The inverse problem of the calculus of variations: The use of geometrical calculus in Douglas's analysis |
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Authors: | W. Sarlet G. Thompson G. E. Prince |
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Affiliation: | Department of Mathematical Physics and Astronomy, Ghent University, Krijgslaan 281, B-9000 Ghent, Belgium ; Department of Mathematics, The University of Toledo, Toledo, Ohio 43606 ; School of Mathematics, University of New South Wales, Sydney NSW 2052, Australia |
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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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Keywords: | Lagrangian systems inverse problem geometrical calculus |
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