Matched pair analysis of the Vlasov plasma

We present the Hamiltonian (Lie-Poisson) analysis of Vlasov plasma, and dynamics its kinetic moments, from matched pair decomposition point view. We express these systems as couplings mutually interacting subdynamics. The mutual interaction is beyond well-known semi-direct product theory. Accordingly, geometric framework discussion, we address matched Lie-Poisson formulation allowing interactions. Moreover, both for moments plasma cases, observe that one constitutive subdynamics compressible isentropic fluid flow, other order \begin{document}$ \geq 2 $\end{document}. In this regard, algebraic/geometric (matched pair) offer, in perfect harmony with physical intuition. To complete a momentum along decomposition.