Mathematical Methods for Physics and Engineering - Matematica.NET

CALCULUS OF VARIATIONSUsing (22.13) and the fact that y does not appear explicitly, we obtain(∂ρ ∂y )− ∂ (τ ∂y )=0.∂t ∂t ∂x ∂xIf, in addition, ρ and τ do not depend on x or t then∂ 2 y∂x = 1 ∂ 2 y2 c 2 ∂t , 2where c 2 = τ/ρ. This is the wave equation for small transverse oscillations of a tautuniform string. ◭22.6 General eigenvalue problemsWe have seen in this chapter that the problem of finding a curve that makes thevalue of a given integral stationary when the integral is taken along the curveresults, in each case, in a differential equation for the curve. It is not a greatextension to ask whether this may be used to solve differential equations, bysetting up a suitable variational problem and then seeking ways other than theEuler equation of finding or estimating stationary solutions.We shall be concerned with differential equations of the form Ly = λρ(x)y,where the differential operator L is self-adjoint, so that L = L † (with appropriateboundary conditions on the solution y) andρ(x) is some weight function, asdiscussed in chapter 17. In particular, we will concentrate on the Sturm–Liouvilleequation as an explicit example, but much of what follows can be applied toother equations of this type.We have already discussed the solution of equations of the Sturm–Liouvilletype 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 ofsuch equations.Suppose we search for stationary values of the integral∫ b []I = p(x)y ′2 (x) − q(x)y 2 (x) dx, (22.22)awith y(a) =y(b) =0andp and q any sufficiently smooth and differentiablefunctions of x. However, in addition we impose a normalisation conditionJ =∫ baρ(x)y 2 (x) dx = constant. (22.23)Here ρ(x) is a positive weight function defined in the interval a ≤ x ≤ b, butwhich may in particular cases be a constant.Then, as in section 22.4, we use undetermined Lagrange multipliers, § and§ We use −λ, rather than λ, so that the final equation (22.24) appears in the conventional Sturm–Liouville form.790