Mathematical Methods for Physics and Engineering : A Comprehensive Guide

22.9 EXERCISES

path of a small test particle is such as to make

∫

dsstationary, find two first
integrals of the equations of motion. From their Newtonian limits, in which
GM/r, ̇r^2 andr^2 φ ̇^2 are allc^2 , identify the constants of integration.
22.16 Use result (22.27) to evaluate

J=

∫ 1

− 1

(1−x^2 )Pm′(x)Pn′(x)dx,

wherePm(x) is a Legendre polynomial of orderm.
22.17 Determine the minimum value that the integral

J=

∫ 1

0

[x^4 (y′′)^2 +4x^2 (y′)^2 ]dx

can have, given thatyis not singular atx=0andthaty(1) =y′(1) = 1. Assume
that the Euler–Lagrange equation gives the lower limit, and verify retrospectively
that your solution makes the first term on the LHS of equation (22.15) vanish.
22.18 Show thaty′′−xy+λx^2 y= 0 has a solution for whichy(0) =y(1) = 0 and
λ≤ 147 /4.
22.19 Find an appropriate, but simple, trial function and use it to estimate the lowest
eigenvalueλ 0 of Stokes’ equation,
d^2 y
dx^2

+λxy=0, withy(0) =y(π)=0.

Explain why your estimate must be strictly greater thanλ 0.
22.20 Estimate the lowest eigenvalue,λ 0 , of the equation

d^2 y

dx^2

−x^2 y+λy=0,y(−1) =y(1) = 0,

using a quadratic trial function.
22.21 A drumskin is stretched across a fixed circular rim of radiusa. Small transverse
vibrations of the skin have an amplitudez(ρ, φ, t)thatsatisfies

∇^2 z=

1

c^2

∂^2 z
∂t^2
in plane polar coordinates. For a normal mode independent of azimuth,z=
Z(ρ)cosωt, find the differential equation satisfied byZ(ρ). By using a trial
function of the formaν−ρν, with adjustable parameterν, obtain an estimate for
the lowest normal mode frequency.
[ The exact answer is (5.78)^1 /^2 c/a.]
22.22 Consider the problem of finding the lowest eigenvalue,λ 0 , of the equation

(1 +x^2 )

d^2 y

dx^2

+2x

dy

dx

+λy=0,y(±1) = 0.

(a) Recast the problem in variational form, and derive an approximationλ 1 to

λ 0 by using the trial functiony 1 (x)=1−x^2.

(b) Show that an improved estimateλ 2 is obtained by usingy 2 (x)=cos(πx/2).

(c) Prove that the estimateλ(γ) obtained by takingy 1 (x)+γy 2 (x)asthetrial

function is

λ(γ)=

64 /15 + 64γ/π− 384 γ/π^3 +(π^2 /3+1/2)γ^2

16 /15 + 64γ/π^3 +γ^2

.

Investigateλ(γ) numerically asγis varied, or, more simply, show that

λ(− 1 .80) = 3.668, an improvement on bothλ 1 andλ 2.