Equations which define a ``consistent'' set of ``boundary'' conditions, and hence a field, for a given set of differential equations are derived from a variational principle. The equivalence of functionals defined over an entire domain and functionals defined over only a subdomain, but with a surface term added to account for the contribution of the excluded subdomain, is exploited. The appropriate surface term is found to satisfy the Hamilton-Jacobi equation. The formalism is specialized to neutron diffusion theory, and it is demonstrated that Stark's double sweep method follows as a natural consequence of this field theoretic formulation. The relation of this formalism to Pontryagin's Maximum Principle and Bellman's Dynamic Programming is demonstrated for problems which can be characterized by minimum variational principles.