Chemistry Reference
In-Depth Information
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Figure 29.
Projection of the conjugate locus is represented by a solid line for
p
r
=
0
.
5. The
conjugate locus of the Grushin model corresponding to
γ
+
=
=
2 is represented in dashed lines.
The horizontal dashed line indicates the position of the cut locus for the Grushin model. Dissipative
parameters are taken to be
=
2
.
5, and
γ
+
=
2.
p
θ
is equal to 2.
for comparison. Note that this locus is only slightly modified with respect to the
one of the Grushin model. However, the projection on the sphere of the conjugate
locus depends on
p
r
(0) for
/
=
γ
.
+
Geometric Interpretation of the Integrable Case
A geometric analysis allows us a complete understanding of the two types of
extremal behaviors. Indeed assuming
γ
−
=
0
,
the restriction of the system to the
two-sphere is
dϕ
dt
sin(2
ϕ
)(
γ
+
−
)
=
+
v
2
2
dθ
dt
=−
(cot
ϕ
)
v
1
,
|
v
|≤
1
We observe that it defines a Zermelo navigation problem [57] on the sphere where
the current is
sin(2
ϕ
)(
γ
+
−
)
∂
∂ϕ
F
0
=
2
