The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations

Fels, M.

doi:10.1090/S0002-9947-96-01720-5

The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations
HTML articles powered by AMS MathViewer

by M. E. Fels PDF

Trans. Amer. Math. Soc. 348 (1996), 5007-5029 Request permission

Abstract:

A simple invariant characterization of the scalar fourth-order ordinary differential equations which admit a variational multiplier is given. The necessary and sufficient conditions for the existence of a multiplier are expressed in terms of the vanishing of two relative invariants which can be associated with any fourth-order equation through the application of Cartan’s equivalence method. The solution to the inverse problem for fourth-order scalar equations provides the solution to an equivalence problem for second-order Lagrangians, as well as the precise relationship between the symmetry algebra of a variational equation and the divergence symmetry algebra of the associated Lagrangian.

References

Anderson I. , The Variational Bicomplex, Academic Press, to appear.
Anderson I. and Fels M., Variational Operators for Differential Equations, in preparation.
Anderson I. and Kamran N., The variational bicomplex for second-order scalar partial differential equations in the plane, Duke Math. J., to appear.
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, DOI 10.1090/memo/0473
Robert L. Bryant, Two exotic holonomies in dimension four, path geometries, and twistor theory, Complex geometry and Lie theory (Sundance, UT, 1989) Proc. Sympos. Pure Math., vol. 53, Amer. Math. Soc., Providence, RI, 1991, pp. 33–88. MR 1141197, DOI 10.1090/pspum/053/1141197
Robert L. Bryant and Phillip A. Griffiths, Characteristic cohomology of differential systems. I. General theory, J. Amer. Math. Soc. 8 (1995), no. 3, 507–596. MR 1311820, DOI 10.1090/S0894-0347-1995-1311820-X
Cartan E., Les sous-groupes des groupes continus de transformations, Annales de l’Ècole Normale, XXV, 1908, 57-194.
Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
Darboux G., Lecon sur la theorie generale des surfaces, Gauthier-Villars, Paris, 1894.
Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
Robert B. Gardner, The method of equivalence and its applications, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 58, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 1062197, DOI 10.1137/1.9781611970135
Artemio González-López, Symmetry bounds of variational problems, J. Phys. A 27 (1994), no. 4, 1205–1232. MR 1269044, DOI 10.1088/0305-4470/27/4/016
Peter J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1986. MR 836734, DOI 10.1007/978-1-4684-0274-2
—, Equivalence, Invariants and Symmetry, Cambridge University Press, 1995.