Formulação variacional para o eletromagnetismo e suas aplicações

Abstract

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.