Mathematical Methods for Physics and Engineering : A Comprehensive Guide

22.9 Exercises

22.9 Exercises

22.1 A surface of revolution, whose equation in cylindrical polar coordinates isρ=
ρ(z), is bounded by the circlesρ=a,z=±c(a>c). Show that the function
that makes the surface integralI=

∫

ρ−^1 /^2 dSstationary with respect to small
variationsisgivenbyρ(z)=k+z^2 /(4k), wherek=[a±(a^2 −c^2 )^1 /^2 ]/2.
22.2 Show that the lowest value of the integral
∫B

A

(1 +y′^2 )^1 /^2

y

dx,

whereAis (− 1 ,1) andBis (1,1), is 2 ln(1 +

√

2). Assume that the Euler–Lagrange
equation gives a minimising curve.
22.3 The refractive indexnof a medium is a function only of the distancerfrom a
fixed pointO. Prove that the equation of a light ray, assumed to lie in a plane
throughO, travelling in the medium satisfies (in plane polar coordinates)

1

r^2

(

dr

dφ

) 2

=

r^2

a^2

n^2 (r)

n^2 (a)

− 1 ,

whereais the distance of the ray fromOat the point at whichdr/dφ=0.
Ifn= [1+(α^2 /r^2 )]^1 /^2 and the ray starts and ends far fromO, find its deviation
(the angle through which the ray is turned), if its minimum distance fromOisa.
22.4 The Lagrangian for aπ-meson is given by

L(x,t)=^12 (φ ̇^2 −|∇φ|^2 −μ^2 φ^2 ),

whereμis the meson mass andφ(x,t) is its wavefunction. Assuming Hamilton’s
principle, find the wave equation satisfied byφ.
22.5 Prove the following results about general systems.

(a) For a system described in terms of coordinatesqiandt, show that iftdoes

not appear explicitly in the expressions forx,yandz(x=x(qi,t), etc.) then

the kinetic energyTis a homogeneous quadratic function of the ̇qi(it may

also involve theqi). Deduce that

∑

i ̇qi(∂T /∂ ̇qi)=2T.

(b) Assuming that the forces acting on the system are derivable from a potential

V, show, by expressingdT /dtin terms ofqiand ̇qi,thatd(T+V)/dt=0.

22.6 For a system specified by the coordinatesqandt, show that the equation of
motion is unchanged if the LagrangianL(q, ̇q, t) is replaced by

L 1 =L+

dφ(q, t)

dt

,

whereφis an arbitrary function. Deduce that the equation of motion of a particle
that moves in one dimension subject to a force−dV(x)/dx(xbeing measured
from a pointO) is unchanged ifOis forced to move with a constant velocityv
(xstill being measured fromO).
22.7 In cylindrical polar coordinates, the curve (ρ(θ),θ,αρ(θ)) lies on the surface of
the conez=αρ. Show that geodesics (curves of minimum length joining two
points) on the cone satisfy

ρ^4 =c^2 [β^2 ρ′^2 +ρ^2 ],

wherecis an arbitrary constant, butβhas to have a particular value. Determine

the form ofρ(θ) and hence find the equation of the shortest path on the cone

between the points (R,−θ 0 ,αR)and(R, θ 0 ,αR).

[ You will find it useful to determine the form of the derivative of cos−^1 (u−^1 ). ]