  Given a matched pair of Lie groups, we show that the tangent bundle of the
matched pair group is isomorphic to the matched pair of the tangent groups. We
thus obtain the Euler-Lagrange equations on the trivialized matched pair of
tangent groups, as well as the Euler-Poincar\'e equations on the matched pair
of Lie algebras. We show explicitly how these equations cover those of the
semi-direct product theory. In particular, we study the trivialized, and the
reduced Lagrangian dynamics on the group $SL(2,\mathbb{C})$.
