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The main objective of this paper is to work out a full-scale application of the inte- grability analysis of the inverse problem of the calculus of variations, as developed in (13, 12). For this purpose, the celebrated work of Douglas on systems with two degrees of freedom is taken as the reference model. It is shown that the coordinate free, geometrical calculus used in the general theoretical developments of (13, 12), provides effective tools also to do the practical calculations. The result is, not only that all subcases distinguished by Douglas can be given a more intrinsic character- ization, but also that in most of the cases, the calculations can be carried out in a more efficient way and often lead to sharper conclusions.

The inverse problem of the calculus of variations: The use of geometrical calculus in Douglas’s analysis