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), 28972919
MSC (2000):
Primary 37J05, 70H03; Secondary 49N45
Published electronically:
March 14, 2002
MathSciNet review:
1895208
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The main objective of this paper is to work out a fullscale 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 coordinatefree, 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
(92k:58070), http://dx.doi.org/10.1090/memo/0473
 2.
M.
Crampin, E.
Martínez, and W.
Sarlet, Linear connections for systems of secondorder ordinary
differential equations, Ann. Inst. H. Poincaré Phys.
Théor. 65 (1996), no. 2, 223–249
(English, with English and French summaries). MR 1411267
(97g:58003)
 3.
Michele
Grassi, Local vanishing of characteristic cohomology, Duke
Math. J. 102 (2000), no. 2, 307–328. MR 1749440
(2001j:58031), http://dx.doi.org/10.1215/S0012709400102256
 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
(95e:49042)
 5.
Jesse
Douglas, Solution of the inverse problem of the
calculus of variations, Trans. Amer. Math.
Soc. 50 (1941),
71–128. MR
0004740 (3,54c), http://dx.doi.org/10.1090/S00029947194100047405
 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
(2000d:49035)
 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
(94h:58004), http://dx.doi.org/10.1016/09262245(92)90007A
 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
(94h:58005), http://dx.doi.org/10.1016/09262245(93)900202
 9.
Eduardo
Martínez, José
F. Cariñena, and Willy
Sarlet, Geometric characterization of separable secondorder
differential equations, Math. Proc. Cambridge Philos. Soc.
113 (1993), no. 1, 205–224. MR 1188830
(93m:58007), http://dx.doi.org/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, (GauthierVillars, Paris) (1910).
 12.
W.
Sarlet and M.
Crampin, Addendum to: “The integrability conditions in the
inverse problem of the calculus of variations for secondorder 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
(2001e:58020), http://dx.doi.org/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 secondorder ordinary
differential equations, Acta Appl. Math. 54 (1998),
no. 3, 233–273. MR 1671779
(99m:58072), http://dx.doi.org/10.1023/A:1006102121371
 14.
W.
Sarlet, A.
Vandecasteele, F.
Cantrijn, and E.
Martínez, Derivations of forms along a map: the framework
for timedependent secondorder equations, Differential Geom. Appl.
5 (1995), no. 2, 171–203. MR 1334841
(96m:58006), http://dx.doi.org/10.1016/09262245(95)00013T
 1.
 I. Anderson and G. Thompson, The inverse problem of the calculus of variations for ordinary differential equations, Memoirs Amer. Math. Soc. 98 No. 473 (1992). MR 92k:58070
 2.
 M. Crampin, E. Martínez and W. Sarlet, Linear connections for systems of secondorder ordinary differential equations, Ann. Inst. H. Poincaré Phys. Théor. 65 (1996), 223249. MR 97g:58003
 3.
 M. Crampin, G.E. Prince, W. Sarlet and G. Thompson, The inverse problem of the calculus of variations: separable systems, Acta Appl. Math. 57 (1999), 239254. MR 2001j:58031
 4.
 M. Crampin, W. Sarlet, E. Martínez, G.B. Byrnes and G.E. Prince, Towards a geometrical understanding of Douglas's solution of the inverse problem of the calculus of variations, Inverse Problems 10 (1994), 245260. MR 95e:49042
 5.
 J. Douglas, Solution of the inverse problem of the calculus of variations, Trans. Amer. Math. Soc. 50 (1941), 71128. MR 3:54c
 6.
 J. Grifone and Z. Muzsnay, Sur le problème inverse du calcul des variations: existence de lagrangiens associées à un spray dans le cas isotrope, Ann. Inst. Fourier 49 (1999), 135. MR 2000d:49035
 7.
 E. Martínez, J.F. Cariñena and W. Sarlet, Derivations of differential forms along the tangent bundle projection, Differential Geometry and its Applications 2 (1992), 1743. MR 94h:58004
 8.
 E. Martínez, J.F. Cariñena and W. Sarlet, Derivations of differential forms along the tangent bundle projection. Part II, Differential Geometry and its Applications 3 (1993), 129. MR 94h:58005
 9.
 E. Martínez, J.F. Cariñena and W. Sarlet, Geometric characterization of separable secondorder differential equations, Math. Proc. Cambridge Philos. Soc. 113 (1993), 205224. MR 93m:58007
 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, (GauthierVillars, Paris) (1910).
 12.
 W. Sarlet and M. Crampin, Addendum to: The integrability conditions in the inverse problem of the calculus of variations for secondorder ordinary differential equations, Acta Appl. Math. 60 (2000), 213224. MR 2001e:58020
 13.
 W. Sarlet, M. Crampin and E. Martínez, The integrability conditions in the inverse problem of the calculus of variations for secondorder ordinary differential equations, Acta Appl. Math. 54 (1998), 233273. MR 99m:58072
 14.
 W. Sarlet, A. Vandecasteele, F. Cantrijn and E. Martínez, Derivations of forms along a map: The framework for timedependent secondorder equations, Differential Geometry and its Applications 5 (1995), 171203. MR 96m:58006
Similar Articles
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, B9000 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/S000299470202994X
PII:
S 00029947(02)02994X
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
