HIGHER EULER OPERATORS AND SOME OF THEIR APPLICATIONS

A set of operators which are associated with the Euler-Lagrange operator is introduced. An analysis of the commutation properties of these new operators, which will be referred to as the higher Euler operators, leads to a generalization of the necessary conditions for an expression to be an Euler-Lagrange expression. A product rule is derived for the higher Euler operators. In the special case of the Euler-Lagrange operator this product rule is basic to simple proofs of sufficiency theorems for the existence of a Lagrangian given the potential Euler-Lagrange expressions. By considering a certain homogeneity property, a characterization of Lagrangians in terms of their Euler-Lagrange expressions is established. Examples of applications of this characterization are given. A general procedure is given for constructing equivalent (not necessarily scalar density) Lagrangians when the field functions are tensorial and the Euler-Lagrange expressions are tensor densities. These results give particular significance to one of the higher Euler operators. © 1979 American Institute of Physics.