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these constants of motion are linked to obvious symmetries of the web: for example,
translational symmetries lead to conservation of momentum and rotational symme-
tries result in angular-momentum conservation. In other cases the constants of motion
are less evident and to find them or to prove their existence is a non-trivial problem.
There does exist a simple and general way to determine whether a phase-space function
F
(
,
)
p
q
is a constant of motion. In fact, taking its time derivative yields
dF
dt
=
∂
q
j
F
·
p
j
+
∂
F
(4.88)
∂
p
j
∂
q
j
j
and using Hamilton's equations gives
dF
dt
=
−
∂
p
j
∂
F
q
l
+
∂
H
q
j
∂
F
H
≡ {
H
,
F
}
,
(4.89)
∂
∂
∂
∂
p
l
j
where
{
A
,
B
}
are Poisson brackets of the indicated functions. Therefore, a function
F
(
p
,
q
)
is a constant of motion if its Poisson bracket with the Hamiltonian
H
(
p
,
q
)
is
identically zero:
dF
dt
= {
H
,
F
} ≡
0
.
(4.90)
A system with
N
degrees of freedom is said to be integrable if there exist
N
constants
of motion,
F
j
(
p
,
q
)
=
c
j
(
j
=
1
,
2
,...,
N
),
(4.91)
where the
c
j
are all constant. Each constant of motion reduces the web's number of
degrees of freedom by one, so, if the web is integrable, the phase-space trajectory is
confined to the surface of an
N
-dimensional manifold in a 2
N
-dimensional phase space:
this particular surface can be proven to have the topology of a torus. Figure
4.12
shows
a simple two-dimensional torus that may restrict the motion of, for example, a web
with two degrees of freedom and two constants of the motion. Two such constants are
the total energy and the total angular momentum which are conserved in a rotationally
symmetric potential. However, typical tori can be bent and twisted in very complicated
ways in the phase space without violating the conservation rules.
A great deal of the time spent in attempting to understand a scientific problem is ded-
icated to determining its most appropriate representation: choosing the variables with
which to model the web. Part of the consideration has to do with the symmetries and
constraints on the network, both of which restrict the possible choices of representation.
In this section we consider the best variables that can be used to describe dynamical
webs that after some period of time return to a previously occupied state. Let us empha-
size that such periodic motion does not imply that the web motion is harmonic. Think
of a roller coaster: the car in which you sit periodically returns to its starting point, but
there can be some exciting dynamics along the way. It is a general result of analytic
dynamics that, if the web returns to a state once, then it returns to that state infinitely
often. This principle is used to construct a very general representation of the dynamics,
namely the action-angle variables.
