  Variational principles are important in the investigation of large classes of
physical systems. They can be used both as analytical methods as well as
starting points for the formulation of powerful computational techniques such
as dynamical optimization methods. Systems with charged objects in dielectric
media and systems with magnetically active particles are important examples. In
these examples and other important cases, the variational principles describing
the system are required to obey a number of constraints. These constraints are
implemented within the variational formulation by means of Lagrange
multipliers. Such constrained variational formulations are in general not
unique. For the application of efficient simulation methods, one must find
specific formulations that satisfy a number of important conditions. An often
required condition is that the functional be positive-definite, in other words,
its extrema be actual minima. In this article, we present a general approach to
attack the problem of finding, among equivalent variational functionals, those
that generate true minima. The method is based on the modification of the
Lagrange multiplier which allows us to generate large families of effective
variational formulations associated with a single original constrained
variational principle. We demonstrate its application to different examples
and, in particular, to the important cases of Poisson and Poisson-Boltzmann
equations. We show how to obtain variational formulations for these systems
with extrema that are always minima.
