Introduction to the calculus of variations and its history

Sa？etak The paper presents the basics of the calculus of variations in case when the extreme searched for depends only on one real variable, and gives an overview of its historical development. We derive a necessary condition for optimality in the form of the Euler-Lagrange equation, and show that it is also a sufficient condition in the case of a convex Lagrangian. The theory is illustrated by the historically important examples of the shortest path, geodesics on the sphere, and Brachistohrone problem