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The language of Lagrangian submanifolds is used to extend a geometriccharacterization of the inverse problem of the calculus of variations ontangent bundles to regular Lie algebroids. Since not all closed sections arelocally exact on Lie algebroids, the Helmholtz conditions on Lie algebroids arenecessary but not sufficient, so they give a weaker definition of the inverseproblem. As an application the Helmholtz conditions on Atiyah algebroids areobtained so that the relationship between the inverse problem and the reducedinverse problem by symmetries can be described. Some examples and comparisonwith previous approaches in the literature are provided.

Inverse problem for Lagrangian systems on Lie algebroids and applications to reduction by symmetries