dynamics of physical multi-level systems, without imposing the approximation to a two-state solution. Our approach offers a platform for accurate steering of the population transfer in physical multi-level systems, which is crucial for fidelity in quantum computation and achieving fundamental excitations in nuclear magnetic resonances and atomic physics. We also introduce and discuss the geometrical trajectories of these dynamics on the Majorana sphere as an interpretation, allowing to gain physical insight on the dynamics of many-body or high-dimensional quantum systems. This work is at the interface between the field of NMR excitation techniques, generally performed with radiofrequency rectangular temporal shaped pulses, and optical ultrashort excitation methods, where the spacings between atomic levels are not necessarily equidistant.