The Inverse Problem of the Calculus of Variations for Scala Fourth Order Ordinary Differential Equations

Document Type

Article

Journal/Book Title/Conference

Trans. Amer. Math Soc.

Volume

348

Issue

12

Publication Date

1996

First Page

5007

Last Page

5029

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 is 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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