Abstract

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.

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