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n
A
A
1
...A
m
x
A
1
... x
A
m
,
L
(
S
B
)
− L
(
S
A
= 0) = lim
n→∞
m
=1
(20.59)
n
A
A
1
...A
m
x
A
1
... x
A
m
.
B
(
S
B
)
−
B
(
S
A
= 0) = lim
n→∞
m
=1
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
{
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
L
B
,B
B
}
:=
{
L
(
S
B
)
,B
(
S
B
)
}
n
x
A
= lim
n→∞
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)
m
=1
The
Lagrange portrait
of a geodesic is based upon
Legen-
dre series
up to order five in terms of series
U
1
,U
2
{
}
=
S
0
,S
1
,...,S
n
{
L, B
}
, power series
{
}
in terms of
dis-
L
0
,B
0
,L
0
,B
0
}
tance functions
. The initial values
con-
stitute the
initial value problem
. In contrast, in the bound-
ary value problem, the homogeneous polynomial in terms
of
{L
A
− L
B
,B
A
− B
B
}
as power series is given, while
the Riemann Cartesian coordinates
{
x
1
,x
2
{
x, y
}
=
{
}
are
completely unknown.
20-22 Hamilton Portrait of a Geodesic: Hamilton Equations,
Initial/Boundary Values
A
1
,A
2
, is based upon the generalized
momenta given by (
20.61
), in particular, for an orthogonal set of coordinates
The
Hamilton portrait
of a
geodesic
, here a submanifold in
E
{
L, B
}
given
by (
20.62
).
P
K
:=
G
KL
U
L
=
√
G
11
cos
α
√
G
22
cos
β
,
(20.61)
P
1
=
G
11
L
=
G
11
cos
α
=
N
(
B
)cos
B
cos
α,
(20.62)
P
2
=
G
22
B
=
G
22
sin
αM
(
B
)sin
α.
The
Hamilton equations
of a
geodesic
as a system of four first order ordinary differential equations
can be written as (
20.63
)fora
Hamilton function
(
20.64
) which is produced by the
Legendre
transformation
of the
Lagrange function
2
2
:=
U
K
G
KL
U
L
.
L
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