An invariant variational principle for canonical flows on Lie groups

In this paper we examine the existence of Lie groups, whose canonical geodesic flows are variational with respect to a left-invariant regular—but not necessarily quadratic (i.e., metric)—Lagrange function. We give effective necessary and sufficient conditions for the existence of an invariant variational principle generating the canonical flow. With these results, taken in conjunction with the classification of Lie algebras, we solve the inverse problem of invariant Lagrangian dynamics in dimensions up to four.