Mathematical Methods for Physics and Engineering : A Comprehensive Guide

CALCULUS OF VARIATIONS

y

a b x

Figure 22.1 Possible paths for the integral (22.1). The solid line is the curve

along which the integral is assumed stationary. The broken curves represent

small variations from this path.

So in general we are led by this type of question to study the value of an

integral whose integrand has a specified form in terms of a certain function

and its derivatives, and to study how that value changes when the form of

the function is varied. Specifically, we aim to find the function that makes the

integralstationary, i.e. the function that makes the value of the integral a local

maximum or minimum. Note that, unless stated otherwise,y′is used to denote

dy/dxthroughout this chapter. We also assume that all the functions we need to

deal with are sufficiently smooth and differentiable.

22.1 The Euler–Lagrange equation

Let us consider the integral

I=

∫b

a

F(y, y′,x)dx, (22.1)

wherea,band the form of the functionFare fixed by given considerations,

e.g. the physics of the problem, but the curvey(x) is to be chosen so as to

make stationary the value ofI, which is clearly a function, or more accurately a

functional, of this curve, i.e.I=I[y(x)]. Referring to figure 22.1, we wish to find

the functiony(x) (given, say, by the solid line) such that first-order small changes

in it (for example the two broken lines) will make only second-order changes in

the value ofI.

Writing this in a more mathematical form, let us suppose thaty(x)isthe

function required to makeIstationary and consider making the replacement

y(x)→y(x)+αη(x), (22.2)

where the parameterαis small andη(x) is an arbitrary function with sufficiently

amenable mathematical properties. For the value ofIto be stationary with respect