Mathematical Methods for Physics and Engineering : A Comprehensive Guide

CALCULUS OF VARIATIONS

22.8 Derive the differential equations for the plane-polar coordinates,randφ,ofa
particle of unit mass moving in a field of potentialV(r). Find the form ofVif
the path of the particle is given byr=asinφ.
22.9 You are provided with a line of lengthπa/2 and negligible mass and some lead
shot of total massM. Use a variational method to determine how the lead shot
must be distributed along the line if the loaded line is to hang in a circular arc
of radiusawhen its ends are attached to two points at the same height. Measure
the distancesalong the line from its centre.
22.10 Extend the result of subsection 22.2.2 to the case of several dependent variables
yi(x), showing that, ifxdoes not appear explicitly in the integrand, then a first
integral of the Euler–Lagrange equations is

F−

∑n

i=1

y′i

∂F

∂yi′

=constant.

22.11 A general result is that light travels through a variable medium by a path which
minimises the travel time (thisis an alternative formulation of Fermat’s principle).
With respect to a particular cylindrical polar coordinate system (ρ, φ, z), the speed
of lightv(ρ, φ) is independent ofz. If the path of the light is parameterised as
ρ=ρ(z),φ=φ(z), use the result of the previous exercise to show that

v^2 (ρ′

2

+ρ^2 φ′

2
+1)
is constant along the path.
For the particular case whenv=v(ρ)=b(a^2 +ρ^2 )^1 /^2 , show that the two Euler–
Lagrange equations have a common solution in which the light travels along a
helical path given byφ=Az+B,ρ=C, provided thatAhas a particular value.
22.12 Light travels in the verticalxz-plane through a slab of material which lies between
the planesz=z 0 andz=2z 0 , and in which the speed of lightv(z)=c 0 z/z 0.
Using the alternative formulation of Fermat’s principle, given in the previous
question, show that the ray paths are arcs of circles.
Deduce that, if a ray enters the material at (0,z 0 ) at an angle to the vertical,
π/ 2 −θ,ofmorethan30◦, then it does not reach the far side of the slab.
22.13 A dam of capacityV(less thanπb^2 h/2) is to be constructed on level ground next
to a long straight wall which runs from (−b,0) to (b,0).Thisistobeachievedby
joining the ends of a new wall, of heighth, to those of the existing wall. Show
that, in order to minimise the lengthLof new wall to be built, it should form
part of a circle, and thatLis then given by
∫b

−b

dx

(1−λ^2 x^2 )^1 /^2

,

whereλis found from

V

hb^2

=

sin−^1 μ

μ^2

−

(1−μ^2 )^1 /^2
μ
andμ=λb.
22.14 In the brachistochrone problem of subsection 22.3.4 show that if the upper end-
point can lie anywhere on the curveh(x, y) = 0, then the curve of quickest descent
y(x)meetsh(x, y) = 0 at right angles.
22.15 The Schwarzchild metric for the static field of a non-rotating spherically sym-
metric black hole of massMis given by

(ds)^2 =c^2

(

1 −

2 GM

c^2 r

)

(dt)^2 −

(dr)^2

1 − 2 GM/(c^2 r)

−r^2 (dθ)^2 −r^2 sin^2 θ(dφ)^2.

Considering only motion confined to the planeθ=π/2, and assuming that the