Necessary and sufficient conditions for the existence of a Lagrangian in field theory. III. Generalized analytic representations of tensorial field equations

Abstract

In this paper we first study the equivalence transformations of class C 2, regular, tensorial, quasi-linear systems of field equations which (a) preserve the continuity, regularity, and quasi-linear structure of the systems; and (b) occur within a fixed system of Minkowski coordinates and field components. We identify, among the transformations of this class, those which either induce or preserve a self-adjoint structure of the field equations and we term them genotopic and isotopic transformations, respectively. We then give the necessary and sufficient conditions for an equivalence transformation of the above type to be either genotopic or isotopic. By using this methodology, we then extend the theorem on the necessary and sufficient condition for the existence of ordered direct analytic representations introduced in the preceding paper to the case of ordered indirect analytic representations in terms of the conventional Lagrange equations; we introduce a method for the construction of a Lagrangian, when it exists, in this broader context; and we explore some implications of the underlying methodology for the problem of the structure of the Lagrangian capable of representing interactions within the framework of the indirect analytic representations. Some of the several aspects which demand an inspection prior to the use of this analytic approach in actual models are pointed out. In particular, we indicate a possible deep impact in the symmetries and conservation laws of the system generated by the use of the concept of indirect analytic representation. As a preparatory step prior to the analysis of these problems, we study some methodological aspects which underlie the generalized Lagrange equations postulated in the first paper of this series for the case when they are regular, namely, when they are simple equivalence transformations of the conventional Lagrange equations. We first introduce a generalization of the action principle capable of inducing the generalized as well as the conventional equations. In this way we establish that the former equations are “bona fide” analytic equations. Finally, as our most general analytic framework for the case of unconstrained field equations, we work out the necessary and sufficient condition for the existence of ordered direct analytic representations of quasi-linear systems in terms of the generalized analytic equations and study their relationship to the conventional representations.