Physics
573 [Nonlinear Mechanics]
Particle in a Uniform Magnetic Field and an Oscillating Electric Wave
Poincaré Sections
The dynamical system is that of a particle of charge 
c (with c the speed of light, and EM equations in
Gaussian units) and mass m moving under the influence of a uniform magnetic field, B0, in the
upward z-direction. Because of the magnetic field it gyrates at frequency 
. It is also affected by a passing, plane
electric wave oscillating at frequency 
.
We investigate the motion of the particle, treating the electric wave as a first-order perturbation, where it is
the magnitude of that wave which determines the size of the perturbation. In three dimensions, the system has three constants of the motion, so that
one may acquire motions that oscillate on a given 3-dimensional surface, for fixed values of those constants.
On such a surface, we choose local coordinates
s, 
, and 
, where s is proportional to the radius of the circles of gyration before the wave arrived, is the perpendicular
portion of the phase of the
oscillating wave, and is the angle in the plane where the particle is
executing its circular motion.
To study this motion further, we pick a particular plane, for a Poincaré section, by choosing a value for = 
. The motions intersect this plane regularly so that after sufficient time, those intersections trace
out a curve in the s, plane, which depends on the value of the constants of
the motion. The equation for that motion may
be taken in the form
Inserting the value = , to define the
section, and inserting a ratio of the two frequencies that is relatively far from an integer, so as to be
not too close to resonance, we may rewrite our equation in the plane in the following form,
where we have re-normalized things so as to have explicitly the argument of the Bessel function as one of
our coordinates, and two constants have been inserted:
The graphs shown below plot horizontally, through all its values,
from 0 to 2, versus s vertically. Each graph has a
fixed value of the constant f0, which amounts to a driving term proportional to the amplitude
of the oscillating electric field, while each displays a selection of
distinct values for h0, which vary from 45 to 50.
The first pair of graphs are taken directly from Lichtenberg's text, which are taken from Karney's thesis.
The one on the left is a graph of the sort already described above, while the one on the right is a
Poincaré section obtained while following a numerical calculation of the motion in time. Since the
numerical calculation is only for some finite time, we can, more or less, see the actual intersections of
the trajectory with the plane where the section is, as time goes on.
| 
|
| calculated, first-order curves in the plane.
| numerical, exact intersects with the plane, in time. |
The next set of graphs are all calculations done by myself, on Maple. They use values of
f0 that vary from 2000 to 7000, and a selection of
distinct values for [h0]1/2, which vary from 45 to 50
at steps of 0.20.
One finds that for f0
below about 1000, the graphs in the plane appear to move rather quietly around the cylinder with very little
change in s. [Remember that although it is locally a plane that the angle variable only runs between 0 to
2, so that it is globally a cylinder.]
On the other hand, by the time f0 gets to 2000 we begin to see some variation in s on any given
curve, and at 4000 the deviation is becoming significant at certain angles.
| 
|
| f0 = 2000
| f0 = 4000
|
In the graphs below we are beginning to see wide variations beginning to develop, although for f0
= 5000, each curve is still going around the entire cylinder. However, at
f0 = 5500, we have already started the very smallest sort of island formation, i.e.,
trajectories that no longer go around the entire cylinder. If they are started in those particular values of
initial conditions, they stay fairly nearby, in angle as well as in action.
| 
|
| f0 = 5000
| f0 = 5500
|
Now at values of 6000 and 7000 we can easily see serious excursions into these islands---not yet any
chains of islands---where we have larger motions that are, however, completely bounded in angle and
definitely do not go all the way around the cylinder.
Such motions are often said to be trapped.
The boundaries between the locations with islands and those without are places where the basic
assumptions underlying perturbation
theory have broken down, and we could expect chaotic responses from the actual system. A different way
of saying the same thing is that the perturbed system is no longer integrable!
| 
|
| f0 = 6000
| f0 = 7000
|
To verify these statements about the lack of integrability, we now show another pair of graphs from
Lichtenberg, for, as he says, "higher amplitude."
The left-hand graph is again a graph from the first-order calculations and is clearly similar to the ones
I did, which were displayed above. On the other hand, the right-hand graph is, again, from an actual
numerical calculation of the exact time dependence of the trajectories intersecting the Poincar&eactute;
section. That graph shows that there are indeed trapped motions, there are also motions that still circle
the entire cylinder; on the other hand, between those motions, there are simply chaotic trajectories
that do not lie on smooth curves at all. This is exactly what the KAM theory would predict about the
boundary area between these two sorts of curves.
Quite near to those chaotic boundary areas, we also see here the chains of islands mentioned briefly
above. These occur at a higher level of perturbation calculation than our first-order ones. We will want to
begin looking for them in an analytic way.
Last updated/modified: 22 February, 2004
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