ABSTRACT The inverse problem of the calculus of variations is the problem of finding variational principles for systems of differential equations. By using the general theory of the varia- tional bicomplex, it is shown that the inverse problem for ordinary differential equations is equivalent to the problem of finding differential two forms, with certain prescribed algebraic properties, which are closed on the prolonged equation manifold. For second order ordi- nary differential equations, this approach leads directly to the fundamental equations of J. Douglas. An algorithm for the analysis of these equations based upon the Cartan-Kahler theorem for exterior differential systems is presented, the inverse problem for higher order ordinary differential equations is solved, and a variety of new examples are considered in detail. KEY WORDS AND PHRASES: inverse problem of the calculus of variations, variational principles for ordinary differential equations, variational bicomplex, exterior differential systems, two dimensional sprays. V I

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