Variational Methods

It would be hopeless to attempt to give a complete account of the history of the calculus of variations. The interest of Greek philosophers in isoperimetric problems underscores the importance of "e;optimal form"e; in ancient cultures, see Hildebrandt-Tromba [1] for a beautiful treatise of this subject. While variatio- nal problems thus are part of our classical cultural heritage, the first modern treatment of a variational problem is attributed to Fermat (see Goldstine [1; p.l]). Postulating that light follows a path of least possible time, in 1662 Fer- mat was able to derive the laws of refraction, thereby using methods which may already be termed analytic. With the development of the Calculus by Newton and Leibniz, the basis was laid for a more systematic development of the calculus of variations. The brothers Johann and Jakob Bernoulli and Johann's student Leonhard Euler, all from the city of Basel in Switzerland, were to become the "e;founding fathers"e; (Hildebrandt-Tromba [1; p.21]) of this new discipline. In 1743 Euler [1] sub- mitted "e;A method for finding curves enjoying certain maximum or minimum properties"e;, published 1744, the first textbook on the calculus of variations.