Mathematical Methods for Physics and Engineering : A Comprehensive Guide

CALCULUS OF VARIATIONS

Using (22.13) and the fact thatydoes not appear explicitly, we obtain

∂

∂t

(

ρ

∂y

∂t

)

−

∂

∂x

(

τ

∂y

∂x

)

=0.

If, in addition,ρandτdo not depend onxortthen

∂^2 y

∂x^2

=

1

c^2

∂^2 y

∂t^2

,

wherec^2 =τ/ρ. This is the wave equation for small transverse oscillations of a taut
uniform string.

22.6 General eigenvalue problems

We have seen in this chapter that the problem of finding a curve that makes the

value of a given integral stationary when the integral is taken along the curve

results, in each case, in a differential equation for the curve. It is not a great

extension to ask whether this may be used to solve differential equations, by

setting up a suitable variational problem and then seeking ways other than the

Euler equation of finding or estimating stationary solutions.

We shall be concerned with differential equations of the formLy=λρ(x)y,

where the differential operatorLis self-adjoint, so thatL=L†(with appropriate

boundary conditions on the solutiony)andρ(x) is some weight function, as

discussed in chapter 17. In particular, we will concentrate on the Sturm–Liouville

equation as an explicit example, but much of what follows can be applied to

other equations of this type.

We have already discussed the solution of equations of the Sturm–Liouville

type in chapter 17 and the same notation will be used here. In this section,

however, we will adopt a variational approach to estimating the eigenvalues of

such equations.

Suppose we search for stationary values of the integral

I=

∫b

a

[

p(x)y′

2

(x)−q(x)y^2 (x)

]

dx, (22.22)

withy(a)=y(b)=0andpandqany sufficiently smooth and differentiable

functions ofx. However, in addition we impose a normalisation condition

J=

∫b

a

ρ(x)y^2 (x)dx= constant. (22.23)

Hereρ(x) is a positive weight function defined in the intervala≤x≤b, but

which may in particular cases be a constant.

Then, as in section 22.4, we use undetermined Lagrange multipliers,§and

§We use−λ, rather thanλ, so that the final equation (22.24) appears in the conventional Sturm–

Liouville form.