E-3 The Newton form of a geodesic in Maupertuis gauge (sphere, ellipsoid-of-revolution) 567
Box E.8 (The coefficients of the Lie series up to the third order, all quantities are to be taken af Taylor point
t = 0).
f 10 = λ =
1
ρ
`
1 +
d 02
2
(q
2 )
2 +
d 12
2
q
1 (q
2 )
2
´
,
f 20 =
1
4ρ 2
`
d 12 (q
2 )
2 + 2d 22 q
1 (q
2 )
2
´
,
f 30 =
1
12ρ 3
`
2d 22 (q
2 )
2 + 6d 32 q
1 (q
2 )
2
´
,
f 03 =
1
12ρ 3
`
2d 12 q
2 + (d 12 d 02 + 4d 14 )(q
2 )
3 + 4d 22 q
1 q
2 + 6d 32 (q
1 )
2 q
2
´
,
f 02 = f 20 , f 21 = f 03 , f 12 = f 30 ,
(E.125)
g 01 = λ =
1
ρ
`
1 +
d 02
2
(q
2 )
2 +
d 12
2
q
1 (q
2 )
2
´
,
g 20 =
1
4ρ 2
`
2d 02 q
2 + 2d 12 q
1 q
2 + 2d 22 (q
1 )
2 q
2 + 4d 04 (q
1 )
2
´
,
g 30 =
1
12ρ 3
`
2d 12 q
2 + (d 12 d 02 + 4d 14 )(q
2 )
3 + 4d 22 q
1 q
2 + 6d 32 (q
1 )
2 q
2
´
,
g 03 =
1
12ρ 3
`
2d 02 + 2d 12 q
1 + 2d 22 (q
1 )
2 + (d
2
02 + 12d 04 )(q
2 )
2 + 2d 32 (q
1 )
3 +
+(2d 12 d 02 + 12d 14 )q
1 (q
2 )
3
´
,
g 02 = g 20 , g 21 = g 03 , g 12 = g 30 ,
(E.126)
r 10 =
1
ρ
`
d 02 q
2 + d 12 q
1 q
2 + d 22 (q
1 )
2 q
2 +
`
2d 04 −
1
2
d
2
02
´
(q
2 )
3
´
,
r 01 = −
1
ρ
` 1
2
d 12 (q
2 )
2 + d 22 q
1 (q
2 )
2
´
,
r 20 =
1
2ρ 2
`
d 12 q
2 + 2d 22 q
1 q
2 + 3d 32 (q
1 )
2 q
2 +
`
2d 14 −
1
2
d 02 d 12
´
(q
2 )
3
´
,
r 11 =
1
2ρ 2
`
d 02 + d 12 q
1 + d 22 (q
1 )
2 + (6d 04 − d 22 )(q
2 )
2 + d 32 (q
1 )
3 +
+ (2d 02 d 12 + 6d 14 − 2d 32 ) q
1 (q
2 )
2
´
,
r 02 = r 20 ,
(E.127)
h 10 =
1
f 10
, h 20 = −
f 20
(f 10 ) 3 ,
h 02 = −
f 02
f 10 (g 01 ) 2 , h 30 =
2(f 20 )
2
(f 10 ) 5 −
f 30
(f 10 ) 4 ,
h 03 =
2f 02 g 02
f 10 (g 01 ) 4 −
f 03
f 10 (g 01 ) 3 , h 21 =
2f 02 g 02
(f 10 ) 3 (g 01 ) 2 −
f 21
(f 10 ) 3 g 01
,
h 12 =
2f 20 g 02
(f 10 ) 3 (g 01 ) 2 −
f 12
(f 10 ) 2 (g 01 ) 2 ,
(E.128)
k 01 =
1
g 01
, k 02 = −
g 02
(g 01 ) 3 ,
k 20 = −
g 20
g 01 (f 10 ) 2 , k 03 =
2(g 20 )
2
(g 01 ) 5 −
g 03
(g 01 ) 4 ,
k 30 =
2f 20 g 20
(f 10 ) 4 g 01
−
g 30
g 01 (f 10 ) 3 , k 21 =
2g 02 g 20
(f 10 ) 2 (g 01 ) 3 −
g 21
(f 10 ) 2 (g 01 ) 2 ,
k 12 =
2f 02 g 20
(f 10 ) 2 (g 01 ) 3 −
g 12
f 10 (g 01 ) 3 .
(E.129)
Last but not least, let us here additionally note that Appendix E-3 is based upon the contribution of
E. Grafarend and R. J. You (1995).
Box E.8 (The coefficients of the Lie series up to the third order, all quantities are to be taken af Taylor point
t = 0).
f 10 = λ =
1
ρ
`
1 +
d 02
2
(q
2 )
2 +
d 12
2
q
1 (q
2 )
2
´
,
f 20 =
1
4ρ 2
`
d 12 (q
2 )
2 + 2d 22 q
1 (q
2 )
2
´
,
f 30 =
1
12ρ 3
`
2d 22 (q
2 )
2 + 6d 32 q
1 (q
2 )
2
´
,
f 03 =
1
12ρ 3
`
2d 12 q
2 + (d 12 d 02 + 4d 14 )(q
2 )
3 + 4d 22 q
1 q
2 + 6d 32 (q
1 )
2 q
2
´
,
f 02 = f 20 , f 21 = f 03 , f 12 = f 30 ,
(E.125)
g 01 = λ =
1
ρ
`
1 +
d 02
2
(q
2 )
2 +
d 12
2
q
1 (q
2 )
2
´
,
g 20 =
1
4ρ 2
`
2d 02 q
2 + 2d 12 q
1 q
2 + 2d 22 (q
1 )
2 q
2 + 4d 04 (q
1 )
2
´
,
g 30 =
1
12ρ 3
`
2d 12 q
2 + (d 12 d 02 + 4d 14 )(q
2 )
3 + 4d 22 q
1 q
2 + 6d 32 (q
1 )
2 q
2
´
,
g 03 =
1
12ρ 3
`
2d 02 + 2d 12 q
1 + 2d 22 (q
1 )
2 + (d
2
02 + 12d 04 )(q
2 )
2 + 2d 32 (q
1 )
3 +
+(2d 12 d 02 + 12d 14 )q
1 (q
2 )
3
´
,
g 02 = g 20 , g 21 = g 03 , g 12 = g 30 ,
(E.126)
r 10 =
1
ρ
`
d 02 q
2 + d 12 q
1 q
2 + d 22 (q
1 )
2 q
2 +
`
2d 04 −
1
2
d
2
02
´
(q
2 )
3
´
,
r 01 = −
1
ρ
` 1
2
d 12 (q
2 )
2 + d 22 q
1 (q
2 )
2
´
,
r 20 =
1
2ρ 2
`
d 12 q
2 + 2d 22 q
1 q
2 + 3d 32 (q
1 )
2 q
2 +
`
2d 14 −
1
2
d 02 d 12
´
(q
2 )
3
´
,
r 11 =
1
2ρ 2
`
d 02 + d 12 q
1 + d 22 (q
1 )
2 + (6d 04 − d 22 )(q
2 )
2 + d 32 (q
1 )
3 +
+ (2d 02 d 12 + 6d 14 − 2d 32 ) q
1 (q
2 )
2
´
,
r 02 = r 20 ,
(E.127)
h 10 =
1
f 10
, h 20 = −
f 20
(f 10 ) 3 ,
h 02 = −
f 02
f 10 (g 01 ) 2 , h 30 =
2(f 20 )
2
(f 10 ) 5 −
f 30
(f 10 ) 4 ,
h 03 =
2f 02 g 02
f 10 (g 01 ) 4 −
f 03
f 10 (g 01 ) 3 , h 21 =
2f 02 g 02
(f 10 ) 3 (g 01 ) 2 −
f 21
(f 10 ) 3 g 01
,
h 12 =
2f 20 g 02
(f 10 ) 3 (g 01 ) 2 −
f 12
(f 10 ) 2 (g 01 ) 2 ,
(E.128)
k 01 =
1
g 01
, k 02 = −
g 02
(g 01 ) 3 ,
k 20 = −
g 20
g 01 (f 10 ) 2 , k 03 =
2(g 20 )
2
(g 01 ) 5 −
g 03
(g 01 ) 4 ,
k 30 =
2f 20 g 20
(f 10 ) 4 g 01
−
g 30
g 01 (f 10 ) 3 , k 21 =
2g 02 g 20
(f 10 ) 2 (g 01 ) 3 −
g 21
(f 10 ) 2 (g 01 ) 2 ,
k 12 =
2f 02 g 20
(f 10 ) 2 (g 01 ) 3 −
g 12
f 10 (g 01 ) 3 .
(E.129)
Last but not least, let us here additionally note that Appendix E-3 is based upon the contribution of
E. Grafarend and R. J. You (1995).