Chapter 2 The First Variation 2.1. The simplest problem Our goal is to minimize (or to maximize) a definite integral of the form J[y] = b a f(x, y(x),y (x)) dx (2.1) subject to the boundary conditions y(a) = ya , y(b) = yb . (2.2) I wrote J[y] rather than J(y) to emphasize that we are dealing with functionals and not just functions. Our definite integral returns a real number for each function y(x). A functional is an operator that maps functions to real numbers. Functional analysis was, originally, the study of functionals. The purpose of the calculus of variations is to maximize or minimize functionals. We will encounter functionals that act on all or part of several well-known function spaces. Function spaces that occur in the calcu- lus of variations include the following: (a) C[a, b], the space of real-valued functions that are continuous on the closed interval [a, b] 27
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