2Variational Methods for Algebraic Equations

Let us begin our exploration of variational methods by a situation which may appear as simple, but contains the fundamental elements necessary to the implementation of variational methods in complex situations: the variational formulation of algebraic equations and its consequences.

Indeed, the fundamental variational standpoint consists of no more saying that x = 0, but that ximg and xy = 0, yimg.

The apparent simplicity of this fundamental change of viewpoint masks a radical transformation in the way of thinking, having profound consequences, which can be measured by the progress performed in engineering with the help of variational tools such as analytical mechanics, hamiltonian mechanics, control theory and modern numerical methods such as finite elements, finite volumes, spectral methods, particle methods and others.

Let us illustrate these consequences by adopting the variational point of view when studying a system of algebraic equations: for instance, let us consider;

images

and the algebraic equations;

images [2.1]

The variational formulation ...

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