The total divergence equation

D. Krupka

Lobachevskii journal of mathematics, Tome 23 (2006), pp. 71-93.

Résumé

In this paper, the total divergence equation is investigated by means of the methods used in the theory of finite order variational sequences. Integrability conditions for this equation are found, and all solutions are described. The correspondence of the solutions with some differential forms on jet spaces is established.

Détail

Keywords: differential equation, integrability, variational sequence
Mots-clés : jet, divergence, Euler–Lagrange expression.

Bibliographie

[1] I. M. Anderson, Introduction to the variational bicomplex, Contemporary Mathematics, 132, Amer. Math. Soc., Providence, RI, 1992, 51–73 | MR

[2] P. Dedecker, W. M. Tulczyjew, “Spectral sequences and the inverse problem of the calculus of variations”, Internat. Colloq. on Diff. Geom. Methods in Math. Phys. (Aix-enProvence, 1979), Lecture Notes in Math., 836, Springer, Berlin, 498–503 | MR

[3] M. Francaviglia, M. Palese, R. Vitolo, “Superpotentials in variational sequences”, Differential Geometry and Applications, Proc. Conf. (Brno, Czech Republic, 1998), Masaryk University, Brno, 1999, 469–480 | MR | Zbl

[4] M. Grassi, “Local vanishing of characteristic cohomology”, Duke Math. J., 102 (2000), 307–328 | DOI | MR | Zbl

[5] D. R. Grigore, “The variational sequence on finite jet bundle extensions and the Lagrange formalism”, Diff. Geom. Appl., 10 (1999), 43–77 | DOI | MR | Zbl

[6] A. Hakova, O. Krupkova, “Variational first-order partial differential equations”, J. Differential Equations, 191 (2003), 67–68 | DOI | MR

[7] I. S. Krasilschik, “Geometry of differential equations: A concise Introduction”, Acta Appl. Math., 72 (2002), 1–17 | DOI | MR

[8] I. S. Krasilschik, V. V. Lychagin, A. M. Vinogradov, Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Gordon and Breach, New York, 1986 | MR

[9] M. Krbek, J. Musilova, “Representation of the variational sequence by differential forms”, Rep. Math. Phys., 51 (2003), 251–258 | DOI | MR | Zbl

[10] D. Krupka, “A geometric theory of ordinary rst order variational problems in fibered manifolds, I. Critical sections”, J. Math. Anal. Appl., 49 (1975), 180–206 | DOI | MR | Zbl

[11] D. Krupka, “The contact ideal”, Diff. Geom. Appl., 5 (1995), 257–276 | DOI | MR | Zbl

[12] D. Krupka, “Variational sequences on finite order jet spaces”, Differential Geometry and its Applications, Proc. Internat. Conf. on Diff. Geom. and Appl. (Brno, August 1989), eds. J. Janyska and D. Krupka, World Scientific, 1990, 236–254 | MR | Zbl

[13] O. Krupkova, The Geometry of Ordinary Variational Equations, Lecture Notes in Math., 1678, Springer, Berlin, 1997 | MR | Zbl

[14] J. F. Pommaret, “Spencer sequence and variational sequence”, Acta Appl. Math., 41 (1995), 285–296 | DOI | MR | Zbl

[15] D. Saunders, The Geometry of Jet Bundles, London Mathematical Society Lecture Note Series, 142, Cambridge University Press, New York, 1989 | MR | Zbl

[16] F. Takens, “A global version of the inverse problem of the calculus of variations”, J. Differential Geometry, 14 (1979), 543–562 | MR | Zbl

[17] W. M. Tulczyjew, “The Lagrange complex”, Bull. Soc. Math. France, 105 (1977), 419–431 | MR | Zbl

[18] A. M. Vinogradov, “The C-spectral sequence, Lagrangian formalism, and conservation laws, I. The linear theory”, J. Math. Anal. Appl., 100 (1984), 1–40 ; “II. The nonlinear theory”, 41–129 | DOI | MR | Zbl | Zbl

[19] R. Vitolo, “Finite order variational bicomplexes”, Math. Proc. Cambridge. Phil. Soc., 125 (1999), 321–333 | DOI | MR | Zbl

[20] T. Tsujishita, “On variational bicomplexes associated with differential equations”, Osaka J. Math., 19 (1982), 311–363 | MR | Zbl