Resumo:
The variational calculus is directly related to the energetic modeling of
physical systems in general. The advantage of such approach in relation to
the classical approaches (Newtonians) is that a system can be subdivided in
many sub-systems whose modeling can be simpler than the one of the
whole. As the total energy of a system is the sum of the energy of the parts,
it is possible to obtain a global result from the partial results. Nevertheless,
in the classic methods, based on force, such flexibility does not exist, the
most possible way is to apply the effect superposition method. However,
this method does not apply to non-linear systems. In general, the
engineering applications are always linked to the solution of a set of
descriptive equations of the phenomena. Starting in these equations and
reaching a functional, which represent them, is the inverse problem of the
variational calculus. This paper presents a description of the methods which
have been employed for the solution of this problem. Morever, this paper
generalize the Vainberg theorem, which was the starting point for all these
methods. From this theorem, this paper proposes methods that apply to a
more general class of problems, such as, for example, the variational
formulation of Navier-Stokes equations. This work approaches, principally,
the variational formulation of Maxwell, including the quaternionic
formulation, which takes to the langrangian classic electromagnetic field
added of a complex term, which allows the functional, when varied, to
result in the four Maxwell equations. Finally, the formulation of Maxwell
equations by means of differential forms is presented. The advantage of the
energetic method on the weak formulation is that the former provides
superior and inferior quotes to the numeric solution of the problems.
