Information Technology Reference
In-Depth Information
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
 
Search WWH ::




Custom Search