Integrability aspects of the inverse problem of the calculus of variations

W. Sarlet


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