Chemistry Reference
In-Depth Information
and the metric is the Grushin metric on the two sphere
dϕ
2
tan
2
ϕdθ
2
g
=
+
The drift can be compensated by a feedback if
<
2. The problem is
symmetric with respect to the equator and we can restrict our analysis to the upper
hemisphere. We observe that the amplitude of the current is maximum for
ϕ
|
γ
+
−
|
π/
4
,
while it is minimum at the North pole and at the equator. More generally, we have
the following proposition.
=
Proposition 11
For the system restricted to the sphere we have two cases:
1. If
<
2
, it defines a Zermelo navigation problem on the whole sphere
minus the equator for the Grushin metric on the sphere.
2. If
|
γ
+
−
|
>
2
, the current can be compensated in the north equator except
in a band centered at
ϕ
|
γ
+
−
|
π/
4
,
hence defining a Zermelo navigation problem
near the equator and near the North pole.
=
To complete the analysis, it is sufficient to describe the following
barrier
phe-
nomenon. Let
ψ
=
π/
2
−
ϕ
and assume that
v
1
=
0. Starting at the equator for
which
ψ
0, when
ψ
increases, we meet a barrier corresponding to the singu-
larity of the vector field. For example, if
γ
=
+
−
>
0, then we have a barrier for
1
)
/
2].
This result explains the two types of extremal behaviors observed in the numer-
ical simulations.
=
[
−
sin(2
ϕ
)(
γ
+
−
•
If
<
2, the extremal curves restricted to the two sphere are periodic
curves, as in the Grushin case.
|
γ
+
−
|
•
If
>
2, we have two types of extremal curves: Near the equator,
since the current can be compensated, we have periodic curves. But if the
trajectory is entering in the band where the current cannot be compensated,
the barrier phenomenon appears and we observe the asymptotic behaviors of
the extremals.
|
γ
+
−
|
2.
The Generic Case
γ
=
/
0
−
In this section, we use mainly numerical simulations to describe extremal curves in
the case
γ
0
.
We concentrate on the description of the generic cases observed
in the numerical simulations. We present numerical results about the behavior of
extremal solutions and conjugate point analysis.
=
/
−
Extremal Trajectories
We begin by analyzing the structure of extremal trajectories. The description is
based on a direct integration of the system. We observe two different asymptotic
