Well-defined equations of motion without constraining external sources

We present a new approach to constrained classical fields that enables the action formalism to dictate how external sources must enter the resulting equations of motion. If symmetries asserted upon the varied fields can be modeled as restrictions in Fourier space, then we prove that these restrictions are automatically applied to external sources in an unambiguous way. In contrast, the typical procedure inserts symmetric Ansätze into the Euler-Lagrange differential equations, even for external sources not being solved. This requires ad hoc constraint of external sources, which can introduce leading-order errors to model systems despite superficial consistency between model field and source terms. To demonstrate, we consider Robertson-Walker cosmologies within general relativity and prove that the influence of pointlike relativistic pressure sources on cosmological dynamics cannot be excluded by theoretical arguments.