  We present the Euler-Lagrange and Hamilton's equations for a system whose
configuration space is a unified product Lie group $G=M\bowtie_{\gamma} H$, for
some $\gamma:M\times M \to H$. By reduction, then, we obtain the Euler-Lagrange
type and Hamilton's type equations of the same form for the quotient space
$M\cong G/H$, although it is not necessarily a Lie group. We observe, through
further reduction, that it is possible to formulate the Euler-Poincar\'{e} type
and Lie-Poisson type equations on the corresponding quotient $\mathfrak{m}\cong
\mathfrak{g}/\mathfrak{h}$ of Lie algebras, which is not a priori a Lie
algebra. Moreover, we realize the $n$th order iterated tangent group $T^{(n)}G$
of a Lie group $G$ as an extension of the $n$th order tangent group $T^nG$ of
the same type. More precisely, $\mathfrak{g}$ being the Lie algebra of $G$,
$T^{(n)}G \cong \mathfrak{g}^{\times \,2^n-1-n} \bowtie_\gamma T^nG$ for some
$\gamma:\mathfrak{g}^{\times \,2^n-1-n} \times \mathfrak{g}^{\times \,2^n-1-n}
\to T^nG$. We thus obtain the $n$th order Euler-Lagrange (and then the $n$th
order Euler-Poincar\'e) equations over $T^nG$ by reduction from those on
$T(T^{n-1}G)$. Finally, we illustrate our results in the realm of the Kepler
problem, and the non-linear tokamak plasma dynamics.
