Mathematical Methods for Physics and Engineering - Matematica.NET

CALCULUS OF VARIATIONSyabxFigure 22.1 Possible paths for the integral (22.1). The solid line is the curvealong which the integral is assumed stationary. The broken curves representsmall variations from this path.So in general we are led by this type of question to study the value of anintegral whose integrand has a specified form in terms of a certain functionand its derivatives, and to study how that value changes when the form ofthe function is varied. Specifically, we aim to find the function that makes theintegral stationary, i.e. the function that makes the value of the integral a localmaximum or minimum. Note that, unless stated otherwise, y ′ is used to denotedy/dx throughout this chapter. We also assume that all the functions we need todeal with are sufficiently smooth and differentiable.Let us consider the integral22.1 The Euler–Lagrange equationI =∫ baF(y, y ′ ,x) dx, (22.1)where a, b and the form of the function F are fixed by given considerations,e.g. the physics of the problem, but the curve y(x) is to be chosen so as tomake stationary the value of I, which is clearly a function, or more accurately afunctional, of this curve, i.e. I = I[y(x)]. Referring to figure 22.1, we wish to findthe function y(x) (given, say, by the solid line) such that first-order small changesin it (for example the two broken lines) will make only second-order changes inthe value of I.Writing this in a more mathematical form, let us suppose that y(x) is thefunction required to make I stationary and consider making the replacementy(x) → y(x)+αη(x), (22.2)where the parameter α is small and η(x) is an arbitrary function with sufficientlyamenable mathematical properties. For the value of I to be stationary with respect776