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.

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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