Matched pair analysis of the Vlasov plasma

We perform Hamiltonian (Lie-Poisson) analysis of the Vlasov plasma, and the dynamics of its kinetic moments, from the matched pair decomposition point of view. We express both of the (Lie-Poisson) systems as couplings of two of their \textit{mutually interacting} (Lie-Poisson) subdynamics. Mutually acting systems are beyond the well-known semi-direct product theory. Accordingly, as the geometric framework of the present discussion, we address \textit{matched pair Lie-Poisson} formulation permitting mutual interactions. Then, all mutual actions, as well as dual and induced cross-actions, are clearly computed for the kinetic moments and the Vlasov plasma. For both cases, we observe that one of the constitutive subdynamics is the compressible isentropic fluid flow, and the other is the higher-order ($\geq 2$) kinetic moments. In this regard, the algebraic/geometric (matched pair) decomposition that we offer, is in perfect harmony with the physical intuition. To complete the discussion, we present a momentum formulation of the Vlasov plasma and, obtain the matched pair decomposition of this realization as well.