![]() In geometry, the simplest, most classic question is without a doubt that of geodetics: how to find the shortest route from one point to another on the earth, or more generally, on a surface. The calculus of variations is a fundamental instrument for analysing the problems posed as problems of optimisation, and as such intervenes in numerous disciplines: geometry, physics, economy, engineering, etc. However, Lagrange had a broad mathematical range and also wrote other articles at that time on different topics, including the vibrating string and differential equations. It is this article that interests us the most because it already contains the foundations of the calculus of variations and the methods of multipliers called “Lagrange multipliers”, ideas that would both be developed over the course of his career. He published at that time numerous articles in the Miscellanea Taurinensia, the first one of which, in 1762, was entitled “Essai d’une nouvelle méthode pour déterminer les maxima et les minima des formules intégrales indéfinies”. In 1758 he co-founded what would become the Academy of Sciences in Turin. The debut of the young Lagrange was a period of great activity. This letter was enough of a recommendation to secure Lagrange a position as a teacher at the Royal School of Artillery in Turin. The importance of the matter has led me to outline, with the aid of your light, an analytical solution to which I will give no publicity until you yourself have published the whole of your research, so that I do not take away any part of the glory that is due to you.) (Your solution to the isoperimetric problem leaves nothing to be desired, and I rejoice that this subject, of which I was almost the only one who dealt with it since the first attempts, has been taken by you to the highest degree of perfection. The reader can also consult for a more advanced epistemological analysis, as well as regarding biographical elements. ![]() A good point of departure is the work of Catherine Goldstein. The history of the calculus of variations and Lagrange’s contribution to it is well documented. Here we will look at how Lagrange was led to his interest in these problems, discuss the simplest elements and principles of his discovery, and finally, show the repercussions they have had up to the present day. Used today almost as much as ordinary differential calculus, with all sorts of domains of application, the calculus of variations forms the basis of the mechanics known as Lagrangian, without which modern physics could not exist. Įuler and Lagrange also studied problems on a conditional extremum.Together with Euler, Lagrange is the inventor of the calculus of variations, a simple and elegant idea that revolutionised the way of solving problems of optimisation, the formulation of classical physics, and had an enormous influence on how partial derivatives equations are viewed. Is a vector function of arbitrary dimension. The principal results concerning the simplest problem of variational calculus are applied to the general case of functionals of the type Implies supplementary conditions to be satisfied by the mobile ends - the so-called transversality condition which, in conjunction with the boundary conditions, yields a closed system of conditions for the boundary value problem. In problems with mobile ends the condition $ \delta J = 0 $ It is required to minimize the functional The following scheme describes a rather wide range of problems of classical variational calculus. ![]() by the method of small perturbations of the arguments and functionals such problems, in the wider sense, are opposite to discrete optimization problems. The term "variational calculus" has a broader sense also, viz., a branch of the theory of extremal problems in which the extrema are studied by the "method of variations" (cf. ![]() This is the framework of the problems which are still known as problems of classical variational calculus. The branch of mathematics in which one studies methods for obtaining extrema of functionals which depend on the choice of one or several functions subject to constraints of various kinds (phase, differential, integral, etc.) imposed on these functions. ![]()
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