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
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by M. E. Fels

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

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

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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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