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In-Depth Information
d
U
K
d
S
=
G
KL
P
L
=
∂H
2
∂P
K
,
(20.63)
∂G
AB
∂U
K
P
A
P
B
=
∂H
2
∂U
K
,
d
P
K
d
S
1
2
=
−
−
H
2
:=
P
K
d
U
K
2
=
1
2
G
KL
P
K
P
L
.
d
S
−L
(20.64)
2
A
1
,A
2
First, let us assume that the differentiable manifold
E
is partially covered by a set of orthog-
onal coordinates
, especially in the sense of
G
12
= 0: the Hamilton equations are firstly
specified towards (
20.65
), (
20.66
), and (
20.67
).
L
=
G
11
P
1
=cos
α/
G
11
,
{
L, B
}
(20.65)
B
=
G
22
P
2
=sin
α/
G
22
,
1
1
P
1
=
P
L
=
2
(
G
11
,L
P
1
+
G
22
,L
P
2
)=
2
(cos
2
αG
11
G
11
,L
+sin
2
αG
22
G
22
−
−
,L
)
,
(20.66)
1
1
P
2
=
P
B
=
2
(
G
11
,B
P
1
+
G
22
,B
P
2
)=
2
(cos
2
αG
11
G
11
,B
+sin
2
αG
22
G
22
−
−
,B
)
.
(20.67)
If we compare (
20.67
)and(
20.62
), differentiated by (
√
G
22
sin
α
)
,G
11
,B
=
G
−
1
11
,B
=
−G
−
2
11
G
11
,B
,
,B
=
G
−
1
22
,B
=
−G
−
2
and
G
22
22
G
22
,B
,weareledto(
20.68
). Of course, the sam
e re
sult would have been
achieved by the comparison of (
20.66
)and(
20.62
), differentiated by (
√
G
11
cos
α
)
.
∂
√
G
22
∂L
sin
α
+
∂
√
G
11
∂B
cos
α
.
1
α
=
√
G
11
G
22
−
(20.68)
Second, we specify the metric tensor
G
KL
by Box
20.1
, 1st chart, in terms of orthogonal coordi-
nates
. Obviously,
P
L
= 0 (which holds for arbitrary surfaces-of-
revolution) generates the
conservation of angular momentum P
1
=
N
(
B
)cos
B
cos
α
=
A
,where
the constant
A
is the
A. C. Clairaut
constant.
2
A
1
,A
2
{
L, B
}
covering partially
E
cos
α
sin
α
M
(
B
)
,
L
=
N
(
B
)cos
B
, B
=
(20.69)
P
1
=
P
L
=[
N
(
B
)cos
B
cos
α
]
=0
,
cos
2
α
d
d
B
ln(
M
2
(
B
)
,
P
2
=
P
B
=
1
d
B
ln(
N
2
(
B
)cos
2
B
)+sin
2
α
d
(20.70)
2
tan
B
N
(
B
)
cos
α.
α
=
−
(20.71)
The ellipsoid-of-revolution
E
A
1
,A
2
is an
analytic manifold
. Thus, there exists the Taylor expansion
in phase space
{
L, B, P
L
,P
B
}
or
{
L, B, α
}
, respectively, namely
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