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Abstract

By investigating perturbations of classical field theories based on variational principles we develop a variety of relations of interest in several fields, general relativity, stellar structure, fluid dynamics, and superfluid theory. The simplest and most familiar variational principles are those in which the field variations are unconstrained. Working at first in this context we introduce the Noether operator, a fully covariant generalization of the socalled canonical stress energy tensor, and prove its equivalence to the symmetric tensor Tμν. By perturbing the Noether operator's definition we establish our fundamental theorem, that any two of the following imply the third (a) the fields satisfy their field equations, (b) the fields are stationary, (c) the total energy of the fields is an extremum against all perturbations. Conversely, a field theory which violates this theorem cannot be derived from an unconstrained principle. In particular both Maxwell's equations for Fμν and Euler's equations for the perfect fluid have stationary solutions which are not extrema of the total energy [(a) + (b) (c)].