Analysis of Simple Charged Particle Systems that Exhibit Chaos

Transcription

1 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

2 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.

3 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.

4 Contents 1 Introduction 2 2 Background Dynamical systems and chaos theory Laws of electromagnetism Selection of simple charged particle systems Methods Numerical methods Measuring chaos with the Lyapunov exponent Visualizing chaos with a Poincaré section Three-body Coulomb Problem Formulation of the three-body Coulomb problem General solution and bounded orbits Langmuir-type orbit equations Levi-Civita regularization Analysis of Langmuir-type orbit solutions Two-Body Coulomb Problem in a Uniform Magnetic Field Statement of general problem General solutions Transformation to relative coordinates Reduced Hamiltonian and equations of motion Dynamics for physical cases Discussion Comparison of the systems by simplicity Potential for further work Conclusion 34 1

5 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

6 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

7 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) = [x 1 (t), x 2 (t),..., x n (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) = [f 1 (x), f 2 (x),..., f n (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, ẋ i = H p i ṗ i = H x i (2.2) where x i (i = 1,..., n) are the position varibles and p i are corresponding momentum variables. Thus there are 2n variables that describe the system, where the pairs (x i, p i ) are called canonically conjugate variables. The space of x and p is called the phase space. 4

8 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 = 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

9 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 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 = 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 v. Hence, F = ṗ = m v (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 q s located at a point r s is E = k eq s (r r s ) r r s 3 (2.5) where k e 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

10 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

11 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 (q 1 /m 1 = q 2 /m 2 ) 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 q 1 + q 2 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

12 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 with x(t 0 ) = x 0. Then the solution at iteration n + 1 is ẋ = f(t, x) (3.1) x n+1 = x n h(k 1 + 2k 2 + 2k 3 + k 4 ) t n+1 = t n + h (3.2) where h is the step size and k i are the following partial step approximations k 1 = f(t n, x n ) k 2 = f(t n + h/2, x n + hk 1 /2) k 3 = f(t n + h/2, x n + hk 2 /2) k 4 = f(t n + h, x n + hk 3 ) (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

13 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 x n+1 to a second solution x n+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

14 x(t) x(0) e λt (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 x i and step size of h, the formula is: N 1 1 λ = lim ln x n+1 (3.5) N Nh x n n=0 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 x 0 by a small amount x 0 in any direction: x 0 = x 0 + x 0 (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: x 1 = x 1 x 1 (3.7) λ 1 = 1 h ln x 1 x 0 (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 x 0, but keeping the displacement in the direction of x 1 so that the orbit orients in the direction of maximum expansion (i.e. the direction of the largest Lyapunov exponent): x 0 = x 1 + x 1 x 0 x 1 (3.9) The procedure is then repeated for the next time steps, as shown in Fig In order to get the largest Lyapunov exponent, the local Lyapunov exponents at each iteration must be averaged, as in Eq 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

15 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 x 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 = x p is the location at which the Poincaré section is taken. Then whenever (x i 1 x p )(x i x p ) < 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 y p = y i 1 + x p x i 1 x i x i 1 (y i y i 1 ) (3.10) The number of points collected for the Poincaré section must be large enough so the 12

16 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

17 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 m i and charge q i are r i = (x i, y i, z i ) and v i = (v x,i, v y,i, v z,i ) respectively, where i = 1, 2, 3. The complete equations of motion are then v 1 = k eq 1 q 2 (r 1 r 2 ) + k eq 1 q 3 (r 1 r 3 ) m 1 r 1 r 2 3 m 1 r 1 r 3 3 v 2 = k eq 1 q 2 (r 2 r 1 ) + k eq 2 q 3 (r 2 r 3 ) m 2 r 1 r 2 3 m 2 r 2 r 3 3 v 3 = k eq 1 q 3 (r 3 r 1 ) + k eq 2 q 3 (r 3 r 2 ) m 3 r 1 r 3 3 m 3 r 2 r 3 3 ṙ 1 = v 1 ṙ 2 = v 2 ṙ 3 = v 3 (4.1) where k e 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 = m 1 v 1 + m 2 v 2 + m 3 v 3 L = m 1 r 1 v 1 + m 2 r 2 v 2 + m 3 r 3 v 3 (4.2) 14

18 In addition, the total energy (Hamiltonian) E is conserved: E = 1 2 m 1 v m 2 v m 3 v k eq 1 q 2 r 1 r k eq 1 q 3 r 1 r k eq 2 q 3 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 z 1 = 0 = z 2 and v z,1 = 0 = v z,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 (x 1, x 2, v x,1, v x,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 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 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

19 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

20 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

21 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 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 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 m 3 = m 1 and q 3 = q 1. 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 x 3 = x 1, y 3 = y 1, y 2 = 0, v x,3 = v x,1, v y,3 = v y,1, and v y,2 = 0. The equations of motion in Eq. 4.1 finally reduce to: 18

22 v x,1 = q 1 q 2 (x 1 x 2 ) m 1 [(x 1 x 2 ) 2 + y 2 1] 3/2 v y,1 = q2 1 q 1 q 2 y 1 + 4m 1 y1 2 m 1 [(x 1 x 2 ) 2 + y1] 2 3/2 2q 1 q 2 (x 2 x 1 ) v x,2 = m 2 [(x 1 x 2 ) 2 + y1] 2 3/2 ẋ 1 = v x,1 ẏ 1 = v y,1 ẋ 2 = v x,2 (4.4) Furthermore, v x,2 and x 2 can be determined from conservation of momentum. Letting the particles coincide at the y-axis gives: v x,2 = 2m 1 m 2 v x,1 x 2 = 2m 1 m 2 x 1 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 order to determine the solution. To reduce the number of parameters, time can be scaled by t m 1 q 1 q 2 t to remove a factor of m 1, q 1, and q 2. The problem is then specified by the following two dimensionless parameters: M = 1 + 2m 1 m 2 Q = q 1 4 q 2 (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: Mx v x = (M 2 x 2 + y 2 ) 3/2 v y = Q y y 2 (M 2 x 2 + y 2 ) 3/2 ẋ = v x ẏ = v y (4.6) where we assume that y > 0. The total energy for the Langmuir-type orbit is 19

23 E = Mv 2 x + v 2 y + 2Q y 2 M 2 x 2 + 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 x 2 + y 2. In this limit, the energy becomes E = Mv 2 x + v 2 y + 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 v y. 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 = ξ 2 v y = η ξ Additionally, the time derivative must be reparametrized according to dt ds equations of motion then become = 2ξ2. The ẋ = 2ξ 2 v x ξ = η v x = 2Mxξ 2 (M 2 x 2 + ξ 4 ) 3/2 η = Eξ Mv 2 xξ + 2M 2 x 2 ξ (M 2 x 2 + ξ 4 ) 3/2 (4.8) where the dot now denotes a derivative with respect to s. The corresponding energy is 20

24 Figure 4.6: The trajectory of the charged particle in a chaotic solution to the Langmuir-type orbit. E = Mv 2 x + η2 ξ 2 + 2Q ξ 2 2 M 2 x 2 + 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 Yes H p e Yes Ps e + e 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 The largest Lyapunov exponent for helium with initial conditions [x, v x, ξ, η] = [0.3, 0, 0.5, 0.4] was computed to be λ The largest Lyapunov 21

25 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, v x, ξ, η] = [0, , 0.02, 0] was computed to be λ A convenient location to take the Poincaré sections is at x = 0. The Poincaré sections for helium and hydrogen are shown in Fig 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 v y. 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

26 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, (x 1, y 1, p 1x, p 1y, x 2, y 2, p 2x, p 2y ). The equations of motion read ṗ x1 = ṗ y1 = k e q 1 q 2 (x 1 x 2 ) [(x 1 x 2 ) 2 + (y 1 y 2 ) 2 ] + q 1B p 3/2 y1 m 1 k e q 1 q 2 (y 1 y 2 ) [(x 1 x 2 ) 2 + (y 1 y 2 ) 2 ] q 1B p 3/2 x1 m 1 ẋ 1 = 1 m 1 p x1 ẏ 1 = 1 m 1 p y1 ṗ x2 = ṗ y2 = k e q 1 q 2 (x 2 x 1 ) [(x 1 x 2 ) 2 + (y 1 y 2 ) 2 ] + q 2B p 3/2 y2 m 2 k e q 1 q 2 (y 2 y 1 ) [(x 1 x 2 ) 2 + (y 1 y 2 ) 2 ] q 2B p 3/2 x2 m 2 ẋ 2 = 1 m 2 p x2 ẏ 2 = 1 m 2 p y2 (5.1) 23

27 where B is the magnetic field strength, m 1 and m 2 are the masses of the particles, q 1 and q 2 are the electrical charges of the particles, and k e is Coulomb s constant. There are three conserved quantities: the x-component of linear momentum P x, the y-component of linear momentum P y, 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 5.2 General solutions P x = p 1x + p 2x q 1 By 1 q 2 By 2 P y = p 1y + p 2y + q 1 Bx 1 + q 2 Bx 2 (5.2) L = (x 1 p 1y y 1 p 1x ) + (x 2 p 2y y 2 p 2x ) Bq 1(x y 2 1) Bq 2(x y 2 2) 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 (q 1 /m 1 = q 2 /m 2 ). 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 (q 1 + q 2 = 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 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 In this section, coordinate transformations are given for the case when charges have a nonzero sum, i.e. q 1 + q 2 0. The case for q 1 + q 2 = 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 m 1 = 1, and q 1 = 1. In order to simplify notation, define the following quantities: 24

28 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. m = 1 + m 2 m 2 q = 1 + q 2 q 2 µ = B(1 + q 2 ) p θ = 2B(1 + q 2 )L + Px 2 + Py 2 ɛ = q 2B = q 1 µ (5.3) 1 + q 2 q 2 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 k e ( 0), and the initial total momentum p θ. The quantity ɛ 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, p r, φ, p φ ) defined as follows: 25

29 x 1 = P y µ + 1 q r cos θ 1 µ p1/2 φ cos ( 2µφ + θ) y 1 = P x µ + 1 q r sin θ 1 µ p1/2 φ sin ( 2µφ + θ) x 2 = P y µ q 1 q y 2 = P x µ q 1 q p 1x = q 1 p 1/2 φ q p 1y = q 1 q p 2x = 1 q p1/2 φ p 2y = 1 q p1/2 φ p 1/2 φ r cos θ 1 µ p1/2 φ cos ( 2µφ + θ) r sin θ 1 µ p1/2 φ sin ( 2µφ + θ) sin ( 2µφ + θ) + p r cos θ (ɛr + p θ p φ ) sin θ µr cos ( 2µφ + θ) + p r sin θ 1 2 (ɛr + p θ p φ ) cos θ µr sin ( 2µφ + θ) p r cos θ 1 2 (ɛr + p θ p φ ) sin θ µr cos ( 2µφ + θ) p r sin θ (ɛr + p θ p φ ) cos θ (5.4) µr The corresponding inverse transformations are: r = (x 1 x 2 ) 2 + (y 1 y 2 ) 2 p r = [p 1x (q 1)p 2x ](x 1 x 2 ) + [p 1y (q 1)p 2y ](y 1 y 2 ) qr φ = 1 ( ) 2µ arctan (p1x + p 2x )(x 1 x 2 ) + (p 1y + p 2y )(y 1 y 2 ) (p 1y + p 2y )(x 1 x 2 ) (p 1x + p 2x )(y 1 y 2 ) p φ = (p 1x + p 2x ) 2 + (p 1y + p 2y ) 2 (5.5) Note that r > 0 and p φ > 0 must hold. There are singularities located at r = 0 and at p φ = 0. Also note that φ is periodic on the interval (0, π ). µ 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, p r, φ, and p φ 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 A far easier way is to derive the equations of motion directly from the Hamiltonian by using Hamilton s equations, 26

30 ṙ = H p r, ṗ r = H r, φ = H p φ, ṗ φ = H φ When the coordinate transformation is applied, the new Hamiltonian reads H = m 2 p2 r + m (p θ p φ ) 2 + mɛ2 8 µ 2 r 2 8 r2 + mɛ 4µ (p θ + p φ ) + ( 1 + (q 1)r + 1 m ) [ q ( 1 m q ) ( ) q 2 2q p r sin (2µφ) 1 2 (ɛr + p θ p φ ) cos (2µφ) µr ] p φ p 1/2 φ (5.6) The equations of motion that result are: ṙ = mp r + (1 m q ṗ r = m 4 ) sin (2µφ)p1/2 φ (p θ p φ ) 2 mɛ2 µ 2 r 3 4 r + 1 (q 1)r (1 m q )(ɛ p θ p φ ) cos (2µφ)p 1/2 µr 2 φ φ = (1 m q )(q 2 2q ) + m 4 ( ɛ µ p θ p φ µ 2 r 2 ) (1 m q ) [ p r sin (2µφ) 1 2 (ɛr + p ] θ 3p φ ) cos (2µφ) µr ṗ φ = 2µ(1 m [ q ) p r cos (2µφ) (ɛr + p ] θ p φ ) sin (2µφ) µr 5.5 Dynamics for physical cases p 1/2 φ p 1/2 φ (5.7) 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, the antiparticles p and d are both included in a case. The case of d-α is very unlikely to occur in nature, but is interesting because of the fact that the gyrofrequencies sum to zero, i.e. q 1 /m 1 + q 2 /m 2 = 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

31 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 p-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-p r plane. These projections are shown in Fig 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-p r plane with a coloring scheme that represents the p φ value at each point. The resulting Poincaré sections are shown in Fig. 5.3 and Fig 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 5.1: Some possible two-body problems with q 1 + q m 2 /m 1 q 2 /q 1 m q Chaotic? p p No, integrable p d No p h 3 2 4/3 1.5 Yes p t 3 1 4/3 2 Yes p α No d t /3 2 Yes d α No, integrable e α No p α Yes d α No Table 5.2: Parameters and initial conditions for chaotic cases 1 2 m q µ p θ k e [r, p r, φ, p φ ] t=0 Remarks h p [2.8, 0, 0, ] λ t p [0.4, 0, 0, ] λ 0.02 d t 5/ [4.5, 0, 0, ] λ p α [3.0, 0, 0, ] Numerically unstable 28

32 Figure 5.2: Poincare 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 Poincare 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

33 Figure 5.4: The projected Poincaré section for the two-body system of a deuteron and triton taken at φ = π 2 µ. The color of each point indicates the p φ position. The Poincaré section for p-α is shown in Fig One difference from the other cases is that the Poincaré section now extends to infinity in p r 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. No chaos was found for case of d-α. This is an interesting case because the gyrofrequencies sum to zero, so q 1 /m 1 + q 2 /m 2 = 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

34 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 p φ position. Note that as r 0, p r. 31