  Many physically important mechanical systems may be described with a Lie
group $G$ as configuration space. According to the well-known Noether's
theorem, underlying symmetries of the Lie group may be used to considerably
reduce the complexity of the problems. However, these reduction techniques,
used without care for general problems (waves, field theory), may lead to
uncomfortable infinite dimensional spaces. As an alternative, the
\emph{covariant} formulation allows to consider a finite dimensional
configuration space by increasing the number of independent variables. But the
geometric elements needed for reduction, adapted to the specificity of
covariant problems which admit Lie groups as configuration space, are difficult
to apprehend in the literature (some are even missing to our knowledge). To
fill this gap, this article reconsiders the historical geometric construction
made by E. Cartan in this particular "covariant Lie group" context. Thus, and
it is the main interest of this work, the Poincar\'e-Cartan and
multi-symplectic forms are obtained for a principal $G$ bundle. It allows to
formulate the \emph{Euler-Poincar\'e equations} of motion and leads to a
Noether's current form defined in the dual Lie algebra.
