562 E Geodesics
Box E.6 (The differential equations of a geodesic in E
2
A,B in terms of conformal coordinates (isometric
coordinates) and Maupertuis gauge: Lagrange portrait, two differential equations of second order, Hamilton
portrait, four differential equations of first order).
(i) Universal Polar Stereographic Projection (UPS):
¨
q
µ +
A
2
(q 1 )
2 +(q 2 )
2
(1−e
2 ) cos f
−1
„ q
(q 1 )
2 +(q 2 )
2
«
sin f
−1
„ q
(q 1 )
2 +(q 2 )
2
«
„
1−e 2 sin 2 f −1
„ q
(q 1 )
2 +(q 2 )
2
«« 2
×
×
∂f
−1
„ q
(q 1 )
2 +(q 2 )
2
«
∂q µ
−
A
2
h
(q 1 )
2 +(q 2 )
2
i 2
cos
2 f
−1
„ q
(q 1 )
2 +(q 2 )
2
«
1−e 2 sin 2 f −1
„ q
(q 1 )
2 +(q 2 )
2
« q
µ = 0
∀ µ = 1, 2 ,
(E.98)
˙
q
µ = δ
µν p ν ∀ µ, ν = 1, 2 ,
(E.99)
˙
p µ = −
A
2
(q 1 ) 2 +(q 2 ) 2
(1−e
2 ) cos f
−1
“ √
(q 1 ) 2 +(q 2 ) 2
”
sin f
−1
“ √
(q 1 ) 2 +(q 2 ) 2
”
„
1−e 2 sin 2 f −1
„ q
(q 1 )
2 +(q 2 )
2
«« 2
×
×
∂f
−1
“ √
(q 1 ) 2 +(q 2 ) 2
”
∂q µ
+
A
2
[(q 1 ) 2 +(q 2 ) 2 ]
2
cos
2 f
−1
“ √
(q 1 ) 2 +(q 2 ) 2
”
1−e 2 sin 2 f −1
„ q
(q 1 )
2 +(q 2 )
2
« q
µ
∀ µ = 1, 2 .
(E.100)
(ii) Universal Mercator Projection (UM):
¨
q
1 = 0 , ¨
q
2 −
(1−e
2 ) sin f
−1 (q
2 /A)
cos 3 f −1 (q 2 /A)
∂f
−1 (q
2 /A)
∂q 2
= 0 ,
(E.101)
˙
q
µ = δ
µν p ν ,
(E.102)
˙
p 1 = 0 (p 1 cyclic) , p 1 = const. ,
(E.103)
˙
p 2 =
(1−e
2 ) sin f
−1 (q
2 /A)
cos 3 f −1 (q 2 /A)
∂f
−1 (q
2 /A)
∂q 2
.
(E.104)
(iii) Universal Transverse Mercator Projection (UTM):
¨
q
1 −
1
2ρ 2 [d 12 (q
2 )
2 + 2d 22 q
1 (q
2 )
2 + 3d 32 (q
1 )
2 (q
2 )
2 + d 14 (q
2 )
4 + O λ 2 ((q
1 )
5 , (q
2 )
6 )] = 0 ,
¨
q
2 −
1
2ρ 2 [2d 02 q
2 + 2d 12 q
1 q
2 + 2d 22 (q
1 )
2 q
2 + 2d 32 (q
1 )
3 q
2 + 4d 04 (q
2 )
3 +
+4d 14 q
1 (q
2 )
3 + O λ 2 ((q
1 )
6 , (q
2 )
5 )] = 0 ,
(E.105)
˙
q
µ = δ
µν p ν ,
(E.106)
˙
p 1 =
1
2ρ 2 [d 12 (q
2 )
2 + 2d 22 q
1 (q
2 )
2 + 3d 32 (q
1 )
2 (q
2 )
2 + d 14 (q
2 )
4 + O λ 2 ((q
1 )
5 , (q
2 )
6 )] ,
˙
p 2 =
1
2ρ 2 [2d 02 q
2 + 2d 12 q
1 q
2 + 2d 22 (q
1 )
2 q
2 + 2d 32 (q
1 )
3 q
2 + 4d 04 (q
2 )
3 +
+4d 14 q
1 (q
2 )
3 + O λ 2 ((q
1 )
6 , (q
2 )
5 )] .
(E.107)
Box E.6 (The differential equations of a geodesic in E
2
A,B in terms of conformal coordinates (isometric
coordinates) and Maupertuis gauge: Lagrange portrait, two differential equations of second order, Hamilton
portrait, four differential equations of first order).
(i) Universal Polar Stereographic Projection (UPS):
¨
q
µ +
A
2
(q 1 )
2 +(q 2 )
2
(1−e
2 ) cos f
−1
„ q
(q 1 )
2 +(q 2 )
2
«
sin f
−1
„ q
(q 1 )
2 +(q 2 )
2
«
„
1−e 2 sin 2 f −1
„ q
(q 1 )
2 +(q 2 )
2
«« 2
×
×
∂f
−1
„ q
(q 1 )
2 +(q 2 )
2
«
∂q µ
−
A
2
h
(q 1 )
2 +(q 2 )
2
i 2
cos
2 f
−1
„ q
(q 1 )
2 +(q 2 )
2
«
1−e 2 sin 2 f −1
„ q
(q 1 )
2 +(q 2 )
2
« q
µ = 0
∀ µ = 1, 2 ,
(E.98)
˙
q
µ = δ
µν p ν ∀ µ, ν = 1, 2 ,
(E.99)
˙
p µ = −
A
2
(q 1 ) 2 +(q 2 ) 2
(1−e
2 ) cos f
−1
“ √
(q 1 ) 2 +(q 2 ) 2
”
sin f
−1
“ √
(q 1 ) 2 +(q 2 ) 2
”
„
1−e 2 sin 2 f −1
„ q
(q 1 )
2 +(q 2 )
2
«« 2
×
×
∂f
−1
“ √
(q 1 ) 2 +(q 2 ) 2
”
∂q µ
+
A
2
[(q 1 ) 2 +(q 2 ) 2 ]
2
cos
2 f
−1
“ √
(q 1 ) 2 +(q 2 ) 2
”
1−e 2 sin 2 f −1
„ q
(q 1 )
2 +(q 2 )
2
« q
µ
∀ µ = 1, 2 .
(E.100)
(ii) Universal Mercator Projection (UM):
¨
q
1 = 0 , ¨
q
2 −
(1−e
2 ) sin f
−1 (q
2 /A)
cos 3 f −1 (q 2 /A)
∂f
−1 (q
2 /A)
∂q 2
= 0 ,
(E.101)
˙
q
µ = δ
µν p ν ,
(E.102)
˙
p 1 = 0 (p 1 cyclic) , p 1 = const. ,
(E.103)
˙
p 2 =
(1−e
2 ) sin f
−1 (q
2 /A)
cos 3 f −1 (q 2 /A)
∂f
−1 (q
2 /A)
∂q 2
.
(E.104)
(iii) Universal Transverse Mercator Projection (UTM):
¨
q
1 −
1
2ρ 2 [d 12 (q
2 )
2 + 2d 22 q
1 (q
2 )
2 + 3d 32 (q
1 )
2 (q
2 )
2 + d 14 (q
2 )
4 + O λ 2 ((q
1 )
5 , (q
2 )
6 )] = 0 ,
¨
q
2 −
1
2ρ 2 [2d 02 q
2 + 2d 12 q
1 q
2 + 2d 22 (q
1 )
2 q
2 + 2d 32 (q
1 )
3 q
2 + 4d 04 (q
2 )
3 +
+4d 14 q
1 (q
2 )
3 + O λ 2 ((q
1 )
6 , (q
2 )
5 )] = 0 ,
(E.105)
˙
q
µ = δ
µν p ν ,
(E.106)
˙
p 1 =
1
2ρ 2 [d 12 (q
2 )
2 + 2d 22 q
1 (q
2 )
2 + 3d 32 (q
1 )
2 (q
2 )
2 + d 14 (q
2 )
4 + O λ 2 ((q
1 )
5 , (q
2 )
6 )] ,
˙
p 2 =
1
2ρ 2 [2d 02 q
2 + 2d 12 q
1 q
2 + 2d 22 (q
1 )
2 q
2 + 2d 32 (q
1 )
3 q
2 + 4d 04 (q
2 )
3 +
+4d 14 q
1 (q
2 )
3 + O λ 2 ((q
1 )
6 , (q
2 )
5 )] .
(E.107)