Decomposing Euler-Poincare Flow on the Space of Hamiltonian Vector Fields

The main result of this paper is a matched-pair decomposition of the space of symmetric contravariant tensors TQ. From this procedure two complementary Lie subalgebras of TQ under mutual interaction arise. Introducing a lift operator, the matched pair decomposition of the space of Hamiltonian vector fields is determined. According to this realization, the Euler-Poincare flows on such spaces are decomposed into two subdynamics: one is the Euler-Poincare formulation of isentropic fluid flows, and the other one corresponds with Euler-Poincare equations on contravariant tensors of order n (sic) 2.