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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**1.**Ian Anderson and Gerard Thompson,*The inverse problem of the calculus of variations for ordinary differential equations*, Mem. Amer. Math. Soc.**98**(1992), no. 473, vi+110. MR**1115829**, 10.1090/memo/0473**2.**M. Crampin, E. Martínez, and W. Sarlet,*Linear connections for systems of second-order ordinary differential equations*, Ann. Inst. H. Poincaré Phys. Théor.**65**(1996), no. 2, 223–249 (English, with English and French summaries). MR**1411267****3.**Michele Grassi,*Local vanishing of characteristic cohomology*, Duke Math. J.**102**(2000), no. 2, 307–328. MR**1749440**, 10.1215/S0012-7094-00-10225-6**4.**M. Crampin, W. Sarlet, E. Martínez, G. B. Byrnes, and G. E. Prince,*Towards a geometrical understanding of Douglas’ solution of the inverse problem of the calculus of variations*, Inverse Problems**10**(1994), no. 2, 245–260. MR**1269007****5.**Jesse Douglas,*Solution of the inverse problem of the calculus of variations*, Trans. Amer. Math. Soc.**50**(1941), 71–128. MR**0004740**, 10.1090/S0002-9947-1941-0004740-5**6.**J. Grifone and Z. Muzsnay,*Sur le problème inverse du calcul des variations: existence de lagrangiens associés à un spray dans le cas isotrope*, Ann. Inst. Fourier (Grenoble)**49**(1999), no. 4, 1387–1421 (French, with English and French summaries). MR**1703093****7.**E. Martínez, J. F. Cariñena, and W. Sarlet,*Derivations of differential forms along the tangent bundle projection*, Differential Geom. Appl.**2**(1992), no. 1, 17–43. MR**1244454**, 10.1016/0926-2245(92)90007-A**8.**E. Martínez, J. F. Cariñena, and W. Sarlet,*Derivations of differential forms along the tangent bundle projection. II*, Differential Geom. Appl.**3**(1993), no. 1, 1–29. MR**1245556**, 10.1016/0926-2245(93)90020-2**9.**Eduardo Martínez, José F. Cariñena, and Willy Sarlet,*Geometric characterization of separable second-order differential equations*, Math. Proc. Cambridge Philos. Soc.**113**(1993), no. 1, 205–224. MR**1188830**, 10.1017/S0305004100075897**10.**Z. Muzsnay,*Sur le problème inverse du calcul des variations*, Thèse de Doctorat, Université Paul Sabatier (Toulouse III) (1997).**11.**C. Riquier,*Les systèmes d'équations aux derivées partielles*, (Gauthier-Villars, Paris) (1910).**12.**W. Sarlet and M. Crampin,*Addendum to: “The integrability conditions in the inverse problem of the calculus of variations for second-order ordinary differential equations” [Acta. Appl. Math. 54 (1998), no. 3, 233–273; MR1671779 (99m:58072)] by Sarlet, Crampin and E. Martínez*, Acta Appl. Math.**60**(2000), no. 3, 213–224. MR**1776705**, 10.1023/A:1006407301149**13.**W. Sarlet, M. Crampin, and E. Martínez,*The integrability conditions in the inverse problem of the calculus of variations for second-order ordinary differential equations*, Acta Appl. Math.**54**(1998), no. 3, 233–273. MR**1671779**, 10.1023/A:1006102121371**14.**W. Sarlet, A. Vandecasteele, F. Cantrijn, and E. Martínez,*Derivations of forms along a map: the framework for time-dependent second-order equations*, Differential Geom. Appl.**5**(1995), no. 2, 171–203. MR**1334841**, 10.1016/0926-2245(95)00013-T

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
37J05,
70H03,
49N45

Retrieve articles in all journals with MSC (2000): 37J05, 70H03, 49N45

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

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