Locally variational systems of differential equations on smooth manifolds, having certain de Rham cohomology group trivial, automatically possess a global Lagrangian. This important result due to Takens is, however, of sheaf-theoretic nature. A new constructive method of finding a global Lagrangian for second-order ODEs on 2-manifolds is described on the basis of the solvability of the exactness equation for the Lepage 2-forms and the top-cohomology theorems. Examples from geometry and mechanics are discussed.

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