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

Sarlet, W.; Thompson, G.; Prince, G.

doi:10.1090/S0002-9947-02-02994-X

The inverse problem of the calculus of variations: The use of geometrical calculus in Douglas’s analysis
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by W. Sarlet, G. Thompson and G. E. Prince PDF

Trans. Amer. Math. Soc. 354 (2002), 2897-2919 Request permission

Abstract:

The main objective of this paper is to work out a full-scale application of the integrability analysis of the inverse problem of the calculus of variations, as developed in recent papers by Sarlet and Crampin. 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 Sarlet and Crampin’s general theoretical developments 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 characterization, 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.

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