  We discuss the problem of the existence of a regular invariant Lagrangian for
a given system of invariant second-order differential equations on a Lie group
$G$, using approaches based on the Helmholtz conditions. Although we deal with
the problem directly on $TG$, our main result relies on a reduction of the
system on $TG$ to a system on the Lie algebra of $G$. We conclude with some
illustrative examples.
