A unified formulation of the construction of variational principles

The use of variational principles as a calculational tool is reviewed, with special emphasis on methods for constructing such principles. In particular, it is shown that for a very wide class of problems it is possible to construct a variational principle (VP) for just about any given quantity Q of interest, by routine procedures which do not require the exercise of ingenuity; the resultant VP will yield an estimate of Q correct to second order whenever the quantities appearing in the VP are known to first order. The only significant requirement for application of the routine procedures is that the entities which enter into the definition of Q be uniquely specified by a given set of equations; the equations may involve difference or differential or integral operators, they may be homogeneous or inhomogeneous, linear or nonlinear, self-adjoint or not, and they may or may not represent time-reversible systems. No numerical calculations are presented, but procedures for the construction of VP's are illustrated for numerous quantities Q of physical interest, particularly those Q arising in quantum-mechanical scattering and transition probability calculations. For pedagogical purposes VP's are also derived for several problems in classical and (simple) mathematical physics which the authors hope will prove instructive and perhaps even amusing. The quantum-mechanical quantities Q whose VP's are examined include various matrix elements and the quantum-mechanical eigenfunctions themselves. Topics examined include some points which have not always been appreciated in the literature, such as the necessity for properly specifying the phase when complex eigenfunctions are involved, and the importance of avoiding, wherever possible, formulations requiring the inversion of singular operators. The basic element of the technique is the recognition that the defining equations of a system can be incorporated into the VP as constraints through the use of (generalized) Lagrange multipliers; these can be constants, scalar, vector, or tensor functions of one or more variables, operators, etc. In the typical problem, these Lagrange multipliers L serve as a new set of adjoint functions or entities, and the construction of the VP simultaneously provides well-defined equations for the L. Moreover, these L often have ready physical significance themselves; for example, they often may be regarded as generalized Green's functions. The construction of the VP also readily yields so-called variational identities for the quantity Q of interest; these identities give explicit (if formal) expressions for the error in the variational estimate of Q. In some cases this error can be shown to have a definite sign, so that the VP actually is an extremum principle, that is, that it yields an upper or lower variational bound for Q; however, our routine procedures for constructing VP's do not routinely yield extremum principles.