A Derivation of Variational Principles for Inhomogeneous Equations

It is shown that variational principles need not be postulated and then correctness proved; they can, in fact, be derived, making their use more a matter of routine than ingenuity. A Lagrange multiplier technique is used to derive a second-order variational principle for estimating an arbitrary functional of the solution to an inhomogeneous equation. The relationship of this principle to a functional Taylor series expansion and to elementary perturbation theory is established. A normalization independent second-order variational principle for an arbitrary functional is derived which reduces to the Schwinger principle if the functional is linear. Two higher order variational principles are derived and shown to be generalizations of the principles of Kostin and Brooks. The Lagrange multiplier technique is applied to the inhomogeneous Sturm-Liouville equation, which leads to a second-order variational principle for estimating an arbitrary functional which allows trial functions that are not continuous and do not satisfy the boundary conditions. This functional is of the type suggested by Buslik plus boundary terms. The differences between a variational principle which can only be used to estimate a functional of interest and one which also acts as a Lagrangian are discussed.