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In this paper we provide a matched pair decomposition of the space ofsymmetric contravariant tensors $\mathfrak{T}\mathcal{Q}$. From this proceduretwo complementary Lie subalgebras of $\mathfrak{T}\mathcal{Q}$ under\textit{mutual} interaction arise. Introducing a lift operator, the matchedpair decomposition of the space of Hamiltonian vector fields is determined.According to these realizations, Euler-Poincar\'{e} flows on such spaces aredecomposed into two subdynamics: one of which is the Euler--Poincar\'{e}formulation of isentropic fluid flows, and the other one corresponds withEuler--Poincar\'{e} equations on higher order contravariant tensors ($n\geq 2$)

Matched Pair Analysis of Euler-Poincar\'{e} Flow on Hamiltonian Vector Fields