The Inverse Problem of the Calculus of Variations for Scala Fourth Order Ordinary Differential Equations
Trans. Amer. Math Soc.
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.
The Inverse Problem of the Calculus of Variations for Scalar Fourth Order Ordinary Deferential Equations, M.E. Fels, Trans. Amer. Math Soc., 348(12), 1996, 5007-5029.