The calculus of variations and optimization methods

Includes bibliographical references (page 53)The calculus of variations is a branch of mathematics concerning the problems of finding maximum and minimum values of an integral. Examples of basic problems discussed in this area are: The problem of finding the plane curve of smallest arc length connecting two points, the problem of finding the shortest arc on the surface of a sphere connecting two points on the surface, the problem of finding a minimal surface of revolution, and the brachistochrone problem, etc. Two basic questions in the calculus of variations are that of finding necessary and sufficient conditions for a minimizing function; one of the four necessary conditions, the Euler differential equation, will be derived and discussed. Next, a parametric approach is presented, followed by a discussion of isoperimetric problems. Fermat's principles of optics\ud stated as a background to Bernoulli's proof; Plateau's problem is illustrated. Finally, we give a brief history of the calculus of variations in the 19th century