Geography Reference
In-Depth Information
Proof.
Once the fundamental solution of the d'Alembert-Euler equations is based on the function space
of homogeneous polynomials
P
r
(
q,p
)=
α
+
β
=
r
C
αβ
q
α
p
β
(0
≤
α
≤
r
)
,
(D.17)
the vectorial Laplace-Beltrami equation has to be fulfilled. (
r
−
1) constraints are given for (
r
+1)
coecients such that for any
r
≥
2 two linear independent harmonic polynomials exist which we
are going to construct.
P
r
(
q,p
)=
c
r,
0
q
r
+
c
r−
1
,
1
q
r−
1
p
+
c
r−
2
,
2
q
r−
2
p
2
+
c
r−
3
,
3
q
r−
3
p
3
+
c
r−
4
,
4
q
r−
4
p
4
+
···
...
+
c
3
,r−
3
q
3
p
r−
3
+
c
2
,r−
2
q
2
p
r−
2
+
c
1
,r−
1
qp
r−
1
+
c
0
,r
p
r
,
(D.18)
ΔP
r
(
q,p
)=(
∂
2
∂q
2
+
∂
2
∂p
2
)
P
r
(
q,p
)=
1)
c
r,
0
q
r−
2
+(
r
3)
c
r−
2
,
2
q
r−
4
p
2
+(
r
5)
c
r−
4
,
4
q
r−
6
p
4
+
=
r
(
r
−
−
2)(
r
−
−
4)(
r
−
···
...
+2
c
r−
2
,
2
q
r−
2
+4
·
3
c
r−
4
,
4
q
r−
4
p
4
+
···
(D.19)
...
+(
r −
1)(
r −
2)
c
r−
1
,
1
q
r−
3
p
+(
r −
3)(
r −
4)
c
r−
3
,
3
q
r−
5
p
3
+
···
...
+3
2
c
r−
3
,
3
q
r−
3
p
+
·
···
.
Obviously, the recurrence relation (
D.20
) connects the coecients of even second index with each
other, similarly the coecients of odd second index according to the following set of coecient
pairs:
c
r,
0
|
c
r−
2
,
2
,c
r−
1
,
1
|
c
r−
3
,
3
,c
r−
2
,
2
|
c
r−
4
,
4
,c
r−
3
,
3
|
c
r−
5
,
5
etc.
c
k,r−k
k
(
k −
1) +
c
k−
2
,r−k
+2
(
r − k
+2)(
r − k
+1)=0
⇔
(D.20)
c
k,r−k
k
(
k −
1)
c
k−
2
,r−k
+2
=
−
k
+1)
.
(
r
−
k
+2)(
r
−
Those harmonic polynomials with coecients of even second index are denoted by
P
r,
1
, while
alternatively those with coecients of odd second index are denoted by
P
r,
2
. Once we chose
c
r,
0
= 1, the recurrence relation leads to (
D.21
)or(
D.22
). In contrast, once we choose
c
r−
1
,
1
=
r
we are led to (
D.23
)or(
D.24
).
r
(
r
−
1)
2
q
r−
2
p
2
+
(
r
−
3)(
r
−
2)(
r
−
1)
4
P
r,
1
(
q,p
)=
q
r
q
r−
4
p
4
+
··· ,
−
(D.21)
∗
3
∗
2
1)
s
r
q
r−
2
s
p
2
s
,
[
2
]
P
r,
1
(
q,p
)=
(
−
(D.22)
2
s
s=0
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