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Integrability aspects of the inverse problem of the
calculus of variations

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*W. Sarlet*

**E-mail:** Willy.Sarlet@rug.ac.be
**Abstract.** For a long time, a paper by J. Douglas of 1941 has
been the only contribution to the question of classifying second-order
ordinary differential equations for which a non-singular multiplier matrix
exists which turns the given system into an equivalent system of Euler-Lagrange
equations. It was based on the Riquier-Janet theory of formal integrability
of partial differential equations and limited to systems with two degrees
of freedom. Quite recently, a geometrical calculus of derivations of tensor
fields along projections has been developed, which in the study of second-order
differential equations is primarily related to the existence of a canonically
defined linear connection on a suitable bundle. It turns out that this
calculus provides the right tools for closely monitoring the process of
Douglas's analysis in a coordinate free way. After a survey of the integrability
analysis which can be carried out this way, we briefly sketch how subcases
belonging to each of the three classes in the main classification scheme
of Douglas can be generalised to an arbitrary number of degrees of freedom.

**AMSclassification.** 58F05, 58G99, 70H35

**Keywords.** Lagrangian systems, inverse problem, integrability