5.9 Spherical Coordinates; Legendre Polynomials 335
5.9 Spherical Coordinates; Legendre Polynomials
After the Cartesian and cylindrical coordinate systems, the one most fre-
quently encountered is the spherical system (Fig. 11), in which
x=ρsin(φ)cos(θ ),
y=ρsin(φ)sin(θ ),
z=ρcos(φ).The variables are restricted by 0≤ρ,0≤θ< 2 π,0≤φ≤π.Inthiscoordi-
nate system the Laplacian operator is
∇^2 u=^1
ρ^2{∂
∂ρ(
ρ^2 ∂u
∂ρ)
+^1
sin(φ)∂
∂φ(
sin(φ)∂u
∂φ)
+^1
sin^2 (φ)∂^2 u
∂θ^2}
.
From what we have seen in other cases, we expect solvable problems in
spherical coordinates to reduce to one of the following.
Problem 1.∇^2 u=−λ^2 uinR, plus homogeneous boundary conditions.
Problem 2.∇^2 u=0inR, plus homogeneous boundary conditions on facing
sides (whereRis a generalized rectangle in spherical coordinates).
Problem 1 would come from a heat or wave equation after separating out
the time variable. Problem 2 is a part of the potential problem.
The complete solution of either of these problems is very complicated, but a
number of special cases are simple, important, and not uncommon. We have
already seen Problem 1 solved (Section 5.8) whenuis a function ofρonly.
A second important case is Problem 2, whenuis independent of the variableθ.
We shall state a complete boundary value problem and solve it by separation
Figure 11 Spherical coordinates.