Chemistry Reference
In-Depth Information
A solution of this system exists except on the set
C
, where
F
and
G
are collinear.
If we write
α
as
α
=
α
y
dy
+
α
z
dz
, then simple algebra shows that
α
y
and
α
z
are
solutions of the system
α
y
(
−
y
)
+
α
z
(
γ
−
−
γ
z
)
=
1
+
α
y
z
=
α
z
y
We obtain
α
y
=
−
y
y
2
z
2
−
γ
z
+
γ
−
+
−
z
α
z
=
y
2
−
γ
−
z
+
γ
+
z
2
From the 1-form
α
, we can define the 2-form
dα
that is given by
∂α
z
∂y
dy
∂α
y
∂z
dα
=
−
∧
dz
and reads after some calculations
2
yz
+
γ
y
−
2
γ
yz
−
+
dα
=
z
2
]
2
dy
∧
dz
[
y
2
−
γ
z
+
γ
−
+
If we write
dα
as
dα
0on
S
and that
the function
g
has a constant sign in the regions delimited by the lines of
S
. This
point is displayed in Fig. 18 for a given set of dissipative parameters.
As suggested by its name, the clock form is a form that allows one to determine
the time taken to travel a path and to compare the extremals. Let
γ
be a path in the
=
g
(
y, z
)
dy
∧
dz
, then one sees that
g
(
y, z
)
=
z
g
<0
g
>0
y
Figure 18.
Sign of the
function
g
in the plane (
y, z
)
for
=
3,
γ
+
=
0
.
4, and
γ
−
=−
0
.
2. The large solid
lines indicate the position of
the set
S
.
g
>0
g
<0
