Chemistry Reference
In-Depth Information
1.
The Integrable Case
For
γ
−
=
0, the Hamiltonian
H
reads
p
θ
cot
2
ϕ
cos
2
ϕ
sin
2
ϕ
)
p
r
+
p
ϕ
H
=−
r
(
γ
+
(
γ
+
−
) sin
ϕ
cos
ϕp
ϕ
+
+
+
Some properties of
H
are invariant when
γ
and
vary.
+
Proposition 8
The Hamiltonian
H
is integrable.
Proof
Using the change of coordinates
ρ
=
ln
r
,
H
can be written
p
θ
cot
2
ϕ
cos
2
ϕ
sin
2
ϕ
)
p
ρ
+
p
ϕ
(48)
where
p
ρ
is the momentum conjugate to the coordinate
ρ
. The Hamiltonian flow
defined by
H
is integrable since
p
θ
and
p
ρ
are constants of the motion.
H
=−
(
γ
+
(
γ
+
−
) sin
ϕ
cos
ϕp
ϕ
+
+
+
The case
|
−
γ
+
|
<
2
Now, we give the main result of this section that is established for
|
γ
+
−
|
<
2.
Proposition 9
Fo r
p
ρ
and
p
θ
>
0
fixed, there exist two trajectories starting
from
(
r
(0)
,ϕ
(0)
,θ
(0))
, which intersect with the same cost at a point such that
ϕ
=
π
−
ϕ
(0)
.
Note that the symmetry of the flow of
H
with respect to the meridian is pre-
served. More precisely, if
p
θ
→−
p
θ
, then we have two extremals with the same
length symmetric with respect to the meridian, so we can assume
p
θ
>
0 in propo-
sition 9. Some lemmas are required before the proof of proposition 9.
Lemma 1
The derivative
dϕ/dθ
along an extremal for
H
=
h
is given by
√
p
θ
cot
2
ϕ
dϕ
dθ
=±
where
is equal to
cos
2
ϕ
sin
2
ϕ
)]
2
=
4[
h
+
p
ρ
(
γ
+
+
)
2
sin
2
(2
ϕ
)
4
]
p
θ
cot
2
ϕ
−
4[1
−
(
γ
+
−
