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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: W. Sarlet, G. Thompson and G. E. Prince
Journal: Trans. Amer. Math. Soc. 354 (2002), 2897-2919
MSC (2000): Primary 37J05, 70H03; Secondary 49N45
Published electronically: March 14, 2002
MathSciNet review: 1895208
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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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Additional Information

W. Sarlet
Affiliation: Department of Mathematical Physics and Astronomy, Ghent University, Krijgslaan 281, B-9000 Ghent, Belgium
Email: willy.sarlet@rug.ac.be

G. Thompson
Affiliation: Department of Mathematics, The University of Toledo, Toledo, Ohio 43606
Email: gthomps@uoft02.utoledo.edu

G. E. Prince
Affiliation: School of Mathematics, University of New South Wales, Sydney NSW 2052, Australia
Address at time of publication: Department of Mathematics, La Trobe University, Victoria 3086, Australia
Email: g.prince@latrobe.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9947-02-02994-X
PII: S 0002-9947(02)02994-X
Keywords: Lagrangian systems, inverse problem, geometrical calculus
Received by editor(s): October 25, 2000
Published electronically: March 14, 2002
Article copyright: © Copyright 2002 American Mathematical Society