Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

y

P

a

θ

−a O

r

z

x

Figure 21.9 The polar axisOzis taken as normal to the plane of the ring of

matter and passing through its centre.

We will illustrate the method for Laplace’s equation in spherical polars and first

assume that the required solution of∇^2 u= 0 can be written as a superposition

in the normal way:

u(r, θ, φ)=

∑∞

=0

∑

m=−

(Ar+Br−(+1))Pm(cosθ)(Ccosmφ+Dsinmφ).

(21.58)

Here, all the constantsA, B, C, Dmay depend uponandm, and we have

assumed that the required solution is finite on the polar axis. As usual, boundary

conditions of a physical nature will then fix or eliminate some of the constants;

for example,ufinite at the origin implies allB= 0, or axial symmetry implies

that onlym= 0 terms are present.

The essence of the method is then to find the remaining constants by determin-

inguat values ofr, θ, φfor which it can be evaluatedby other means, e.g. by direct

calculation on an axis of symmetry. Once the remaining constants have been fixed

by these special considerations to have particular values, the uniqueness theorem

can be invoked to establish that they must have these values in general.

Calculate the gravitational potential at ageneral point in space due to a uniform ring of

matter of radiusaand total massM.

Everywhere except on the ring the potentialu(r) satisfies the Laplace equation, and so if
we use polar coordinates with the normal to the ring as polar axis, as in figure 21.9, a
solution of the form (21.58) can be assumed.
We expect the potentialu(r, θ, φ)totendtozeroasr→∞, and also to be finite atr=0.
At first sight this might seem to imply that allAandB, and henceu, must be identically
zero, an unacceptable result. In fact, what it means is that different expressions must apply
to different regions of space. On the ring itself we no longer have∇^2 u= 0 and so it is not