Thesis (Ph.D.)--Hong Kong University of Science and Technology, 2004. / Includes bibliographical references (leaves 131-137). Also available in electronic version. Access restricted to campus users.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1970. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

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M.Sc. / Important ideas from dynamical systems theory and the restricted three-body problem are introduced. The intention is the application of dynamical systems theory techniques to the restricted three-body problem to better understand the phenomenon of gravitational capture. Chapter 1 gives a much deeper review of the purpose of this dissertation. Chapter 2 introduces and reviews important concepts from dynamical systems. Chapter 3 reviews the restricted three-body problem and all important aspects of the problem. In chapter 4 we define and study the phenomenon of gravitational capture. We take a novel approach by applying a symplectic method, namely the implicit midpoint method, to model trajectories in the restricted three-body problem. As far as we know, this is the first time such a method has actually been applied, with other authors preferring to apply explicit methods in trajectory modelling. In the closing of this chapter we review our whole discourse and suggest topics for future research. The disseration is concluded with two appendix chapters. In the first chapter we list all the computer code we have written for this dissertation. The second appendix chapter reviews the n-body problem and we show a full solution of the two-body problem.

Los problemas de dispersión de partículas, como son los de dos y tres cuerpos, tienen una relevancia crucial en física atómica, pues permiten describir diversos procesos de colisiones. Hoy en día, los casos de dos cuerpos pueden ser resueltos con el grado de precisión numérica que se desee. Los problemas de dispersión de tres partículas cargadas son notoriamente más difíciles pero aún así algo similar, aunque en menor medida, puede establecerse. El objetivo de este trabajo es contribuir a la comprensión de procesos Coulombianos de dispersión de tres cuerpos desde un punto de vista analítico. Esto no solo es de fundamental interés, sino que también es útil para dominar mejor los enfoques numéricos que se actualmente se desarrollan dentro de la comunidad de colisiones atómicas. Para lograr este objetivo, proponemos aproximar la solución del problema con desarrollos en series de funciones adecuadas y expresables analíticamente. Al hacer esto, desarrollamos una serie de herramientas matemáticas relacionadas con funciones Coulombianas, ecuaciones diferenciales de segundo orden homogéneas y no homogéneas, y funciones hipergeométricas en una y dos variables. En primer lugar, trabajamos con las funciones de onda Coulombianas radiales y revisamos sus principales propiedades. Así, extendemos los resultados conocidos para dar expresiones analíticas de los coeficientes asociados al desarrollo, en serie de funciones de tipo Laguerre, de las funciones Coulombianas irregulares. También establecemos una nueva conexión entre los coeficientes asociados al desarrollo de la función Coulombiana regular y los polinomios de Meixner-Pollaczek. Esta relación nos permite deducir propiedades de ortogonalidad y clausura para estos coeficientes al considerar la carga como variable. Luego, estudiamos las funciones hipergeométricas de dos variables. Para algunas de ellas, como las funciones de Appell o las confluentes de Horn, presentamos expresiones analíticas de sus derivadas respecto de sus parámetros. También estudiamos un conjunto particular de funciones Sturmianas Generalizadas de dos cuerpos construidas considerando como potencial generador el potencial de Hulthén. Contrariamente al caso habitual, en el que las funciones Sturmianas se construyen numéricamente, las funciones Sturmianas de Hulthén poseen forma analítica. Sus propiedades matem´aticas pueden ser analíticamente estudiadas proporcionando una herramienta única para comprender y analizar los problemas de dispersión y sus soluciones. Además, proponemos un nuevo conjunto de funciones a las que llamamos funciones Quasi-Sturmianas. Estas funciones se presentan como una alternativa para expandir la solución buscada en procesos de dispersi´on de dos y tres cuerpos. Se definen como soluciones de una ecuación diferencial de tipo-Schrödinger, no homogénea. Por construcción, incluyen un comportamiento asintótico adecuado para resolver problemas de dispersión. Presentamos diferentes expresiones analíticas y exploramos sus propiedades matemáticas, vinculando y justificando los desarrollos realizados previamente. Para finalizar, utilizamos las funciones estudiadas (Sturmianas de Hulthén y Quasi-Sturmianas) en la resolución de problemas particulares de dos y tres cuerpos. La eficacia de estas funciones se ilustra comparando los resultados obtenidos con datos provenientes de la aplicación de otras metodologías. / Two and three-body scattering problems are of crucial relevance in atomic physics as they allow to describe different atomic collision processes. Nowadays, the two-body cases can be solved with any degree of numerical accuracy. Scattering problem involving three charged particles are notoriously difficult but something similar –though to a lesser extentcan be stated. The aim of this work is to contribute to the understanding of three-body Coulomb scattering problems from an analytical point of view. This is not only of fundamental interest, it is also useful to better master numerical approaches that are being developed within the collision community. To achieve this aim we propose to approximate scattering solutions with expansions on sets of appropriate functions having closed form. In so doing, we develop a number of related mathematical tools involving Coulomb functions, homogeneous and non-homogeneous second order differential equations, and hypergeometric functions in one and two variables. First we deal with the two-body radial Coulomb wave functions, and review their main properties. We extend known results to give in closed form the Laguerre expansions coefficients of the irregular solutions, and establish a new connection between the coefficients corresponding to the regular solution and Meixner-Pollaczek polynomials. This relation allows us to obtain an orthogonality and closure relation for these coefficients considering the charge as a variable. Then we explore two-variable hypergeometric functions. For some of them, such as Appell and confluent Horn functions, we find closed form for the derivatives with respect to their parameters. We also study a particular set of two-body Generalized Sturmian functions constructed with a Hulth´en generating potential. Contrary to the usual case in which Sturmian functions are numerically constructed, the Hulth´en Sturmian functions can be given in closed form. Their mathematical properties can thus be analytically studied providing a unique tool to investigate scattering problems. Next, we introduce a novel set of functions that we name Quasi-Sturmian functions. They constitute an alternative set of functions, given in closed form, to expand the sought after solution of two- and three-body scattering processes. Quasi-Sturmian functions are solutions of a non-homogeneous second order Schr¨odinger-like differential equation and have, by construction, the appropriate asymptotic behavior. We present different analytic expressions and explore their mathematical properties, linking and justifying the developed mathematical tools described above. Finally we use the studied Hulth´en Sturmian and Quasi-Sturmian functions to solve some particular two- and three-body scattering problems. The efficiency of these sets of functions is illustrated by comparing our results with those obtained by other methods

This dissertation expands across three topics the geometric theory of phase space transit in the circular restricted three-body problem (CR3BP) and its generalizations. The first topic generalizes the low energy transport theory that relies on linearizing the Lagrange points in the CR3BP to time-periodic perturbations of the CR3BP, such as the bicircular problem (BCP) and the elliptic restricted three-body problem (ER3BP). The Lagrange points are no longer invariant under perturbation and are replaced by periodic orbits, which we call Lagrange periodic orbits. Calculating the monodromy matrix of the Lagrange periodic orbit and transforming into eigenbasis coordinates reveals that the transport geometry is a discrete analogue of the continuous transport geometry in the unperturbed problem. The second topic extends the theory of low energy phase space transit in periodically perturbed models using a nonlinear analysis of the geometry. This nonlinear analysis relies on calculating the monodromy tensors, which generalize monodromy matrices in order to encode higher order behavior, about the Lagrange periodic orbit. A nonlinear approximate map can be obtained which can be used to iterate initial conditions within the linear eigenbasis, providing a computationally efficient means of distinguishing transit and nontransit orbits that improves upon the predictions of the linear framework. The third topic demonstrates that the recently-discovered "arches of chaos" that stretch through the solar system, causing substantial phase space divergence for high energy particles, may be identified with the stable and unstable manifolds to the singularities of the CR3BP. We also study the arches in terms of particle orbital elements and demonstrate that the arches correspond to gravity assists in the two-body limit. / Doctor of Philosophy / Suppose that we have a spacecraft and we want to model its motion under gravity. Depending upon what trade-offs we are willing to make between accuracy and complexity, we have several options at our disposal. For example, the restricted three-body problem (R3BP) and its generalizations prove useful in many real-world situations and are rich in theoretical power despite seeming mathematically simple. The simplest restricted three-body problem is the circular restricted three-body problem (CR3BP). In the CR3BP, two masses (like a star and a planet or a planet and a moon) orbit their common center of gravity in circular orbits, while a much smaller body (like a spacecraft) moves freely, influenced by the gravitational fields that the two masses create. If we add in an extra force that acts on the spacecraft in a periodic, cycling way, the regular CR3BP becomes a periodically-perturbed CR3BP. Examples of periodically-perturbed CR3BP's include the bicircular problem (BCP), which adds in a third mass that appears to orbit the center of the system from a distance, and the elliptic restricted three-body problem (ER3BP), which allows the two masses to orbit more realistically as ellipses rather than circles. The purpose of this dissertation is to determine how to select trajectories that move spacecraft between places of interest in restricted three-body models. We generalize existing theories of CR3BP spacecraft motion to periodically-perturbed CR3BP's in the first two topics, and then we investigate some new areas of research in the unperturbed CR3BP in the third topic. We utilize numerical computations and mathematical methods to perform these analyses.

The three-body bound state problem is considered using configuration-space Faddeev equations within the framework of the total-angular-momentum representation. Different three-body systems are considered, the main concern of the investigation being the i) calculation of binding energies for weakly bounded trimers, ii) handling of systems with a plethora of states, iii) importance of three-body forces in trimers, and iv) the development of a numerical technique for reliably handling three-dimensional integrodifferential equations. In this respect we considered the three-body nuclear problem, the 4He trimer, and the Ozone (16 0 3 3) system. In practice, we solve the three-dimensional equations using the orthogonal collocation method with triquintic Hermite splines. The resulting eigenvalue equation is handled using the explicitly Restarted Arnoldi Method in conjunction with the Chebyshev polynomials to improve convergence. To further facilitate convergence, the grid knots are distributed quadratically, such that there are more grid points in regions where the potential is stronger. The so-called tensor-trick technique is also employed to handle the large matrices involved. The computation of the many and dense states for the Ozone case is best implemented using the global minimization program PANMIN based on the well known MERLIN optimization program. Stable results comparable to those of other methods were obtained for both nucleonic and molecular systems considered. / Physics / D.Phil. (Physics)

The three-body bound state problem is considered using configuration-space Faddeev equations within the framework of the total-angular-momentum representation. Different three-body systems are considered, the main concern of the investigation being the i) calculation of binding energies for weakly bounded trimers, ii) handling of systems with a plethora of states, iii) importance of three-body forces in trimers, and iv) the development of a numerical technique for reliably handling three-dimensional integrodifferential equations. In this respect we considered the three-body nuclear problem, the 4He trimer, and the Ozone (16 0 3 3) system. In practice, we solve the three-dimensional equations using the orthogonal collocation method with triquintic Hermite splines. The resulting eigenvalue equation is handled using the explicitly Restarted Arnoldi Method in conjunction with the Chebyshev polynomials to improve convergence. To further facilitate convergence, the grid knots are distributed quadratically, such that there are more grid points in regions where the potential is stronger. The so-called tensor-trick technique is also employed to handle the large matrices involved. The computation of the many and dense states for the Ozone case is best implemented using the global minimization program PANMIN based on the well known MERLIN optimization program. Stable results comparable to those of other methods were obtained for both nucleonic and molecular systems considered. / Physics / D.Phil. (Physics)

Bound state systems consisting of three nonrelativistic particles are numerically studied. Calculations are performed employing two-body and three-body forces as input in the Hamiltonian in order to study the role or contribution of three-body forces to the binding in these systems. The resulting differential Faddeev equations are solved as three-dimensional equations in the two Jacobi coordinates and the angle between them, as opposed to the usual partial wave expansion approach. By expanding the wave function as a sum of the products of spline functions in each of the three coordinates, and using the orthogonal collocation procedure, the equations are transformed into an eigenvalue problem. The matrices in the aforementioned eigenvalue equations are generally of large order. In order to solve these matrix equations with modest and optimal computer memory and storage, we employ the iterative Restarted Arnoldi Algorithm in conjunction with the so-called tensor trick method. Furthermore, we incorporate a polynomial accelerator in the algorithm to obtain rapid convergence. We applied the method to obtain the binding energies of Triton, Carbon-12, and Ozone molecule. / Physics / M.Sc (Physics)