Formulation of variational principles via Lagrange multipliers

Abstract

Recent work on variational principles in mathematical physics enables one to construct, in a novel and systematic way, stationary expressions for a wide class of functionals P(E), where E is an unknown (vector) function whose defining equation cannot be solved exactly. The method involves the use of Lagrange multipliers, which can be a constant λ, a function F(x,y,z), and a dyadic operator Γ, to account for each of the equations (constraints) that define E. As illustrations, we consider vector and scalar scattering problems, and eigenvalue problems such as the determination of resonant frequencies and propagation constants of waveguides.