20-2 Lagrange portrait, Hamilton portrait, Lie series, Clairaut constant 427
Box 20.3 (The Legendre recurrence of U
(m)A in terms of U
A , index set A 1 , A 2 , . . . , A m−1 , A m ∈ {1, 2}
(Legendre 1806)).
U
A = −{
A
A 1 A 2 }U
A 1 U
A 2
`
geodesic, (20.42)
´
,
(20.55)
U
A =
= −{
A
A 1 A 2 } A 3 U
A 3 U
A 2 U
A 1 + 2{
A
A 1 A 2 }{
A 1
A 3 A 4
}U
A 2 U
A 3 U
A 4 ,
U
(4)A =
= −{
A
A 1 A 2 }, A 3 A 4 U
A 4 U
A 3 U
A 2 U
A 1 + 3{
A
A 1 A 2 }, A 3 {
A 3
A 4 A 5
}U
A 1 U
A 2 U
A 4 U
4 5
+{
A
A 1 A 2 }, A 3 {
A 1
A 4 A 5
}A
A 2 U
A 3 U
A 4 U
A 5 + 6{
A
A 1 A 2 }{
A 1
A 3 A 4
}{
A 3
A 5 A 6
}U
A 2 U
A 4 U
A 5 U
A 6 ,
etc.
(20.56)
If we replace U
(m)A in the Taylor expansion (20.54) by means of the Legendre recurrence of Box 20.3,
we have solved the initial value problem of the geodesic (20.42) in terms of power series U
A 1
0 , U
A 1
0 U
A 2
0 ,
U
A 1
0 U
A 2
0 U
A 3
0
etc. generating an exponential map, in particular
U
A (S) =
= U
A
0 + SU
A
0 + S
2 A
A
A 1 ,A 2
U
A 1
0 U
A 2
0
+ · · · + lim
n→∞
n
m=3
S
m A
A
A 1 A 2 ...A m
U
A 1
0 U
A 2
0
. . . U
A m
0
.
(20.57)
With respect to the first chart, part of the minimal atlas of E
2
A 1 ,A 2
, namely the orthogonal coordinates
{L, B}, we transform U
A
0 via
L
0 = U
1
0 =
x
S
√
G 11
, B
0 = U
2
0 =
y
S
√
G 22
(20.58)
into (Riemann) normal coordinates {x, y} (inverse Riemann mapping, inverse Riemann cha-cha-cha).
Accordingly, we succeed to represent the Taylor series (20.54) in terms of (Riemann) normal coordinates {x, y}, in particular, x
1 := x/
√
G 11 and x
2 := y/
√
G 22
L(S B ) − L(S A = 0) = lim
n→∞
n
m=1
A
L
A 1 ...A m x
A 1 . . . x
A m ,
B(S B ) − B(S A = 0) = lim
n→∞
n
m=1
A
B
A 1 ...A m x
A 1 . . . x
A m .
(20.59)
By standard series inversion of the homogeneous two-dimensional polynomial (20.59), we have
solved the boundary value problem for given values {L A , B A } := {L(S A = 0), B(S A = 0)} and
{L B , B B } := {L(S B ), B(S B )} coordinating the points P A = P (L A , B A ) and P B = P (L B , B B ), respectively, particularly in the form of the polynomial for
x
A = lim
n→∞
n
m=1
A
A
A 1 ...A m
[U
A 1 (S B ) − U
A 1 (S A )] . . . [U
A m (S B ) − U
A m (S A )] .
(20.60)