ABSTRACT

It is human nature to optimize. Typically, we try to maximize profit, minimize cost, travel to a destination in the quickest time, and put out the least effort to get our work done. Nature, too, is seen to strive for efficiency. By Fermat’s principle, light travels a route of the least time between two points, and by Hamilton’s principle, mass particles move with the “least action” (giving Newton’s second law of motion as a consequence). Our natural propensity to optimize has led to a long-standing effort to systematically determine the optimal realization of a variety of activities in science and engineering. This continuing effort has created a body of mathematical methods called (the mathematics of) optimization. We are concerned in these pages with a class of problems in optimization which involves minimizing or maximizing the value of an integral. In the simplest setting, we seek a function y(x) on the interval a ≤ x ≤ b which maximizes or minimizes the definite integral () J [ y ] ≡ ∫ a b F ( x , y ( x ) , y ′ ( x ) )   d x , (     ) ′ ≡ d (     ) d x . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749821/5a9bd1b0-aa97-4dfe-adb3-8f69a65a6308/content/eq1.tif"/>