Variational Methods for Engineers with Matlab

Introduction

Variational methods may be seen as a new language leading to new formulations and new methods of solution of equations, where the central concept is the physical notion of work. In this language, we do not say that a quantity is null, but we say that its work is null for all the possible values of a connected variable. For instance, we do not say that a displacement is null, but that the associated work is null for any force. However, we do not say that a force is null, but that its work is null for any displacement. In the simplest situation, we do not say that x = 0, but we say that “x is a real number such that its product for any other real number is equal to zero”, e.g. ∈ and xy = 0, ∀y ∈ . Although this modification and the equivalence between both the formulations seem trivial, it implies a deep conceptual change, as it will be seen in the following. Moreover, despite the view expressed by Richard Feynman – from the standpoint of Physics – in [FEY 85], these two formulations may not be equivalent – from the standpoint of Mathematics – in certain situations: it may become necessary to adopt a complex theoretical framework in order to obtain such an equivalence – that will never be complete.

The beginning of the history of variational methods is often brought to ...