Chemistry Reference
In-Depth Information
Proof
The proof is standard. Assuming optimality beyond the intersecting point,
we can construct a broken minimizer that is an extremal with nonzero
p
θ
.
The case
|
−
γ
+
|≥
2
Extremal curves have a more complex behavior for
|
−
γ
+
|≥
2. We proceed
as before by fixing
p
θ
,
p
ρ
, and
H
h
. The projections of the extremals on the
sphere in the coordinates (
ϕ, θ
) are either aperiodic or periodic according to the
values of
p
θ
,
p
ρ
, and
h
. Some general characteristics of the extremals can be
determined and are described by the following results.
Aperiodic extremals are extremals such that
ϕ
is not periodic. They have an
asymptotic fixed point (
r
f
,ϕ
f
,θ
f
) and
p
ϕ
→±∞
=
, when
t
→+∞
. The possible
fixed points are described by the following proposition:
Proposition 10
The projection of the asymptotic fixed point of the extremal on the
sphere in the coordinates
(
θ, ϕ
)
is located on one of the parallels
ϕ
f
=
α
,
ϕ
f
=
π/
2
−
α
,
ϕ
f
=
π/
2
+
α
, and
ϕ
f
=
π
−
α
, where
α
=
arcsin[2
/
|
−
γ
+
|
]
/
2
.If
>γ
(respectively,
<γ
), then only the fixed points such that
ϕ
f
=
α
or
ϕ
f
=
+
+
π
−
α
(respectively,
ϕ
f
=
π/
2
−
α
or
ϕ
f
=
π/
2
+
α
) can be reached depending
on the initial value
ϕ
(0)
.
Proof
Proposition 10 can be shown by solving the following equation deduced
from the system (42):
)
2
sin
2
(2
ϕ
)
4
1
−
(
γ
+
−
=
0
)
2
sin
2
(2
ϕ
)
/
4as
a function of
ϕ
and we deduce that the derivative
dϕ/dθ
has no zero in
]
α, π/
2
Then, we determine the sign of the expression 1
−
(
γ
+
−
>
2, a direct inspection of
dϕ/dθ
also
shows that
dϕ/dθ <
0 (respectively,
dϕ/dθ >
0) if
ϕ
−
α
[
∪
]
π/
2
+
α, π
−
α
[. For
−
γ
+
∈
]
α, π/
2
−
α
[ (respectively,
ϕ
α
[), which indicates the parallel that can be reached. The same
analysis can be undertaken for
∈
]
π/
2
+
α, π
−
−
γ
<
2, which completes the proof.
+
Note that the aperiodic extremals have no conjugate point and are always optimal.
Periodic extremals occur in a band near the equator. They have the same properties
as the extremals for
<
2. In particular, two extremals of the same length
intersect on the antipodal parallel.
The characteristics of the extremals are summarized in Fig. 26. Two periodic
trajectories intersecting with the same length on the antipodal parallel are dis-
played. Other extremals are aperiodic with fixed points located on the parallel
ϕ
f
|
−
γ
+
|
=
arcsin[2
/
|
−
γ
+
|
]
/
2or
ϕ
f
=
π
−
arcsin[2
/
|
−
γ
+
|
]
/
2.
