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Variational principles are important in the investigation of large classes ofphysical systems. They can be used both as analytical methods as well asstarting points for the formulation of powerful computational techniques suchas dynamical optimization methods. Systems with charged objects in dielectricmedia and systems with magnetically active particles are important examples. Inthese examples and other important cases, the variational principles describingthe system are required to obey a number of constraints. These constraints areimplemented within the variational formulation by means of Lagrangemultipliers. Such constrained variational formulations are in general notunique. For the application of efficient simulation methods, one must findspecific formulations that satisfy a number of important conditions. An oftenrequired condition is that the functional be positive-definite, in other words,its extrema be actual minima. In this article, we present a general approach toattack the problem of finding, among equivalent variational functionals, thosethat generate true minima. The method is based on the modification of theLagrange multiplier which allows us to generate large families of effectivevariational formulations associated with a single original constrainedvariational principle. We demonstrate its application to different examplesand, in particular, to the important cases of Poisson and Poisson-Boltzmannequations. We show how to obtain variational formulations for these systemswith extrema that are always minima.

Generating true minima in constrained variational formulations via modified Lagrange multipliers