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Figure 4.21.
A schematic illustration of the intersection between the domain of
(
E
)
and the plane
ζ
=
ζ
∗
(the Poincaré plane, which is the
domain) [
7
]. Reproduced with permission.
invariant measure on the manifold
and almost any distribution function relaxes to the
flat distribution. Thus, the mean value of the booster doorway variable
ξ
is given by
ξ
d
ξ
d
v
ξ
=
(4.117)
d
ξ
d
v
and the mean-square value is
2
d
ξ
ξ
d
v
2
ξ
=
,
(4.118)
d
ξ
d
v
both of which are relatively easy to obtain numerically.
Response to a weak external perturbation
Let us now investigate in some detail the effect of the perturbation on the booster.
Consider the unperturbed Hamiltonian
H
0
given by
1
2
m
1
π
1
2
m
2
v
2
2
H
0
=
+
+
U
0
(ξ,ζ),
(4.119)
where the unperturbed potential is defined by
2
2
2
+
ξζ
3
3
+
ξ
)
=
ξ
2
+
ζ
−
ζξ
2
4
4
U
0
(ξ,ζ)
≡
(ξ,ζ,
=
ζ
,
U
0
(4.120)
so the web with
=
0 has the zero subscripts. Assuming that the perturbation
H
1
=
εξ
is switched on at
t
=
0 and using the unit step function
θ(
t
)
=
0for
t
<
0 and
θ(
t
)
=
1
for
t
≥
0
,
the total Hamiltonian can be written as