E-3 The Newton form of a geodesic in Maupertuis gauge (sphere, ellipsoid-of-revolution) 561
Box E.5 (The Lagrangean version versus the Hamiltonian version of a geodesic in E
2
A,B in terms of conformal
coordinates (isometric coordinates) and Maupertuis gauge).
(i) Universal Polar Stereographic Projection (UPS):
L
2 (q(t), ˙
q(t)) =
1
2
h `
˙
q
1
´ 2 +
`
˙
q
2
´ 2
i
+
A
2 cos
2 f
−1
„ q
(q 1 )
2 +(q 2 )
2
«
2
„
1−e 2 sin 2 f 1
„ q
(q 1 )
2 +(q 2 )
2
««
1
(q 1 )
2 +(q 2 )
2 ,
(E.90)
H
2 (q(t), p(t)) =
1
2
`
p
2
1 + p
2
2
´ −
A
2 cos
2 f
−1
„ q
(q 1 )
2 +(q 2 )
2
«
2
„
1−e 2 sin 2 f 1
„ q
(q 1 )
2 +(q 2 )
2
««
1
(q 1 )
2 +(q 2 )
2 .
(E.91)
(ii) Universal Mercator Projection (UM):
L
2 (q(t), ˙
q(t)) =
1
2
h `
˙
q
1
´ 2 +
`
˙
q
2
´ 2
i
+
1−e
2 sin
2 f
1 (q
2 /A)
2 cos 2 f −1 (q 2 /A) ,
(E.92)
H
2 (q(t), p(t)) =
1
2
`
p
2
1 + p
2
2
´ −
1−e
2 sin
2 f
−1 (q
2 /A)
2 cos 2 f 1 (q 2 /A) .
(E.93)
(iii) Universal Transverse Mercator Projection (UTM):
L
2 (q(t), ˙
q(t)) =
1
2
h `
˙
q
1
´ 2 +
`
˙
q
2
´ 2
i
+
1
2ρ 2 (1 + d 02 (q
2 )
2 + d 12 q
1 (q
2 )
2 + d 22 (q
1 )
2 (q
2 )
2 +
+d 32 (q
1 )
3 (q
2 )
2 + d 04 (q
2 )
4 + d 14 q
1 (q
2 )
4 + O λ 2 ((q
1 )
6 , (q
2 )
6 )) ,
(E.94)
H
2 (q(t), p(t)) =
1
2
`
p
2
1 + p
2
2
´ −
1
2ρ 2 (1 + d 02 (q
2 )
2 + d 12 q
1 (q
2 )
2 + d 22 (q
1 )
2 (q
2 )
2 +
+d 32 (q
1 )
3 (q
2 )
2 + d 04 (q
2 )
4 + d 14 q
1 (q
2 )
4 + O λ 2 ((q
1 )
6 , (q
2 )
6 )) .
(E.95)
(iv) Universal Conic Projection (UC) (variant one and variant two):
L
2 (q(t), ˙
q(t)) =
1
2
h `
˙
q
1
´ 2 +
`
˙
q
2
´ 2
i
+
A
2 cos
2 r
−1
“
(q
1 )
2 +(q
2 )
2
”
2c 2 n 2
„
1−e 2 sin 2 r 1
„ q
(q 1 )
2 +(q 2 )
2
«« ×
×
0
@ tan
2
0
@ π
4
−
r
−1
„ q
(q 1 )
2 +(q 2 )
2
«
2
1
A
0
@
1+e sin r
−1
„ q
(q 1 )
2 +(q 2 )
2
«
1−e sin r −1
„ q
(q 1 )
2 +(q 2 )
2
«
1
A
e1
A
−n
,
(E.96)
H
2 (q(t), p(t)) =
1
2
`
p
2
1 + p
2
2
´ −
A
2 cos
2 r
−1
“
(q
1 )
2 +(q
2 )
2
”
2c 2 n 2
„
1−e 2 sin 2 r 1
„ q
(q 1 )
2 +(q 2 )
2
«« ×
×
0
@ tan
2
0
@ π
4
−
r
−1
„ q
(q 1 )
2 +(q 2 )
2
«
2
1
A
0
@
1+e sin r
−1
„ q
(q 1 )
2 +(q 2 )
2
«
1−e sin r −1
„ q
(q 1 )
2 +(q 2 )
2
«
1
A
e1
A
−n
.
(E.97)
Box E.5 (The Lagrangean version versus the Hamiltonian version of a geodesic in E
2
A,B in terms of conformal
coordinates (isometric coordinates) and Maupertuis gauge).
(i) Universal Polar Stereographic Projection (UPS):
L
2 (q(t), ˙
q(t)) =
1
2
h `
˙
q
1
´ 2 +
`
˙
q
2
´ 2
i
+
A
2 cos
2 f
−1
„ q
(q 1 )
2 +(q 2 )
2
«
2
„
1−e 2 sin 2 f 1
„ q
(q 1 )
2 +(q 2 )
2
««
1
(q 1 )
2 +(q 2 )
2 ,
(E.90)
H
2 (q(t), p(t)) =
1
2
`
p
2
1 + p
2
2
´ −
A
2 cos
2 f
−1
„ q
(q 1 )
2 +(q 2 )
2
«
2
„
1−e 2 sin 2 f 1
„ q
(q 1 )
2 +(q 2 )
2
««
1
(q 1 )
2 +(q 2 )
2 .
(E.91)
(ii) Universal Mercator Projection (UM):
L
2 (q(t), ˙
q(t)) =
1
2
h `
˙
q
1
´ 2 +
`
˙
q
2
´ 2
i
+
1−e
2 sin
2 f
1 (q
2 /A)
2 cos 2 f −1 (q 2 /A) ,
(E.92)
H
2 (q(t), p(t)) =
1
2
`
p
2
1 + p
2
2
´ −
1−e
2 sin
2 f
−1 (q
2 /A)
2 cos 2 f 1 (q 2 /A) .
(E.93)
(iii) Universal Transverse Mercator Projection (UTM):
L
2 (q(t), ˙
q(t)) =
1
2
h `
˙
q
1
´ 2 +
`
˙
q
2
´ 2
i
+
1
2ρ 2 (1 + d 02 (q
2 )
2 + d 12 q
1 (q
2 )
2 + d 22 (q
1 )
2 (q
2 )
2 +
+d 32 (q
1 )
3 (q
2 )
2 + d 04 (q
2 )
4 + d 14 q
1 (q
2 )
4 + O λ 2 ((q
1 )
6 , (q
2 )
6 )) ,
(E.94)
H
2 (q(t), p(t)) =
1
2
`
p
2
1 + p
2
2
´ −
1
2ρ 2 (1 + d 02 (q
2 )
2 + d 12 q
1 (q
2 )
2 + d 22 (q
1 )
2 (q
2 )
2 +
+d 32 (q
1 )
3 (q
2 )
2 + d 04 (q
2 )
4 + d 14 q
1 (q
2 )
4 + O λ 2 ((q
1 )
6 , (q
2 )
6 )) .
(E.95)
(iv) Universal Conic Projection (UC) (variant one and variant two):
L
2 (q(t), ˙
q(t)) =
1
2
h `
˙
q
1
´ 2 +
`
˙
q
2
´ 2
i
+
A
2 cos
2 r
−1
“
(q
1 )
2 +(q
2 )
2
”
2c 2 n 2
„
1−e 2 sin 2 r 1
„ q
(q 1 )
2 +(q 2 )
2
«« ×
×
0
@ tan
2
0
@ π
4
−
r
−1
„ q
(q 1 )
2 +(q 2 )
2
«
2
1
A
0
@
1+e sin r
−1
„ q
(q 1 )
2 +(q 2 )
2
«
1−e sin r −1
„ q
(q 1 )
2 +(q 2 )
2
«
1
A
e1
A
−n
,
(E.96)
H
2 (q(t), p(t)) =
1
2
`
p
2
1 + p
2
2
´ −
A
2 cos
2 r
−1
“
(q
1 )
2 +(q
2 )
2
”
2c 2 n 2
„
1−e 2 sin 2 r 1
„ q
(q 1 )
2 +(q 2 )
2
«« ×
×
0
@ tan
2
0
@ π
4
−
r
−1
„ q
(q 1 )
2 +(q 2 )
2
«
2
1
A
0
@
1+e sin r
−1
„ q
(q 1 )
2 +(q 2 )
2
«
1−e sin r −1
„ q
(q 1 )
2 +(q 2 )
2
«
1
A
e1
A
−n
.
(E.97)