The EulerLagrange equation

CALCULUS OF VARIATIONS
y
b
a
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
integral stationary, 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/dx throughout 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
F(y, y , x) dx,
(22.1)
a
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 to
make stationary the value of I, 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 function y(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 of I.
Writing this in a more mathematical form, let us suppose that y(x) is the
function required to make I stationary and consider making the replacement
y(x) → y(x) + αη(x),
(22.2)
where the parameter α is small and η(x) is an arbitrary function with sufficiently
amenable mathematical properties. For the value of I to be stationary with respect
776