Download Analysis of Simple Charged Particle Systems that Exhibit Chaos

Analysis of Simple Charged Particle Systems
that Exhibit Chaos
by
Vladimir Zhdankin
A thesis submitted in partial fulfillment of the
requirements for the degree of
Bachelor of Science
(Engineering Physics)
at the
UNIVERSITY OF WISCONSIN-MADISON
2011
Abstract
Few-body charged particle systems that exhibit chaos with a small number of degrees of
freedom are explored. In particular, the dynamics of two classical systems are investigated
in order to determine under what circumstances chaos can occur. The two systems are the
three-body Coulomb problem and the two-body Coulomb problem in a uniform magnetic
field. It is found that both systems can exhibit chaos in a four-dimensional phase space.
The three-body Coulomb problem is found to require a special symmetry (in the form of the
Langmuir-type orbit) in order to remain bounded and allow chaos. The two-body Coulomb
problem is found to exhibit chaos when the particles move in a plane with an orthogonal
magnetic field. For both systems, parameters are chosen to represent common physical cases.
It is argued that these systems are among the simplest charged particle systems that can
exhibit chaos.
Acknowledgements
I am grateful for the guidance, support, and friendly discussions provided by my advisor
Clint Sprott. Without his help, my project and thesis would not have been possible. I would
also like to thank the rest of the UW-Madison faculty that have contributed to my scientific
growth at various times.
Contents
1 Introduction
2
2 Background
2.1 Dynamical systems and chaos theory . . . . . . . . . . . . . . . . . . . . . .
2.2 Laws of electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Selection of simple charged particle systems . . . . . . . . . . . . . . . . . .
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3 Methods
3.1 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Measuring chaos with the Lyapunov exponent . . . . . . . . . . . . . . . . .
3.3 Visualizing chaos with a Poincaré section . . . . . . . . . . . . . . . . . . . .
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4 Three-body Coulomb Problem
4.1 Formulation of the three-body Coulomb problem .
4.2 General solution and bounded orbits . . . . . . .
4.3 Langmuir-type orbit equations . . . . . . . . . . .
4.4 Levi-Civita regularization . . . . . . . . . . . . .
4.5 Analysis of Langmuir-type orbit solutions . . . . .
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6 Discussion
6.1 Comparison of the systems by simplicity . . . . . . . . . . . . . . . . . . . .
6.2 Potential for further work . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7 Conclusion
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5 Two-Body Coulomb Problem in a Uniform Magnetic
5.1 Statement of general problem . . . . . . . . . . . . . .
5.2 General solutions . . . . . . . . . . . . . . . . . . . . .
5.3 Transformation to relative coordinates . . . . . . . . .
5.4 Reduced Hamiltonian and equations of motion . . . . .
5.5 Dynamics for physical cases . . . . . . . . . . . . . . .
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Chapter 1
Introduction
The complexity of nature can be encapsulated by what is popularly known as the butterfly
effect. This is the notion that the flap of a butterfly’s wings in one part of the world can
determine whether a tornado develops elsewhere. More generally, the butterfly effect suggests
that an initially small effect can build up over time to have tremendous implications in the
future. This has long been known to be a feature of weather systems. Forecasters have
difficulty making accurate weather predictions more than a week in advance because of the
large number of small effects that are difficult to take into account. There are countless other
experiences in daily life that show a similar sensitivity to small changes. Examples include
the roll of a die, the fall of a piece of paper to the floor, and the turbulent flow of water [1].
Chaos theory is the branch of science that deals with dynamical systems that have a
sensitive dependence to initial conditions. Accordingly, such systems are said to be chaotic.
The laws that govern chaotic systems are deterministic; that is, the outcome of the future is
entirely determined by the state at an earlier time. Nevertheless, the dynamics are irregular,
and the state of the system is unpredictable over long periods of time.
The examples of chaotic systems described above are rather complicated since they are
composed of a large number of particles, and thus require a large number of variables to
completely describe the system. Thus, it may not be surprising that the behavior of the
system is difficult to predict since there are many different factors that contribute to the
result. However, chaos can also exist in systems with only a few degrees of freedom. A
famous example of this is the three-body problem of classical mechanics, in which three
massive objects interact through Newton’s law of universal gravitation. Three-body systems
are widespread in the universe, an example being triple star systems. For a two-body system,
there exists an analytic solution that requires bounded orbits to follow elliptical paths. Upon
the addition of a third body, no analytic solution exists, and the dynamics are chaotic
in general. The presence of chaos in the three-body problem and other simple physical
systems suggests that chaos is not something that emerges with complexity, but instead is a
fundamental property of nature. Investigating the simplest systems that can exhibit chaos
offers the potential to better understand how chaos emerges in nature.
In this thesis, the emergence of chaos in classical electrodynamics is investigated by
considering the dynamics of two basic charged particle problems: the three-body Coulomb
2
problem and the two-body Coulomb problem in a uniform magnetic field. These systems are
governed by Coulomb’s law and the Lorentz force, which are both fundamental components of
electrodynamics. Determining the emergence of chaos in these systems can lead to important
applications in plasma physics [2], astrophysics [3], and atomic scattering [4].
Both systems are studied numerically and found to exhibit chaos in a four-dimensional
phase space with parameters chosen to represent physical cases. The three-body problem
is found to require a special symmetry (called the Langmuir-type orbit) in order to remain
bounded and exhibit chaos. The two-body problem in a magnetic field is studied only in the
restricted case where the two particles undergo planar motion with a perpendicular magnetic
field and the charges do not sum to zero. Chaos is observed in a variety of cases with physical
parameters. Finally, it is argued that these two systems are among the simplest charged
particle problems that can exhibit chaos. Although simplicity is not a well-defined quantity,
some criteria include a low-dimensional phase space and a small number of parameters.
Although chaos was previously known to exist in both of these problems under certain
conditions [5, 6], the analysis done here goes beyond previous work mainly by studying
different combinations of parameters. In particular, chaos is observed for the two-body
problem in a magnetic field with charges of equal sign, which was unknown before [7].
Section 2 provides an overview of dynamical systems, chaos theory, and electrodynamics.
The selected charged particle systems are also discussed in detail. Section 3 describes the
numerical methods used to solve the equations and the tools used to determine whether a
system is chaotic. Sections 4 and 5 describe the two systems in detail and discuss the results
of the numerical analysis. Then section 6 compares the two systems, particularly in terms
of simplicity. Additionally, potential paths for future work are discussed. Finally, section 7
provides a conclusion.
3
Chapter 2
Background
2.1
Dynamical systems and chaos theory
A dynamical system is a system that evolves in time according to a set of fixed mathematical
laws. At any time t, the state of a dynamical system can be completely described by a vector
x(t) = [x1 (t), x2 (t), . . . , xn (t)], where n is the number of degrees of freedom. The space of all
possible values of x for a given system is known as the state space. Solutions with a given
set of initial conditions can be described in the state space [1].
The evolution of a continuous dynamical system is governed by differential equations.
Mathematically,
ẋ(t) = f (x(t))
(2.1)
where the dot denotes a time derivative and the functions f (x) = [f1 (x), f2 (x), . . . , fn (x)]
describe the time rate of change of the variables. In addition to the differential equations,
the set of initial conditions x(0) must be given in order for the problem to be specified.
There may also be a number of constants that must be specified in f . These constants are
known as parameters, and their values can affect the dynamics of the system.
In this thesis, the dynamical systems that are discussed have the additional property of
being Hamiltonian. These systems conserve a quantity known as the Hamiltonian H (which
physically represents the total energy of the system), and the equations of motion satisfy
Hamilton’s equations,
∂H
∂pi
∂H
ṗi = −
∂xi
ẋi =
(2.2)
where xi (i = 1, ..., n) are the position varibles and pi are corresponding momentum variables.
Thus there are 2n variables that describe the system, where the pairs (xi , pi ) are called
canonically conjugate variables. The space of x and p is called the phase space.
4
Figure 2.1: The chaotic trajectory of one body in the gravitational three-body problem. The
left shows the original trajectory, while the right shows a perturbed trajectory that results
when the initial position of one body is changed from x = 0.5 to x = 0.5005.
The features of a given dynamical system can be ascertained from the phase plot. A
resting state is represented as a single point in phase space because no change occurs. A
periodic solution is represented as a closed curve since the solution returns to the original
state after a finite time. A quasiperiodic solution shows up on the phase plot as a torus.
Quasiperiodicity is defined by the existence of two frequencies in the solution that are incommensurate. Although such a solution is not periodic, it is also not chaotic because nearby
trajectories do not diverge exponentially.
A chaotic solution has the most remarkable manifestation on a phase plot: it fills out a
region of the phase space known as the chaotic sea. Any solution whose initial state starts
inside the chaotic sea will exhibit chaos. Given enough time, the orbit will come arbitrarily
close to any point in the chaotic sea. The boundary of the chaotic sea often has a fractal
structure.
Chaotic systems are bounded dynamical systems characterized by a number of properties
[1]. The first is that they are aperiodic, so the dynamics are irregular and never repeat. The
second is that they exhibit a sensitive dependence on initial conditions, which makes longterm prediction impossible. The third is that the governing equations are nonlinear and
usually have no analytic solution. Thus the most effective approach to studying chaotic
systems is by numerical methods.
A classic example of a chaotic system is the three-body problem. In this problem, three
massive bodies interact through Newton’s law of universal gravitation. This can be a model
for a triple star system, among other things. The problem has been studied throughout
history by a number of famous mathematicians. In 1887, King Oscar II of Sweden offered a
prize for a solution to the problem, which was awarded to Henri Poincaré despite the fact
that he did not solve the original problem [1].
5
The irregularity and divergence of initial conditions that define chaos can be seen by
plotting the motion of one body in the three-body problem, as shown in Fig. 2.1. The first
image shows the original trajectory, while the second image shows the trajectory when initial
conditions are perturbed from x = 0.5 to x = 5.005.
2.2
Laws of electromagnetism
Newton’s second law describes the effect of the net force F acting on a particle. It states
that the net force equals the time derivative of the momentum. Ignoring relativistic effects,
this is equal to the mass m times the acceleration v̇. Hence,
F = ṗ = mv̇
(2.3)
The description of the electromagnetic force acting on a charged particle is known as the
Lorentz force. The Lorentz force on a particle with charge q interacting with electric field E
and magnetic field B is
F = q(E + v × B)
(2.4)
A complete classical description of electric and magnetic fields is given by Maxwell’s
equations. In general, the evolution of electric and magnetic fields is complicated. Maxwell’s
equations state that a magnetic field is generated by the motion of charged particles. Another
result is that changing electric fields induce magnetic fields, and vice versa. Furthermore, the
magnetic and electric fields can have a number of configurations depending on the physical
situation that generates them.
For the problems in this thesis, the effects of the full Maxwell’s equations are ignored in
favor of a simple view of electric and magnetic fields. The only magnetic field that will be
considered is a uniform one generated by an external source. On the other hand, the electric
fields will be solely generated by charged particles. The electric field at a point r generated
by a particle with charge qs located at a point r s is
E=
ke qs (r − r s )
|r − r s |3
(2.5)
where ke is Coulomb’s constant. When Eq. 2.5 is plugged into Eq. 2.4, the resulting force
describes Coulomb’s law.
Coulomb’s law states that between any pair of charged particles, there is a force that
is inversely proportional to the square of the distance between the particles. The force is
also proportional to the product of charges, which makes particles with opposite signs of
charge attract and particles with same signs of charge repel. Hence, the Coulomb force is
mathematically similar to Newton’s law of universal gravitation except for the fact that it
can be repulsive.
6
The equations introduced here are classical approximations. There are a number of effects
that arise with more realistic models of physics. First, accelerated charged particles emit
radiation as given by the Larmor formula. This causes energy to be lost from the particles,
so the system is no longer stable. Second, there are a number of relativistic effects that
must be considered when velocities are large. Third, the laws of quantum mechanics must
be introduced when interactions occur at small scales. This is especially important when the
problems are meant to model atoms.
2.3
Selection of simple charged particle systems
The aim of this thesis is to study some of the simplest charged particle systems with the
possibility of exhibiting chaos. The goal is not to realistically model any particular physical
situation, but to ascertain where chaos emerges in ideal systems with few degrees of freedom.
The question of how to measure simplicity will be discussed in more detail in section 4, but
for now it can be assumed that simplicity implies a small number of degrees of freedom,
particles, and interactions. The two systems chosen here are simple in the sense that no
aspect of the problem can be removed without allowing an analytic solution.
The two-body Coulomb problem has an analytic solution, and therefore chaos cannot
exist. If charges have opposite signs, then the problem is mathematically identical to the
two-body gravitational problem. Therefore, the bounded cases are periodic with particles
undergoing elliptical orbits. If the charges have like signs, then the particles repel and the
system is unbounded.
The three-body Coulomb problem is a natural step up in complexity from the two-body
case. Much like the three-body gravitational problem, there is no analytic solution, and
so the possibility of chaos exists. The problem has mostly been studied in the context
of atoms. In particular, most researchers study the case of the helium atom [8], but less
common atoms such as positronium have also been considered [9]. One might expect that
such classical models of the atom would be of little interest since the effects of quantum
mechanics dominate on the atomic scale. However, quantum wave functions for high energy
states can sometimes be linked to classical orbits [10]. Furthermore, this correspondence
between the solutions of classical dynamics and those of quantum mechanics is essential for
the study of quantum chaos [11]. The three-body Coulomb problem can also be a coarse
approximation for more general situations such as the interactions between particles in a
plasma or ions in any substance.
There has been a large amount of work done on the three-body Coulomb problem in the
last two decades. An extensive review of the field has been written by Tanner et al. [11].
In particular, chaotic solutions of the Langmuir-type orbit for helium had been discovered
previously by Richter et al. [5]. This thesis expands that work by investigating the solutions
for other physical cases and computing Lyapunov exponents.
A second way to increase the complexity of the two-body Coulomb problem is to add a
uniform magnetic field. The problem of a single charged particle in a uniform magnetic field
has an analytic solution in which the particle follows a helix in the direction of the magnetic
7
field. In two dimensions with a perpendicular field, this reduces to a circular orbit. Upon
the addition of a second charged particle, there is no analytic solution to the problem [7].
Therefore, the possibility of chaos exists. In fact, the two-dimensional case in which the two
particles move in a plane with a perpendicular magnetic field allows for chaos.
The two-body Coulomb problem in a uniform magnetic field can be relevant in a number
of physical situations. One common case that is studied is the classical hydrogen atom in
a magnetic field [6], although in reality the laws of quantum mechanics dominate at that
scale. Regardless, it presents a valuable opportunity to study quantum chaos [12]. Another
relevant case is for ion pairs that compose the plasma in a planetary or stellar magnetic
field. This is especially pertinent in the vicinity of white dwarfs and neutron stars where
strong magnetic fields exist [13]. The large-scale magnetic field is approximately uniform at
the level of particle interactions, so the uniform magnetic field approximation is justified.
Curilef and Claro obtained analytic solutions for the two-dimensional case where particles
have equal mass and equal magnitude of charge [14]. Both the two-dimensional and threedimensional problems were studied analytically by Pinheiro and MacKay in a recent series
of two papers [7, 15]. In the two-dimensional case, they found that cases with equal gyrofrequencies (q1 /m1 = q2 /m2 ) are integrable. Furthermore, they inferred that chaos exists in
cases with opposite signs of charge unless gyrofrequencies sum to zero. However, they were
unable to establish what happens when the gyrofrequencies sum to zero and whether there is
chaos for cases with equal signs of charge. With the exception of those papers, the problem
has been studied numerically. In particular, the case of hydrogen in three-dimensional space
was found to be chaotic by Schmelcher and Cederbaum [16] and Friedrich and Wintgen [6].
However, the problem has barely been studied for other choices of particles. This thesis
focuses on the two-dimensional case and numerically studies common physical systems with
q1 + q2 6= 0. Chaos is verified by computing the largest Lyapunov exponent and observing a
chaotic sea in the Poincaré sections. The problem with equal signs of charge is found to be
capable of exhibiting chaos.
There are other charged particle systems of comparable importance and simplicity to
the two chosen here. One such problem is a single charge in a nonuniform magnetic field.
Possible magnetic fields relevant to physical situations include the magnetotail of a planet [3]
and the dipole field [17]. Bonfim et al. [17] found chaos for the problem of a charged particle
in a magnetic dipole field. One thing that this could model is an ion from the solar wind
entering the magnetic field of the earth. Other interesting magnetic field configurations
might include the toroidal magnetic field found in plasma confinement devices or a magnetic
mirror. Alternatively, the problem of a charged particle in an external electric field could be
interesting. These problems and other similar ones would naturally fit into this thesis, but
are ignored in favor of a more concentrated analysis of the three-body problem and two-body
problem in a magnetic field.
8
Chapter 3
Methods
3.1
Numerical methods
Since chaotic systems generally do not have analytic solutions, the optimal way to study
them is by using numerical methods. The premise behind numerical methods is to treat
time as a discrete rather than continuous quantity. The differential equation can then be
treated as an algebraic equation and solved for future time steps. For the problems in this
thesis, MATLAB was used to numerically solve the equations.
The integration method used is the fourth-order Runge-Kutta method [1]. Suppose we
have the differential equation
ẋ = f (t, x)
(3.1)
with x(t0 ) = x0 . Then the solution at iteration n + 1 is
1
xn+1 = xn + h(k1 + 2k2 + 2k3 + k4 )
6
tn+1 = tn + h
(3.2)
where h is the step size and ki are the following partial step approximations
k1
k2
k3
k4
= f (tn , xn )
= f (tn + h/2, xn + hk1 /2)
= f (tn + h/2, xn + hk2 /2)
= f (tn + h, xn + hk3 )
(3.3)
It is straightforward to extend this algorithm to multiple variables. In order to further
improve the performance of the algorithm, an adaptive step size is used. In such an implementation, the step size changes throughout time depending on the predicted error. This
allows the step size to decrease during times when the trajectory is undergoing large changes,
which reduces the overall error. It also allows the step size to increase during times when
9
the trajectory is relatively straight, which decreases the computational time required to obtain the solution. These features make the adaptive step size particularly suitable for stiff
problems, in which numerical solutions are unstable unless the step size is taken to be very
small.
The error in the numerical solution can be approximated by comparing the original
solution xn+1 to a second solution x0n+1 taken at the same time but calculated by using
two half-steps. If the error is greater than the acceptable amount, then the computation is
performed again using the smaller step size and the error criterion is checked again. If the
error is less than the acceptable amount, then that solution is used and the next iteration is
computed with a twice larger step size.
One method to check the overall accuracy of the numerical solution for a Hamiltonian
system is to check if the total energy H is conserved. Numerical error can cause the energy to
drift from the original value. A symplectic integrator can be used to help conserve energy in
Hamiltonian systems, but this requires the kinetic energy to be only a function of momenta
and the potential energy to be only a function of positions [18].
3.2
Measuring chaos with the Lyapunov exponent
Measuring the Lyapunov exponents is one of the standard ways of classifying the dynamics
of a system. The spectrum of Lyapunov exponents is a set of numbers that describe the
rate at which nearby trajectories diverge in a dynamical system. Each Lyapunov exponent
corresponds to a dimension in phase space and can be measured numerically by using a
method developed by Wolf et al. [19]. However, knowing only the largest Lyapunov exponent
λ is enough to determine the dynamics of the system [1]. Since the largest Lyapunov exponent
is easier to measure, that is what is computed here.
It is important to distinguish between two kinds of Lyapunov exponents. The first type is
known as the local Lyapunov exponent, which measures the rate of divergence between two
nearby orbits at an instant of time. This value can vary throughout an orbit since nearby
orbits may diverge in one region of space and then converge back together at another region.
The second type of Lyapunov exponent is known as the global Lyapunov exponent, which is
the average of the local Lyapunov exponents in the limit as time approaches infinity.
The global Lyapunov exponent is what determines the dynamics of a system. If λ > 0,
then nearby solutions diverge exponentially from the original solution, which is the defining
feature of chaotic systems. If λ = 0, then nearby solutions remain close to the original
solution, and so the orbit is periodic or quasiperiodic. If λ < 0, then the solution decays to
a stable equilibrium. Thus, all three distinct outcomes can be distinguished by measuring
the global Lyapunov exponent.
Consider a solution x(t) that contains the position coordinates and corresponding velocity
components for all of the particles. Let the initial conditions of the system be x(0). Now
consider a second orbit that is slightly perturbed from the original orbit. Let the difference
between the original orbit and perturbed orbit be ∆x(t). Then the distance between the
two orbits grows as:
10
|∆x(t)| ≈ |∆x(0)|eλt
(3.4)
Eq. 3.4 can be rearranged and made discrete to give a numerical formula for the largest
Lyapunov exponent. In a simulation with a discrete solution xi and step size of h, the
formula is:
N −1
1 X ∆xn+1 λ = lim
ln N →∞ N h
∆x
n
n=0
(3.5)
The procedure for calculating the largest Lyapunov exponent is as follows. First, a
perturbed trajectory x∗0 is created by displacing the original initial conditions x0 by a small
amount ∆x0 in any direction:
x∗0 = x0 + ∆x0
(3.6)
Then an integrator is used to advance both the actual solution and the perturbed solution.
Afterwards, the new separation and resulting local Lyapunov exponent can be computed:
∆x1 = x∗1 − x1
1 ∆x1 λ1 = ln h
∆x0
(3.7)
(3.8)
The perturbed trajectory must be then be readjusted to stay near the actual orbit. This
is done by resetting the distance between the orbits to |∆x0 |, but keeping the displacement
in the direction of ∆x1 so that the orbit orients in the direction of maximum expansion (i.e.
the direction of the largest Lyapunov exponent):
x∗0
∆x 0
= x1 + ∆x1 ∆x1
(3.9)
The procedure is then repeated for the next time steps, as shown in Fig. 3.1. In order
to get the largest Lyapunov exponent, the local Lyapunov exponents at each iteration must
be averaged, as in Eq. 3.5. The computation must be carried out until the average has
converged to the desired accuracy. For periodic systems, this required time may be only a
few periods. For chaotic systems, this time may be much longer depending on the shape of
the chaotic sea.
11
Figure 3.1: Numerical calculation of an orbit x and the perturbed orbit x∗ , where distance
between the two orbits is used to compute the largest Lyapunov exponent. After each time
step, the distance between the two orbits is reset to ∆x0 .
3.3
Visualizing chaos with a Poincaré section
A Poincaré section is a plot of the phase space for a dynamical system with reduced dimensions. If the phase space has n dimensions, then the Poincaré section will have n − 1
dimensions. This is accomplished by plotting the solution when one variable is set to a fixed
value. Numerically, this is done by plotting the solution in the remaining n − 1 dimensional
space whenever the variable crosses the specified value during a time step.
The lower dimensionality of the Poincaré section makes the solution easier to visualize.
This is particularly useful for systems that already have a small number of dimensions since
reduction by one dimension can make it possible to present the solution in two-dimensional
or three-dimensional space. In the Poincaré section, a periodic orbit appears as a set of
points rather than a closed loop. Furthermore, a quasiperiodic orbit appears as a set of
closed loops rather than a torus. This process is shown in Fig. 3.2, where the Poincaré
section is simply a slice of the complete solution.
The numerical method to construct a Poincaré section is straightforward. Suppose x = xp
is the location at which the Poincaré section is taken. Then whenever (xi−1 −xp )(xi −xp ) < 0,
linear interpolation can be used between the two time steps to obtain the point in the
Poincaré section. Explicitly, if y is any other variable in the system, then the y coordinate
of the point in the Poincaré section is
yp = yi−1 +
xp − xi−1
(yi − yi−1 )
xi − xi−1
(3.10)
The number of points collected for the Poincaré section must be large enough so the
12
Figure 3.2: The lower-dimensional representations of a solution x(t) gained by taking the
Poincaré section. Since a periodic orbit is a closed loop in phase space, the Poincaré section
reduces it to a point, as shown to the left. Likewise, since a quasiperiodic orbit is a torus in
phase space, the Poincaré section reduces it to a closed loop, as shown to the right.
structures become clear. This can require many orbits in order to get fine resolution on the
boundaries of the chaotic sea, although the fact that the solution is chaotic could be evident
much earlier. In order to get a Poincaré section of the entire system, the procedure must be
repeated with different sets of initial conditions that show representative orbits.
The two systems studied in this thesis have chaotic solutions that exist in a fourdimensional phase space. Therefore, the Poincaré section shows the solution in a threedimensional space. However, there is one more trick that can be used to further reduce the
dimension of the system by one. Since both systems are Hamiltonian, the fact that total
energy is conserved can be used to constrain the solutions to have a fixed energy. In other
words, only consider initial conditions with the same energy. This restricts the Poincaré
section to the surface of constant energy. Therefore, starting with a four-dimensional Hamiltonian system, a two-dimensional surface has been obtained that reveals the dynamics for a
given energy. The topology of the surface of constant energy can be rather complicated, so
it may not be possible to have a one-to-one projection into a plane. Regardless, the fact that
the dynamics can be shown on a two-dimensional surface in three-dimensional space gives a
convenient way to represent the dynamics.
13
Chapter 4
Three-body Coulomb Problem
4.1
Formulation of the three-body Coulomb problem
The three-body Coulomb problem describes a system of three electrically charged particles
that interact through Coulomb’s law. In order to have any possibility of being bounded,
one charge must have the opposite sign of the other two. This results in an attractive force
between two pairs of the particles and a repulsive force between the third pair. Among
other things, this can be used to model the classical helium atom or the interactions inside
a plasma [8].
In the most general form, the three-body Coulomb problem has an eighteen-dimensional
phase space consisting of the positions and velocities of the three bodies. In a Cartesian
coordinate system, the position and velocity vectors corresponding to the body with mass
mi and charge qi are r i = (xi , yi , zi ) and v i = (vx,i , vy,i , vz,i ) respectively, where i = 1, 2, 3.
The complete equations of motion are then
ke q1 q2 (r 1 − r 2 ) ke q1 q3 (r 1 − r 3 )
+
m1 |r 1 − r 2 |3
m1 |r 1 − r 3 |3
ke q1 q2 (r 2 − r 1 ) ke q2 q3 (r 2 − r 3 )
+
=
m2 |r 1 − r 2 |3
m2 |r 2 − r 3 |3
ke q1 q3 (r 3 − r 1 ) ke q2 q3 (r 3 − r 2 )
=
+
m3 |r 1 − r 3 |3
m3 |r 2 − r 3 |3
= v1
= v2
= v3
v̇ 1 =
v̇ 2
v̇ 3
ṙ 1
ṙ 2
ṙ 3
(4.1)
where ke is Coulomb’s constant. In addition, there are several conserved quantities. The
vectors describing total linear momentum P and total angular momentum L are conserved:
P = m1 v 1 + m2 v 2 + m3 v 3
L = m1 r 1 × v 1 + m2 r 2 × v 2 + m3 r 3 × v 3
14
(4.2)
In addition, the total energy (Hamiltonian) E is conserved:
1
1
1
ke q1 q2
ke q1 q3
ke q2 q3
E = m1 |v 1 |2 + m2 |v 2 |2 + m3 |v 3 |2 +
+
+
2
2
2
|r 1 − r 2 |2 |r 1 − r 3 |2 |r 2 − r 3 |2
(4.3)
The equations of motion as written in Eq. 4.1 can be simplified by rescaling to set some
parameters to unity and using the conservation laws (Eq. 4.2) to determine the position and
velocity of the third particle. This process reduces the number of phase space dimensions to
twelve and is done explicitly by Pérez and Mahecha [9].
Several restricted cases of the problem exist. One of these is the planar problem, in which
the three bodies move in a plane. Then z1 = 0 = z2 and vz,1 = 0 = vz,2 , so the number of
dimensions in phase space is further reduced from twelve to eight. Another special case is
the colinear problem, in which the three bodies move along a line. In that case, only four
variables (x1 , x2 , vx,1 , vx,2 ) remain. However, bound solutions to the colinear case necessarily
have collisions, so the equations would have to be reparametrized in order to remove the
singularity [20, 21].
4.2
General solution and bounded orbits
As mentioned in section 2.3, the three-body Coulomb problem has no analytic solution in
general. Therefore numerical methods must be used to acquire solutions for any given case.
A numerical search shows that solutions with random initial conditions are unbounded [8,9].
An example of a solution is shown in Fig. 4.1. The solution has a transient chaotic phase
followed by the particles becoming unbounded. Typically, two of the particles escape as a
two-body system in one direction while the third particle escapes in the opposite direction.
This process is often referred to as autoionization [8].
An argument for why autoionization occurs can be obtained from energy conservation.
In order to have a bounded system, the energy must be negative for all particles (in other
words, potential energy is stronger than kinetic energy). However, any pair of attracting
particles in a chaotic system can come arbitrary close together, giving them an arbitrarily
large negative energy. By total energy conservation, the third particle will have positive
energy and then escape [5].
There are several types of bounded solutions, but they require special initial conditions
that impose certain symmetries [9, 11]. One of the simplest periodic orbits occurs when the
two negative charges have circular orbits around the positive charge in a way that gives
no net force on the positive charge, as shown in Fig. 4.2. However, this solution demands
specific parameters and is unstable to perturbations in parameters or initial conditions.
A similar orbit is the Wannier orbit [5], in which the positive charge is stationary while
identical negative charges orbit from diametrically opposite sides in the same direction
around the positive charge. In order for the forces to balance on the positive charge, the
negative charges always have equal distances from the positive charge. The resulting orbits
are elliptical for both negative charges, as shown in Fig. 4.3.
15
Figure 4.1: A solution to the three-body Coulomb problem with equal masses, equal magnitudes of charge, and random initial conditions. The two negative charged particles are
colored red and green, while the positive charged particle is colored blue.
Figure 4.2: A periodic solution in which the negative charges orbit on circles.
16
Figure 4.3: A periodic solution in which the identical particles orbit on ellipses. At any
given moment, the positively charged particle experiences no net force because the negative
charges are located on opposite sides at equal distance.
Figure 4.4: A periodic solution in which the identical particles have a reflectional symmetry
across the positively charged particle.
17
Figure 4.5: The planar Langmuir-type orbit. Note the symmetry with respect to reflection
across the x-axis.
A less obvious periodic solution occurs when two identical negative charges maintain
equal distances from the positive charge with no net angular momentum. This gives the
positive charge one degree of freedom in which to move, with the negative charges satisfying
reflectional symmetry across this line. The resulting orbit is shown in Fig. 4.4. This is
known as the Langmuir orbit because it was first discovered by Irving Langmuir in 1921 for
the case of helium [22].
It turns out that the general solutions to the three-body Coulomb problem that satisfy
the symmetry of the Langmuir orbit are not periodic. Instead, the solutions are quasiperiodic
or chaotic. Such an orbit is called a Langmuir-type orbit [9] and will be discussed in detail
in the following sections.
4.3
Langmuir-type orbit equations
The Langmuir-type orbit is a bounded solution to the three-body Coulomb problem in which
two identical particles act symmetrically on an oppositely charged particle as shown in Fig.
4.5. If this symmetry is enforced in the equations of motion, then the number of dimensions
in phase space can be reduced. First of all, the symmetry requires that two charges are
identical, so m3 = m1 and q3 = q1 . Second, all three particles can be taken to be located
on the xy-plane since the only difference in three-dimensions is rotation around the axis of
symmetry with a constant angular momentum. Third, the x-axis can be taken as the axis
of symmetry. Then the symmetry requires that x3 = x1 , y3 = −y1 , y2 = 0, vx,3 = vx,1 ,
vy,3 = −vy,1 , and vy,2 = 0. The equations of motion in Eq. 4.1 finally reduce to:
18
q1 q2 (x1 − x2 )
m1 [(x1 − x2 )2 + y12 ]3/2
q12
q1 q2 y 1
v̇y,1 =
+
2
4m1 y1 m1 [(x1 − x2 )2 + y12 ]3/2
2q1 q2 (x2 − x1 )
v̇x,2 =
m2 [(x1 − x2 )2 + y12 ]3/2
ẋ1 = vx,1
ẏ1 = vy,1
ẋ2 = vx,2
v̇x,1 =
(4.4)
Furthermore, vx,2 and x2 can be determined from conservation of momentum. Letting
the particles coincide at the y-axis gives:
2m1
vx,1
m2
2m1
x1
x2 = −
m2
vx,2 = −
After these simplifications are applied, the phase space has four dimensions remaining.
In other words, only the motion of one particle must be known in orderq
to determine the
solution. To reduce the number of parameters, time can be scaled by t → |qm1 q12 | t to remove
a factor of m1 , q1 , and q2 . The problem is then specified by the following two dimensionless
parameters:
M =1+
Q=
2m1
m2
|q1 |
4|q2 |
(4.5)
Note that M > 1 and Q > 0 must be true for physical cases. Applying these changes to
Eq. 4.4, we obtain the equations for the Langmuir-type orbit:
−M x
+ y 2 )3/2
y
Q
v̇y = 2 −
2
2
y
(M x + y 2 )3/2
ẋ = vx
ẏ = vy
v̇x =
(M 2 x2
where we assume that y > 0. The total energy for the Langmuir-type orbit is
19
(4.6)
E = M vx2 + vy2 +
2Q
2
−p
2
y
M x2 + y 2
(4.7)
An important feature of the Langmuir-type orbit is that it is bounded if E < 0, which
occurs when the potential energy is larger than the kinetic energy. A proof that it remains
bounded is as follows [23]: If the particle is to escape, then M 2 x2 + y 2 → ∞. In this limit,
the energy becomes
E∞ = M vx2 + vy2 +
2Q
y
Note that E∞ > 0 since terms are all positive because y > 0. But if we start the particle
with E < 0 and assume that energy is conserved, this scenario is impossible. Therefore the
particle cannot escape.
4.4
Levi-Civita regularization
An inconvenient feature of the Langmuir-type orbit is the potential triple collisions. This
results in a singularity at y = 0 since v̇y → ∞. Singularities are a major source of error
in numerical solutions because the small numerical errors are multiplied by large numbers.
Long simulations are required to get an accurate Lyapunov exponent, so it is essential to
avoid the singularity. Fortunately, the singularity can be removed by reparametrizing the
equations through a method known as Levi-Civita regularization [23].
The first step is to introduce variables ξ and η, defined by the transformation
y = ξ2
η
vy =
ξ
Additionally, the time derivative must be reparametrized according to
equations of motion then become
dt
ds
= 2ξ 2 . The
ẋ = 2ξ 2 vx
ξ˙ = η
v̇x =
−2M xξ 2
(M 2 x2 + ξ 4 )3/2
η̇ = Eξ − M vx2 ξ +
2M 2 x2 ξ
(M 2 x2 + ξ 4 )3/2
(4.8)
where the dot now denotes a derivative with respect to s. The corresponding energy is
20
Figure 4.6: The trajectory of the charged particle in a chaotic solution to the Langmuir-type
orbit.
E = M vx2 +
2
η 2 2Q
+ 2 −p
2
2
ξ
ξ
M x2 + ξ 4
(4.9)
Note that E has replaced Q as a parameter in the equations of motion. With Levi-Civita
regularization, the terms in the differential equations do not get as large when ξ → 0. Hence,
Eq. 4.8 is used in the numerical simulations.
4.5
Analysis of Langmuir-type orbit solutions
The parameters are chosen to represent three interesting physical systems, as listed in Table
4.1. The classical helium atom has been studied extensively in the literature, but the negative
hydrogen ion has not. Although the ionized positronium atom is not a common physical
system, it does represent a case in which all three masses are equal.
Table 4.1: Three-body charged particle systems
System 1
2
M
Q
Chaotic?
−
He
α e
1.00025 0.125
Yes
−
−
H
p e
1.001
0.25
Yes
Ps−
e+ e−
3
0.25
No
No chaos was observed for the case of positronium, but there was chaos for helium
and hydrogen. This was established by computing positive values of the largest Lyapunov
exponent and observing a chaotic sea in the Poincaré sections. The trajectory for the chaotic
solution of helium is shown in Fig. 4.6. The largest Lyapunov exponent for helium with initial
conditions [x, vx , ξ, η] = [0.3, 0, 0.5, 0.4] was computed to be λ ≈ 0.016. The largest Lyapunov
21
Figure 4.7: The Poincaré section for the Langmuir-type orbit at x = 0. The left plot is for
the case of helium, while the right plot is for the case of negative hydrogen ion.
exponent for ionized hydrogen with initial conditions [x, vx , ξ, η] = [0, 61.1985, 0.02, 0] was
computed to be λ ≈ 0.04.
A convenient location to take the Poincaré sections is at x = 0. The Poincaré sections
for helium and hydrogen are shown in Fig. 4.7. The Poincaré sections for both cases are
qualitatively similar, with a mixed phase space of chaotic and quasiperiodic solutions. There
is a robust chaotic sea which extends up to ξ = 0. If Levi-Civita regularization had not been
used, the Poincaré section would extend up to vy → ∞.
One important question about the Langmuir-type orbit is whether it is stable with respect
to perturbations away from symmetry. It has been shown by Richter and Wintgen that the
periodic Langmuir orbit has a small island of stability for the case of helium [24]. For the
cases which are chaotic, the stability was tested by perturbing the initial conditions away
from symmetry in the full equations of motion. No stable solutions were found. Therefore,
it is unlikely that the Langmuir-type would exist in nature. This implies that real bounded
three-body charged particle systems are unlikely to be chaotic.
22
Chapter 5
Two-Body Coulomb Problem in a
Uniform Magnetic Field
5.1
Statement of general problem
The problem of two charged particles interacting through the Coulomb force in the presence
of a uniform magnetic field has a twelve-dimensional phase space in the most general form.
This consists of the positions and momenta of both particles in three-dimensional space.
However, only the restricted problem with two spatial dimensions is studied here. Therefore
the problem is restated as that of two planar charged particles interacting via the Coulomb
force in the presence of a perpendicular magnetic field. In the general form, this problem
has an eight-dimensional phase space which consists of the positions and momenta of the
particles, (x1 , y1 , p1x , p1y , x2 , y2 , p2x , p2y ). The equations of motion read
ṗx1 =
ṗy1 =
ẋ1 =
ẏ1 =
ṗx2 =
ṗy2 =
ẋ2 =
ẏ2 =
ke q1 q2 (x1 − x2 )
[(x1 − x2 )2 + (y1 − y2 )2 ]3/2
ke q1 q2 (y1 − y2 )
[(x1 − x2 )2 + (y1 − y2 )2 ]3/2
1
px1
m1
1
py1
m1
ke q1 q2 (x2 − x1 )
[(x1 − x2 )2 + (y1 − y2 )2 ]3/2
ke q1 q2 (y2 − y1 )
[(x1 − x2 )2 + (y1 − y2 )2 ]3/2
1
px2
m2
1
py2
m2
23
q1 B
py1
m1
q1 B
−
px1
m1
+
q2 B
py2
m2
q2 B
−
px2
m2
+
(5.1)
where B is the magnetic field strength, m1 and m2 are the masses of the particles, q1
and q2 are the electrical charges of the particles, and ke is Coulomb’s constant. There
are three conserved quantities: the x-component of linear momentum Px , the y-component
of linear momentum Py , and angular momentum L. These conserved quantities are the
usual momentum equations with additional terms due to the magnetic field. Explicitly, the
conserved quantities are
Px = p1x + p2x − q1 By1 − q2 By2
Py = p1y + p2y + q1 Bx1 + q2 Bx2
L = (x1 p1y − y1 p1x ) + (x2 p2y − y2 p2x )
1
1
+ Bq1 (x21 + y12 ) + Bq2 (x22 + y22 )
2
2
5.2
(5.2)
General solutions
Curilef and Claro [14] have shown that there are analytic solutions for the case of two
particles of equal mass with equal magnitude of charge. This means that chaos cannot
occur in systems such as proton-proton, electron-electron, or electron-positron. Furthermore,
Pinheiro and MacKay [15] have found that the equations are integrable for the case of equal
gyrofrequencies (q1 /m1 = q2 /m2 ).
Pinheiro and MacKay [15] have established a large number of mathematical properties of
the system and derived reduced equations of motion that will be shown in the next section.
In particular, they proved that the system is bounded (in space) unless the charges sum to
zero (q1 + q2 = 0). If the charges do sum to zero, then there is a drift for the center of mass.
They also established that chaos exists in cases with opposite signs of charge.
The trajectories of the particles in the case of a deuteron and triton, which will be
discussed in more detail later, is shown in Fig. 5.1. It is clear that the solution is a mixture of
gyration due to the magnetic field and interaction between the particles due to the Coulomb
force.
5.3
Transformation to relative coordinates
Although the general equations of motion can be used to numerically study the system,
it is useful to solve the reduced equations instead. Coordinate transformations derived by
Pinheiro and MacKay [7] can be used to reduce the number of phase space dimensions from
eight to four. This is done by using symmetry and the conservation laws in Eq. 5.3. In
this section, coordinate transformations are given for the case when charges have a nonzero
sum, i.e. q1 + q2 6= 0. The case for q1 + q2 = 0 must be treated differently since the center
of mass undergoes a drift, but equations are also derived by Pinheiro and MacKay although
they will not be used here. Units can be chosen such that m1 = 1, and q1 = 1. In order to
simplify notation, define the following quantities:
24
Figure 5.1: An example of trajectories in a solution of the two-body Coulomb problem in a
uniform magnetic field. The parameters are taken to represent the chaotic case of a deuteron
(blue) and triton (red) that is discussed later.
1 + m2
m2
1 + q2
q=
q2
µ = B(1 + q2 )
pθ = −2B(1 + q2 )L + Px2 + Py2
q−1
q2 B
=
µ
=
1 + q2
q2
m=
(5.3)
The five parameters are the relative mass m (≥ 1), the relative charge q, the strength
of the magnetic field µ, the strength of the Coulomb force ke (≥ 0), and the initial total
momentum pθ . The quantity is derived from q and µ. If the first particle is chosen to be
the less massive of the pair, then only 1 ≤ m ≤ 2 must be considered. The phase space is
reduced to four dimensions with a transformation to relative polar coordinates (r, pr , φ, pφ )
defined as follows:
25
Py 1
1 1/2
+ r cos θ − pφ cos (−2µφ + θ)
µ
q
µ
1 1/2
−Px 1
+ r sin θ − pφ sin (−2µφ + θ)
y1 =
µ
q
µ
1 1/2
Py q − 1
−
r cos θ − pφ cos (−2µφ + θ)
x2 =
µ
q
µ
−Px q − 1
1 1/2
y2 =
−
r sin θ − pφ sin (−2µφ + θ)
µ
q
µ
1
pθ − pφ
q − 1 1/2
pφ sin (−2µφ + θ) + pr cos θ + (r +
) sin θ
p1x = −
q
2
µr
q − 1 1/2
1
pθ − p φ
pφ cos (−2µφ + θ) + pr sin θ − (r +
) cos θ
p1y =
q
2
µr
1
pθ − pφ
1 1/2
) sin θ
p2x = − pφ sin (−2µφ + θ) − pr cos θ − (r +
q
2
µr
1 1/2
1
pθ − p φ
) cos θ
p2y = pφ cos (−2µφ + θ) − pr sin θ + (r +
q
2
µr
x1 =
(5.4)
The corresponding inverse transformations are:
p
(x1 − x2 )2 + (y1 − y2 )2
[p1x − (q − 1)p2x ](x1 − x2 ) + [p1y − (q − 1)p2y ](y1 − y2 )
pr =
qr
1
(p1x + p2x )(x1 − x2 ) + (p1y + p2y )(y1 − y2 )
φ =
arctan
2µ
(p1y + p2y )(x1 − x2 ) − (p1x + p2x )(y1 − y2 )
2
pφ = (p1x + p2x ) + (p1y + p2y )2
r =
(5.5)
Note that r > 0 and pφ > 0 must hold. There are singularities located at r = 0 and at
pφ = 0. Also note that φ is periodic on the interval (0, πµ ).
Physically, the variable r represents the distance between the particles. The angle θ,
which drops out of the Hamiltonian and equations of motion, is the angle that the vector
between the two particles makes with the horizontal. The physical interpretation of the
angle φ is less clear, but it encodes the relative angular momentum of the particles.
5.4
Reduced Hamiltonian and equations of motion
One way to obtain the equations of motion is to take the time derivative of r, pr , φ, and pφ
in the original coordinates by using Eq. 5.5, use the original equations of motion to replace
the time derivatives of the original coordinates, and then return to the reduced coordinates
by substitution of Eq. 5.4. A far easier way is to derive the equations of motion directly
from the Hamiltonian by using Hamilton’s equations,
26
ṙ =
∂H
,
∂pr
ṗr = −
∂H
,
∂r
φ̇ =
∂H
,
∂pφ
ṗφ = −
∂H
∂φ
When the coordinate transformation is applied, the new Hamiltonian reads
m 2 m (pθ − pφ )2 m2 2 m
m
q−2
H =
p +
r +
(pθ + pφ ) + 1 −
pφ
+
2 r
8
µ2 r 2
8
4µ
q
2q
1
m
1
pθ − pφ
1/2
+
+ 1−
pr sin (2µφ) − (r +
) cos (2µφ) pφ
(q − 1)r
q
2
µr
(5.6)
The equations of motion that result are:
m
1/2
) sin (2µφ)pφ
q
m (pθ − pφ )2 m2
1
1
m
pθ − p φ
1/2
ṗr =
−
r
+
+
(1
−
)(
−
) cos (2µφ)pφ
4
µ2 r 3
4
(q − 1)r2 2
q
µr2
m q−2
m p θ − pφ
φ̇ = (1 − )(
)+ ( −
)
q
2q
4 µ
µ2 r 2
1
m
1
pθ − 3pφ
−1/2
+ (1 − ) pr sin (2µφ) − (r +
) cos (2µφ) pφ
2
q
2
µr
m
1
pθ − pφ
1/2
ṗφ = −2µ(1 − ) pr cos (2µφ) + (r +
) sin (2µφ) pφ
(5.7)
q
2
µr
ṙ = mpr + (1 −
5.5
Dynamics for physical cases
In order to reduce the parameter space that must be explored, the analysis is limited to
physically common choices of m and q. The particles and corresponding parameters are
shown in Table 5.1, along with whether chaotic solutions were found in the search. These
cases cover a reasonable range of parameter values, although are other combinations of
particles that may be interesting but were not considered here.
The particles shown in Table 5.1 are the proton p, electron e, deuteron d, triton t (tritium
nucleus), helion h (helium-3 nucleus), and alpha particle α (helium-4 nucleus). Additionally,
¯ is very unlikely to
the antiparticles p̄ and d¯ are both included in a case. The case of d-α
occur in nature, but is interesting because of the fact that the gyrofrequencies sum to zero,
i.e. q1 /m1 + q2 /m2 = 0. Pinheiro and MacKay were unable to establish what happens in
this case [7].
Out of the cases listed in Table 5.1, four were found to exhibit chaos. These cases are
listed in Table 5.2 with one possible set of parameters and initial conditions that gives a
27
chaotic solution. The estimated value of the largest Lyapunov exponent for cases with the
same signs of charge are also included.
The largest Lyapunov exponent could not be computed accurately for p̄-α, where charges
have opposite signs. This is due to numerical error that results from collisions between the
particles. Since the Coulomb force is attractive, the particles can come arbitrarily close to
the singularity at r = 0, and this causes numerical instability over long simulations. This
makes it difficult to compute the Lyapunov exponents since it requires averaging over a long
simulation time. The error is inferred by observing the energy drift away from the conserved
value. Regardless, it is still possible to construct a Poincaré section by collecting points until
energy drift becomes too large, and that is how chaos in inferred.
The surface of constant energy for p-h has an ellipsoidal shape, so it is easy to split the
Poincaré section into two projections onto the r-pr plane. These projections are shown in
Fig. 5.2. The Poincaré section predominantly consists of a chaotic sea, but there are also
some islands of quasiperiodicity.
The surface of constant energy is more complicated for the cases of p-t and d-t. To help
visualize the Poincaré sections, the projection is taken onto the r-pr plane with a coloring
scheme that represents the pφ value at each point. The resulting Poincaré sections are shown
in Fig. 5.3 and Fig. 5.4. The case of p-t has a phase space with a roughly equal amount
of chaotic and quasiperiodic regions, while the case of d-t almost exclusively consists of a
chaotic sea. It is possible that the complex shape of the chaotic sea in the p-t case makes it
difficult to get an accurate value of the largest Lyapunov exponent.
Table
1
p
p
p
p
p
d
d
e
p̄
d¯
1 2
h p
t p
d t
p̄ α
5.1: Some possible two-body problems with q1 + q2 6= 0
2 m2 /m1 q2 /q1
m
q
Chaotic?
p
1
1
2
2 No, integrable
d
2
1
1.5
2
No
h
3
2
4/3
1.5
Yes
t
3
1
4/3
2
Yes
α
4
2
1.25
1.5
No
t
1.5
1
5/3
2
Yes
α
2
2
1.5
1.5 No, integrable
α
8000
-2
1.000125 0.5
No
α
4
-2
1.25
0.5
Yes
α
2
-2
1.5
0.5
No
Table 5.2: Parameters
m
q
µ pθ
4
3
-1
1
4
2
-1
2
5/3
2
1
3
1.25 0.5 -0.5 1
and
ke
1
1
1
3
initial conditions for chaotic cases
[r, pr , φ, pφ ]t=0
Remarks
[2.8, 0, 0, 0.9370]
λ ≈ 0.0155
[0.4, 0, 0, 3.1109]
λ ∼ 0.02
[4.5, 0, 0, 4.8038]
λ ≈ 0.0175
[3.0, 0, 0, 1.4102] Numerically unstable
28
Figure 5.2: Poincaré section for the two-body system of a helion and proton taken at φ = π/2.
The surface of constant energy can be conveniently separated into two projections onto the
r-pr plane.
Figure 5.3: The projected Poincaré section for the two-body system of a proton and triton
taken at φ = 0. The color of each point indicates the pφ position.
29
Figure 5.4: The projected Poincaré section for the two-body system of a deuteron and triton
π
taken at φ = 2|µ|
. The color of each point indicates the pφ position.
The Poincaré section for p̄-α is shown in Fig. 5.5. One difference from the other cases is
that the Poincaré section now extends to infinity in pr as r approaches zero. This is because
the Coulomb force is attractive in this case, which allows the particles to come arbitrarily
close to each other. Therefore, this Poincaré section confirms that the system is numerically
unstable.
¯ This is an interesting case because the gyrofrequencies
No chaos was found for case of d-α.
sum to zero, so q1 /m1 + q2 /m2 = 0 or equivalently q = 2 − m. To answer the question posed
by Pinheiro and MacKay about what happens in this case, a search for chaos in general cases
with q = 2 − m was made. No cases that exhibited chaos were found, which suggests that
the solutions are all periodic or quasiperiodic.
30
Figure 5.5: The projected Poincaré section for the two-body system of an antiproton and
alpha particle taken at φ = 0. The color of each point indicates the pφ position. Note that
as r → 0, pr → ∞.
31
Chapter 6
Discussion
6.1
Comparison of the systems by simplicity
Both the three-body Coulomb problem and the two-body Coulomb problem in a uniform
magnetic field are relatively simple physical systems. This is because of the small number
of particles and small number of laws describing their interaction. However, this does not
necessarily mean that the problems are both mathematically simple.
The simplicity of a dynamical system is not a well-defined quantity, but there are several
factors that can be taken in account to characterize it [25]. One of the most fundamental
considerations is the number of degrees of freedom (or equivalently, the number of dimensions
in the phase space). The number of degrees of freedom sets the number of differential
equations and the number of initial conditions required to specify the problem. For a complex
system, there are many degrees of freedom. One example is an N -body problem, where the
number of degrees of freedom is 6N , arrived at by counting the number of position and
velocity coordinates required to specify all particles. Both of the chaotic problems studied
here have a four-dimensional phase space, so they are both simple in that regard.
The number of parameters is another component of simplicity. The choices of values
for parameters can affect the dynamics, so the entire parameter space must be explored to
fully understand a problem. With a large number of parameters, it becomes time consuming
to explore this space. The three-body Coulomb problem only required two parameters to
be described in the case of the Langmuir-type orbit. However, the two-body problem in a
magnetic field required five parameters, although some of the parameters could be set equal
to one without losing the possibility of chaos. This would suggest that the two-body problem
in a magnetic field is more complicated.
The decisive factor in distinguishing these two cases is the complexity of the actual
equations of motion. This can be measured by counting the number of operations used in
the differential equations. The reduced equations for the two-body problem in a magnetic
field are far more complicated than the Langmuir-type orbit equations in this regard.
Despite the differences in mathematical complexity, the important point is that both
systems are physically simple and can exhibit chaos. However, the case of the Langmuir-
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type orbit appears to be unstable, so it might not be relevant in the real world. The two-body
problem in a magnetic field with equal signs of charge is also unstable in three dimensions,
but the problem with opposite signs of charge is stable and therefore can be more interesting
to study for physical purposes.
6.2
Potential for further work
The question that this thesis attempted to answer is: what are the simplest charged particle systems that can exhibit chaos? Investigating the three-body Coulomb problem and
two-body Coulomb problem in a uniform magnetic field were steps toward answering this
question. However, more work should be done to obtain a conclusive answer.
The first thing to note is that the numerical analysis done here is not complete. Although
some of the most relevant physical cases have been studied, there are regions of the parameter
space that have not been explored. In particular, it would be interesting to fill in the gaps of
the parameter space for the magnetic two-body problem. It would also be useful to obtain
equations in which the singularities are removed from the magnetic two-body problem in
order to facilitate numerical analysis for cases of opposite signs of charge.
In the three-body Coulomb problem, it would be interesting to determine if the chaotic
solutions of the Langmuir-type orbit are ever stable with respect to perturbations away from
the symmetry.
For the two-body problem in a uniform magnetic field, the cases studied here were for
the restricted problem of two spatial dimensions. It would be interesting to determine what
happens in the analogous problem with three spatial dimensions. The solutions with like
signs of charge must be unbounded, but it is unclear how the solution with opposite charges
would behave. The equations in a reduced six-dimensional phase space have been derived
by Pinheiro and MacKay [15], so a good starting point would be to observe the dynamics for
some of the physical cases listed in Table 5.1 using those equations. Additionally, only the
cases where q1 + q2 6= 0 were studied here, but there are some important cases with q1 = −q2
that should be studied. These cases would require a different transformation derived by
Pinheiro and MacKay [7].
Another issue is that the two systems here are in the classical form. One question is:
how do things change when effects from special relativity and other effects of the fields in
Maxwell’s equations are included [26]? The drawback is that such effects would increase the
complexity of the system. Also, the fact that accelerated particles radiate energy would have
to be addressed to obtain well-behaved solutions.
Finally, it would be interesting to determine ways in which the problems can be tweaked
so that the unstable cases become stable. Such changes would make the systems more likely
to be observed in nature.
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Chapter 7
Conclusion
In this thesis, the dynamics of two simple charged particle systems have been investigated in
order to determine under which conditions chaos can occur. The goal was not to realistically
model a given physical system, but rather to determine the simplest classical situations in
which chaos arises. The presence of chaos was confirmed by numerically computing the
largest Lyapunov exponent and observing a chaotic sea in the Poincaré sections. In order to
reduce the size of the parameter space to search, the values of mass and charge were chosen
to resemble plausible physical systems.
The first system studied was the three-body Coulomb problem. The solution with random
initial conditions has a transient chaotic phase and then becomes unbounded. However,
there is a special symmetry known as the Langmuir-type orbit that prevents the system
from becoming unbounded and allows chaos to occur. Reduced equations of motion for the
Langmuir-type orbit were derived, and it was found that the dynamics take place in a fourdimensional phase space. The Langmuir-type orbit appears to be unstable with respect to
perturbations away from the symmetry for the chaotic solution, so it is of limited practical
interest.
The second system studied was the two-body Coulomb problem in a uniform magnetic
field. In particular, the restricted problem where particles are located on a plane with an
orthogonal magnetic field was studied. Reduced equations for the case where the charges do
not sum to zero (q1 + q2 6= 0) were used to find solutions in a four-dimensional phase space.
Chaotic solutions were found to exist in several physical cases, including the previously unexplored problem of charges with the same sign. A Poincaré section for the antideuteron-alpha
particle case suggests that chaotic solutions exist for cases where charges have opposite signs,
but numerical instability prevented a converging Lyapunov exponent from being computed.
Finally, the two systems were compared in terms of simplicity. It was argued that these
two systems represent some of the simplest classical charged particle systems that can exhibit
chaos.
34
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