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

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\u27s 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