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real-world problems are very important. In dynamics a strategy was devised for solving
non-integrable Hamiltonian webs by defining integrable Hamiltonians that differed only
slightly from non-integrable ones. The
real
problem is then said to be a perturbation of
the
idealized
problem that can be rigorously solved. The difference between the two
Hamiltonians is called the perturbation and this approach relies on the assumption that
the perturbation is in some sense small.
4.4.1
The surface of section
In real complex physical webs it is often the case that the web is characterized by a
large number of variables and the dimensionality of the phase space can exceed three.
In such situations how can the trajectory be visualized? Here we have recourse to a tech-
nique briefly discussed previously that was originally suggested by Poincaré, called the
Poincaré surface of section. Consider a web described by two spatial variables and their
canonical momenta, so that the phase space is four-dimensional with axes
x
p
x
and
p
y
. If energy is conserved the motion takes place on the surface of a three-dimensional
surface of constant energy:
,
y
,
p
x
+
p
y
E
=
H
=
+
V
(
x
,
y
).
(4.96)
2
m
One variable, say
p
y
, can be expressed in terms of the constant energy and therefore
eliminated from the dynamical description:
2
m
E
p
x
p
y
=±
−
2
m
−
V
(
x
,
y
)
.
(4.97)
Following motion on a three-dimensional surface can often be difficult in practice, so
one needs a way to accurately characterize that motion on a two-dimensional plane such
as a sheet of paper. This is where the surface of section becomes useful.
The trajectories determined by (
4.96
) are recorded on a surface of section along the
x
coordinate, denoted by
S
x
, given by the intersection of the three-dimensional manifold
on which the dynamics unfold with the
y
.
This is a “slice” through the trajectories in phase space. If we specify the coordinates of
the web on this slice
S
x
we have completely specified the state of the web. Apart from
an overall sign, since
x
=
0 plane and have the coordinates
(
x
,
p
x
)
0 are specified on
S
x
, the momentum value
p
y
is
given by (
4.97
). It is customary to take the positive sign and to define the
y
surface of
section
S
y
in a similar way.
If the potential
V
,
p
x
and
y
=
supports bounded motion in the phase space, the trajectories
will repeatedly pass through
S
x
with every traversal of the orbit. If the locations at
which the orbit pierces
S
x
are recorded with each traversal we can build up a succession
of intercept points as shown in Figure
4.15
. The coordinates
(
x
,
y
)
of one point of
interception can be related to those of the next interception by means of a map. Thus, an
initial point
X
0
=
(
(
x
,
p
x
)
x
0
,
p
x
0
)
on
S
x
will generate a sequence of points
X
1
,
X
2
,...,
X
N
.
It is this sequence that constitutes the motion on the surface of section.
