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Definition E.3 (Local polar/normal coordinates).
2
, we introduce local polar/normal coordinates by (
E.118
), where
t
is
the parameter of the Maupertuis gauged geodesic and
α
is the azimuth of the geodesic passing
the point
q
In the tangent plane
T
q
M
2
,g
{
μν
∈{
M
}
.
u
1
=
u
:=
t
cos
α,
(E.118)
u
2
=
u
:=
t
sin
α.
End of Definition.
E-36 Maupertuis Gauged Geodesics (Lie Series, Hamilton Portrait)
2
A,B
and the geodesic “Maupertuis gauged” in their Hamilton form are analytic. Accord-
ingly, we can solve (
E.113
) by the Taylor expansion (
E.119
).
M
2
:=
E
n
q
1
=
q
0
+
d
q
1
d
2
q
1
d
(
m
)
q
1
d
t
m
d
t
(
t
=0)
t
+
1
1
m
!
d
t
2
(
t
=0)
t
2
+ lim
(
t
=0)
t
m
,
2!
n→∞
m
=3
(E.119)
n
q
2
=
q
0
+
d
q
2
d
t
(
t
=0)
t
+
1
d
2
q
2
1
m
!
d
(
m
)
q
2
d
t
m
d
t
2
(
t
=0)
t
2
+ lim
(
t
=0)
t
m
.
2!
n→∞
m
=3
By means of (
E.113
)and(
E.118
), we here are able to take
advantage of the Lie recurrence (“Lie series”) which is sum-
marized in the following Box
E.7
in order to formulate
the solution of the
initial value problem
(“erste geodatische
Hauptaufgabe”). By standard series inversion of the homo-
geneous polynomial
q
μ
q
0
, we have finally solved the
boundary value problem
(“zweite geodatische Hauptauf-
gabe”) in terms of local polar/normal coordinates.
−
n
q
μ
=
q
0
+
λ
0
u
μ
+
α
μγ
u
μ
u
γ
+ lim
n→∞
α
μ
1
...μ
m
u
μ
1
...u
μ
m
,
(E.120)
m
=3
n
q
0
)+
b
μγ
(
q
μ
q
0
)(
q
γ
q
0
) + lim
n→∞
u
μ
=
b
0
(
q
μ
b
μ
1
...μ
m
−
−
−
m
=3
(
q
μ
1
q
μ
0
)
...
(
q
μ
m
q
μ
0
)
.
−
−
(E.121)
BymeansoftheLieseriestosolve(
E.113
), which is generated by the Universal Transverse
Mercator Projection (UTM), we are led to (
E.122
)and(
E.123
), with
Δq
1
:=
q
1
q
0
and
Δq
2
:=
−
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