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ABSTRACT
Much has been said and written on the calculus of variations, a traditional subject of mathematics, and the offered models have often been expressed in terms of minimality or maximality. The traditional methods (Euler-Lagrange equation , Hamiltonian formulation and Hamilton-Jacobi equation) and the direct methods are explained. The vectorial case of direct methods is then presented via the different notions of convexity and an existence result. Non-convex problems of the calculus of variations conclude this article: different envelopes, relaxation theorem and various examples.
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Bernard DACOROGNA: Professor, Mathematics Section EPFL (École polytechnique fédérale de Lausanne), Switzerland
INTRODUCTION
The calculus of variations is one of the classic subjects of mathematics. It has attracted many famous mathematicians. Before presenting the most important model case, we'll start with an informal discussion. In mathematics, physics, engineering or even economics or ecology, models are often expressed in terms of a minimality or maximality principle. This is precisely the central issue in calculating variations. For example, in mathematics, we may be interested in finding, under certain constraints, a curve of minimum length or a surface of minimum area. In physics, a typical example is the principle of least action; other examples will be given in more detail in this presentation. Conservation laws, which correspond mathematically to differential equations, are often derived from a variational principle. The solutions of the variational problem are then solutions of associated differential equations.
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Calculating variations
Examples of variational problems
References
In this dossier, we have followed our book very closely (especially in the first three paragraphs). Numerous books on the subject exist, and we particularly recommend the following (precise references to works not mentioned in this bibliography can be found at ).
For classic methods: Akhiezer , Bliss , Bolza...
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