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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
MathSciNet review: 1373634
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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.


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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: http://dx.doi.org/10.1090/S0002-9947-96-01720-5
PII: S 0002-9947(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