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The reason why action-angle variables are such a good representation of the dynam-
ics of conservative Hamiltonian webs has to do with how complexity has traditionally
been introduced into the dynamics of mechanical webs. Historically a complex physi-
cal network is divided into a simple part plus a perturbation. The simple part undergoes
oscillatory motion and the deviation from this periodic behavior can be described by
perturbation theory. If this is a correct picture of what occurs then action-angle vari-
ables are the best description, as shown below. A Hamiltonian can be written in terms
of action-angle variables as
H
0
(
J
)
and the equations of motion take the simple form
dJ
j
dt
=−
∂
H
0
∂θ
j
=
0
,
(4.92)
d
dt
=
∂
θ
j
H
0
J
j
=
ω
j
(
J
)
=
constant
.
(4.93)
∂
We note in passing that the original set of canonically conjugate variables (
p
,
q
)is
replaced by the new set of canonically conjugate variables (
J
, so that Hamilton's
equations are valid in both representations. Equation (
4.93
) results from the fact that
J
is a constant as determined by (
4.92
), so the angle increases linearly with time. The
solutions to Hamilton's equations in the action-angle representation are given by
,θ)
J
j
=
constant
,
θ
j
(
t
)
=
ω
j
(
J
)
t
+
θ
j
(
0
)(
j
=
1
,
2
,...,
N
).
(4.94)
In the simple torus example shown in Figure
4.12
it is easy to realize that, if the two
angular frequencies
ω
2
are commensurable, that is, their ratio is rational, then the
trajectory eventually returns to its starting point and the motion is periodic. On the other
hand, if the ratio of the two angular frequencies is irrational the trajectory never returns
to its starting point and the motion is aperiodic. Complex webs can be separated into
a piece described by action-angle variables, which is the simple part, and the deviation
from this simple periodic behavior, which can be described using perturbation theory.
The unperturbed Hamiltonian in the action-angle representation is given by
ω
1
and
N
H
0
(
J
)
=
J
j
ω
j
(4.95)
j
=
1
for an integrable web with
N
degrees of freedom.
If the web is integrable then the phase space is filled with invariant tori and a given
trajectory remains on the particular torus selected by the initial conditions and its
motion can be periodic or aperiodic. Complex webs deviate from such behavior and are
described in analytic dynamics by perturbation theory. Perturbation theory arises from
the notion that nonlinear effects can be treated as small deviations away from exactly
solvable problems. For example, in the real world one can solve the two-body Kepler
problem; that is, the universe consists of two bodies interacting under mutual gravita-
tion only. This exactly solvable situation cannot be extended to a universe with three
interacting bodies. The evolution of three bodies interacting through their mutual grav-
itational fields still eludes us. Thus, systematic approximation techniques for solving
