The calculus of variations

A mathematical theory in continuous evolution

Abstract

We give a short history of the mathematical theory of the Calculus of variations. After the illustration of isoperimetric, geodetic and brachystochrone problems, we expound the method of Lagrange variations, which leads to Eulero-Lagrange fundamental equation. We provide then a brief glance at the direct method and regularity of the solutions of problems in Calculus of variations. This work also contains some brief biographical notes on mathematicians who have contributed to the development of the Calculus of variations.