Chemistry Reference
In-Depth Information
behaviors corresponding to stationary points of the dynamics that are described
by the following results.
Proposition 12
In the case denoted (a), where
,
the asympto
tic sta
tionary points
(
ρ
f
,ϕ
f
,θ
f
)
of the dynamics are given by
ρ
f
|
p
ϕ
(
t
)
|→+∞
when
t
→+∞
√
1
2
/
(1
=|
γ
−
|
+
+
γ
)
and
ϕ
f
=
arctan(1
/
)
if
γ
>
0
or
ϕ
f
=
π
−
+
−
arctan(1
/
)
if
γ
<
0
.
−
Proof
We assume that
and that cot(
ϕ
) remains finite
in this limit. One deduces from the Hamiltonian system that (
ρ
f
,ϕ
f
) satisfy the
following equations:
|
p
ϕ
(
t
)
|→+∞
as
t
→+∞
cos
2
ϕ
f
+
sin
2
ϕ
f
)
γ
cos
ϕ
f
=
ρ
f
(
γ
−
+
γ
sin
ϕ
f
ρ
f
−
=
(
γ
+
−
) cos
ϕ
f
sin
ϕ
f
+
ε
where
ε
=±
1 according to the sign of
p
ϕ
. The quotient of the two equations
leads to
cos
2
ϕ
f
+
sin
2
ϕ
f
)
(
γ
+
−
) cos
ϕ
f
sin
ϕ
f
+
ε
=
tan
ϕ
f
(
γ
+
which simplifies into
ε
tan
ϕ
f
=
Using the fact that
ϕ
f
∈
]0
,π
[ and
γ
cos
ϕ
f
≥
0, one arrives to
ϕ
f
=
arctan(1
/
)
−
if
γ
>
0 and
ϕ
f
=
π
−
arctan(1
/
)if
γ
<
0. From the equation
−
−
cos
2
ϕ
f
+
sin
2
ϕ
f
)
γ
cos
ϕ
f
=
ρ
f
(
γ
−
+
one finally obtains that
√
1
2
γ
+
−
ρ
f
=
1
+
γ
+
Proposition 13
In the case denoted (b), where
lim
t
→+∞
ϕ
(
t
)
=
0
or
π
, the asymp-
totic limit of the dynamics is characterized by
ρ
f
=|
γ
−
|
/γ
and
ϕ
f
=
0
if
γ
>
0
+
−
or
ϕ
f
=
π
if
γ
<
0
.
−
Proof
Using the relation
cos
2
ϕ
f
+
sin
2
ϕ
f
)
γ
cos
ϕ
f
=
ρ
f
(
γ
−
+
one deduces that
γ
cos
ϕ
f
≥
0 and that
ρ
f
=|
γ
−
|
/γ
if
ϕ
f
=
0or
π
.
−
+
