The inverse problem of the calculus

of variations for scalar fourth-order ordinary

differential equations

Author:
M. E. Fels

Journal:
Trans. Amer. Math. Soc. **348** (1996), 5007-5029

MSC (1991):
Primary 53B50, 49N45

DOI:
https://doi.org/10.1090/S0002-9947-96-01720-5

MathSciNet review:
1373634

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Abstract | References | Similar Articles | Additional Information

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.

**1.**Anderson I. ,*The Variational Bicomplex*, Academic Press, to appear.**2.**Anderson I. and Fels M.,*Variational Operators for Differential Equations*, in preparation.**3.**Anderson I. and Kamran N.,*The variational bicomplex for second-order scalar partial differential equations in the plane*, Duke Math. J., to appear.**4.**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**, https://doi.org/10.1090/memo/0473**5.**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**, https://doi.org/10.1090/pspum/053/1141197**6.**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**, https://doi.org/10.1090/S0894-0347-1995-1311820-X**7.**Cartan E.,*Les sous-groupes des groupes continus de transformations*, Annales de l'Ècole Normale,**XXV**, 1908, 57-194.**8.**Chern S.S.,*The geometry of the differential equation*, Sci. Rep. Tsiing Hua Univ.,**4**, 1940, 97-111. MR**3:21c****9.**Darboux G.,*Lecon sur la theorie generale des surfaces*, Gauthier-Villars, Paris, 1894.**10.**Douglas J.,*Solution to the inverse problem of the calculus of variations*, Trans. Amer. Math. Soc.,**50**, 1941, 71-128. MR**3:54c****11.**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****12.**Artemio González-López,*Symmetry bounds of variational problems*, J. Phys. A**27**(1994), no. 4, 1205–1232. MR**1269044****13.**Peter J. Olver,*Applications of Lie groups to differential equations*, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1986. MR**836734****14.**-,*Equivalence, Invariants and Symmetry*, Cambridge University Press, 1995. CMP**95:14**

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Additional Information

**M. E. Fels**

Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Email:
fels@math.umn.edu

DOI:
https://doi.org/10.1090/S0002-9947-96-01720-5

Keywords:
Inverse problem of the calculus of variations,
variational principles for scalar ordinary differential equations,
variational bicomplex,
equivalence method,
divergence symmetries

Received by editor(s):
June 15, 1995

Article copyright:
© Copyright 1996
American Mathematical Society