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⎡
e
2
)cos
2
r
−
1
(
q
1
)
2
+(
q
2
)
2
1+sin
r
−
1
(
q
1
)
2
+(
q
2
)
2
−
1
A
2
(1
−
⎣
p
μ
=
2
c
2
n
1
e
2
sin
2
r
−
1
(
q
1
)
2
+(
q
2
)
2
1
e
sin
r
−
1
(
q
1
)
2
+(
q
2
)
2
2
×
−
−
2
⎞
⎠
r
−
1
(
q
1
)
2
+(
q
2
)
2
1+
e
sin
r
−
1
(
q
1
)
2
+(
q
2
)
2
⎞
⎞
−
2
n−
1
e
×
tan
π
⎠
⎠
e
sin
r
−
1
(
q
1
)
2
+(
q
2
)
2
4
−
−
2
1
−
e
2
)cos
r
−
1
(
q
1
)
2
+(
q
2
)
2
sin
r
−
1
(
q
1
)
2
+(
q
2
)
2
c
2
n
2
1
A
2
(1
−
−
×
(E.110)
e
2
sin
2
r
−
1
(
q
1
)
2
+(
q
2
)
2
2
−
r
−
1
(
q
1
)
2
+(
q
2
)
2
⎞
tan
π
⎠
×
4
−
2
−
2
n
⎤
⎦
×
2
⎞
⎠
1+
e
sin
r
−
1
(
q
1
)
2
+(
q
2
)
2
1
− e
sin
r
−
1
(
q
1
)
2
+(
q
2
)
2
⎞
e
⎠
∂r
−
1
(
q
1
)
2
+(
q
2
)
2
∂q
μ
×
.
E-35 Maupertuis Gauged Geodesics (Normal Coordinates, Local
Tangent Plane)
The unit tangent vector (Darboux one-leg)
d
at a point
q
of a geodesic can be represented in the
local base
{
g
1
,
g
2
}
, the local tangent vectors (Gauss two-leg), which spans the tangent space
T
q
M
2
3
,δ
ij
}
.
Note that from now on, we assume
{
g
1
,
g
2
}
to be an orthogonal, but not normalized Gauss
two-leg which can be materialized by orthogonal coordinates
at a point
q
,namelyby(
E.111
), where
x
(
q
1
,q
2
) denotes the immersion
{
M
2
,g
μν
}→{
R
{
q
1
,q
2
}
.
d
=
x
(
q
1
,q
2
)=
∂
x
∂q
μ
q
μ
=
g
1
q
1
+
g
2
q
2
(E.111)
Alternatively, we can represent
d
by polar coordinates as follows: by means of the scalar products
(inner products) (
E.112
), we introduce the azimuth
α
with respect to the first Gauss two-leg
g
1
. Inserting (
E.111
)into(
E.112
) and differentiating (
E.112
), we obtain (
E.113
) or, in terms of
conformal coordinates, we obtain (
E.114
). The parameter change
s
→
t
,(
E.112
) implemented,
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