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];

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http://dx.doi.org/10.1090/stml/072/02