The inverse problem for invariant Lagrangians on a Lie group

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.