566 E Geodesics
Box E.7 (Lie recurrence up to the third derivatives, all quantities are to be taken at Taylor point t = 0,
higher derivatives are available from the authors).
˙
q
1 = λ cos α ,
˙
q
1 t = λu = λu
1 ,
˙
q
2 = λ sin α ,
˙
q
2 t = λv = λu
2 ,
˙
α = λ .2 cos α − λ .1 sin α ,
˙
αt = λ .2 u − λ .1 v ,
¨
q
1 = ˙
λ cos α − λ ˙
α sin α = λλ .1 (cos
2 α + sin
2 α) ,
¨
q
1 t
2 = λλ .1 (u
2 + v
2 ) ,
¨
q
2 = ˙
λ sin α + λ ˙
α cos α = λλ .2 (cos
2 α + sin
2 α) ,
¨
q
2 t
2 = λλ .2 (u
2 + v
2 ) ,
¨
α = ˙
λ .2 cos α − ˙
λ .1 sin α − λ .2 ˙
α sin α − λ .1 ˙
α sin α ,
¨
αt
2 = (λλ .12 − λ .1 λ .2 )u
2 − (λλ .12 − λ .1 λ .2 )v
2 + (λλ .22 − λ ˙
λ .11 + (λ .1 )
2 − (λ .2 )
2 )uv ,
···
q
1 = ˙
λλ .1 (cos
2 α + sin
2 α) + λ ˙
λ .1 (cos
2 α + sin
2 α) ,
···
q
1 t
3 = (λ(λ .1 )
2 + λ
2 λ .11 )u
3 + (λλ .1 λ .2 +
+λ
2 λ .12 )u
2 v + (λ(λ .1 )
2 + λ
2 λ .11 )uv
2 + (λλ .1 λ .2 + λ
2 λ .12 )v
3 ,
···
q
2 = ˙
λλ .2 (cos
2 α + sin
2 α) + λ ˙
λ .2 (cos
2 α + sin
2 α) ,
···
q
2 t
3 = (λ(λ .2 )
2 + λ
2 λ .22 )v
3 + (λλ .1 λ .2 + λ
2 λ .12 )uv
2 +
+(λ(λ .2 )
2 + λ
2 λ .22 )u
2 v + (λλ .1 λ .2 + λ
2 λ .12 )u
3 ,
···
α = −4(λλ .12 − λ .1 λ .2 ) ˙
α cos α sin α+
+(λλ .22 − λλ .11 + (λ .1 )
2 − (λ .2 )
2 )(cos
2 α − sin
2 α) ˙
α+
+(cos
2 α − sin
2 α)
d
dt
(λλ .12 − λ .1 λ .2 )+
+ cos α sin α
d
dt
((λλ .22 − λλ .11 + (λ .1 )
2 − (λ .2 )
2 ) ,
···
αt
3 = (λλ .2 λ .22 − 2λλ .2 λ .11 + (λ .1 )
2 λ .2 − (λ .2 )
3 + λ
2 λ .121 )u
3 +
+(5λ .1 (λ .2 )
2 − (λ .1 )
3 − 6λλ .2 λ .12 + 2λλ .1 λ .11 − λλ .1 λ .22 + 2λ
2 λ .122 − λ
2 λ .111 )u
2 v+
+(−5(λ .1 )
2 λ .2 − (λ .2 )
3 + 6λλ .1 λ .12 − 2λλ .2 λ .22 + λλ .2 λ .11 − 2λ
2 λ .112 + λ
2 λ .222 )uv
2 +
+(−λλ .1 λ .11 + 2λλ .1 λ .22 + (λ .1 )
3 − λ .1 (λ .2 )
2 − λ
2 λ .122 )v
3 .
(E.124)
Note that λ .µ , λ .µν , and λ .µνγ denote the first derivative with respect to q
µ , the second derivative with
respect to q
µ and q
ν , and the third derivative with respect to q
µ , q
ν , and q
γ .