  The language of Lagrangian submanifolds is used to extend a geometric
characterization of the inverse problem of the calculus of variations on
tangent bundles to regular Lie algebroids. Since not all closed sections are
locally exact on Lie algebroids, the Helmholtz conditions on Lie algebroids are
necessary but not sufficient, so they give a weaker definition of the inverse
problem. As an application the Helmholtz conditions on Atiyah algebroids are
obtained so that the relationship between the inverse problem and the reduced
inverse problem by symmetries can be described. Some examples and comparison
with previous approaches in the literature are provided.
