Doctor of Philosophy / Department of Physics / Brett D. Esry / The Efimov effect plays a central role in few-body systems at ultracold temperature and has thus accelerated a lot of studies on its manifestation in the collisional stability of the quantum degenerate gases. Near broad Feshbach resonances, Efimov physics has been studied both theoretically and experimentally through the zero-energy scattering observables. We have extended the theoretical studies of Efimov physics to a much broader extent. In particular, we have investigated the three-body Efimov physics near narrow Feshbach resonances and have also identified the Efimov features beyond the zero temperature limit. We have found, near a narrow Feshbach resonance, the non-trivial contribution from both of the resonance width and the short-range physics to the three-body recombination and vibrational dimer relaxation. Remarkably, the collisional stability of the Feshbach molecules are found to be opposite to that near the broad resonances: an increased stability for molecules made by bosons and a decreased stability for those made by fermions. The universal physics observed near the narrow Feshbach resonances is further found not to be limited to the zero temperature observables. We have found that the general features of Efimov physics and those pertaining to a narrow resonance are manifested in different energy ranges above zero temperature. This opens the opportunity to observe Efimov physics by changing the collisional energy while keeping the atomic interaction fixed. The landscape of the universal Efimov physics is thus delineated in both of the interaction and the energy domain. We have also investigated Efimov physics in heteronuclear four-body systems where the complexity can be reduced by approximations. In particular, we have proposed ways for controllable production of the Efimov tri-atomic molecules by three-body or four-body recombinations involving four atoms. We have also confirmed the existence of four-body Efimov effect in a system of three heavy particles and one light particle, which has resolved a decade-long controversy on this topic. Finally, we have studied the collisional properties of four identical bosons in 1D, which is important to the experiments on the quantum gases confined in the 1D optical lattices.

The double hypernuclear systems are studied within the context of the hyperspherical approach. Possible bound states of these systems are sought as zeros of the corresponding three-body Jost function in the complex energy plane. Hypercentral potentials for the system are constructed from known potentials in order to determine bound states of the system. Calculated binding energies for double- hypernuclei having A = 4 − 20, are presented. / Physics / M.Sc. (Physics)

An integral part of NASA's vision for the coming years is a sustained infrastructure in cislunar space. The current baseline trajectory for this facility is a Near Rectilinear Halo Orbit (NRHO), a periodic orbit in the Circular Restricted Three-Body Problem. One of the goals of the facility is to serve as a proving ground for human spaceflight operations in deep space. Thus, this investigation focuses on transfers between the baseline NRHO and a family of periodic orbits that originate from a period-doubling bifurcation along the halo family. This new family of orbits has been termed the ``butterfly" family. This investigation also provides an overview of the evolution for a large subset of the butterfly family. Transfers to multiple subsets of the family are found by leveraging different design strategies and techniques from dynamical systems theory. The different design strategies are discussed in detail, and the transfers to each of these regions are compared in terms of propellant costs and times of flight.

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The double hypernuclear systems are studied within the context of the hyperspherical approach. Possible bound states of these systems are sought as zeros of the corresponding three-body Jost function in the complex energy plane. Hypercentral potentials for the system are constructed from known potentials in order to determine bound states of the system. Calculated binding energies for double- hypernuclei having A = 4 − 20, are presented. / Physics / M.Sc. (Physics)

Arnold a démontré l'existence de solutions quasipériodiques dans le problème planétaire à trois corps plan, sous réserve que la masse de deux des corps, les planètes, soit petite par rapport à celle du troisième, le Soleil. Cette condition de petitesse dépend de façon cachée de la largeur d'analyticité de l'hamiltonien du problème, dans des coordonnées transcendantes. Hénon ex- plicita un rapport de masses minimal nécessaire à l'application du théorème de Arnold. L'objectif de cette thèse sera de donner une condition suffisante sur les rapports de masses. Une première partie de mon travail consiste à estimer cette largeur d'analyticité, ce qui passe par l'étude précise de l'équation de Kepler dans le complexe, ainsi que celle des singularités complexes de la fonction perturbatrice. Une deuxième partie consiste à mettre l'hamiltonien sous forme normale, dans l'optique d'une application du théorème KAM (du nom de Kolmogorov-Arnold-Moser). Il est nécessaire d'étudier le hamiltonien séculaire pour le mettre sous une forme normale adéquate. On peut alors quantifier la non-dégénérescence de l'hamiltonien séculaire, ainsi qu'estimer la perturbation. Enfin, il faut démontrer une version quantitative fine du théorème KAM, inspirée de Pöschel, avec des constantes explicites. A l'issue de ce travail, il est montré que le théorème KAM peut être appliqué pour des rapports de masses entre planètes et étoile de l'ordre de 10^(-85). / Arnold showed the existence of quasi-periodic solutions in the plane planetary three-body prob- lem, provided that the mass of two of the bodies, the planets, is small compared to the mass of the third one, the Sun. This smallness condition depends in a sensitive way on the analyticity widths of the Hamiltonian of the three-body problem, expressed with the help of some tran- scendental coordinates. Hénon gave a minimal ratio of masses necessary to the application of Arnold’s theorem. The main objective of this thesis is to determine a sufficient condition on this ratio. A first part of this work consists in estimating these analyticity widths, which requires a precise study of the complex Kepler equation, as well as the complex singularities of the disturb- ing function. A second part consists in reworking the Hamiltonian to put it under normal form, in order to apply the KAM theorem (KAM standing for Kolmogorov-Arnold-Moser). In this aim, it is essential to work with the secular Hamiltonian to put it under a suitable normal form. We can then quantify the non-degeneracy of the secular Hamiltonian, as well as estimate the perturbation. Finally, it is necessary to derive a quantitative version of the KAM theorem, in order to identify the hypotheses necessary for its application to the plane three-body problem. After this work, it is shown that the KAM theorem can be applied for a ratio of masses that is close to 10^(−85) between the planets and the star.

Durant le milieu des années quatre-vingt, le développement des faisceaux radioactifs a permis l’étude de noyaux exotiques situés à la limite de la stabilité nucléaire. Parmi ceux-ci, l’6He possède une structure particulière constituée d’un cœur α et de deux neutrons de valence faiblement liés. Il fait partie des noyaux à halo. Il s’agit également d’un système possédant au moins un état lié à trois amas (α+n+n) alors qu’aucun des sous-systèmes à deux amas (α+n ou n+n) n’est lié. Un tel système est dit borroméen. Deux autres exemples de systèmes borroméens sont le 9Be=n+α+α et le 12C=α+α+α. Les noyaux borroméens possèdent un ou deux états liés dont les propriétés sont maintenant bien connues. En revanche, les propriétés des états du continu le sont moins. Il existe aussi des systèmes à trois amas non liés mais caractérisés par une ou plusieurs résonances étroites à basse énergie. Par exemple, nous pouvons citer le 6Be=α+p+p et le 9B=p+α+α qui sont les noyaux miroirs de l’6He et du 9Be. L’étude des états du continu à trois corps est un sujet important de la recherche en physique nucléaire actuelle.<p><p>D’un point de vue théorique, les modèles en amas permettent de décrire les noyaux possédant une structure à plusieurs corps. En particulier, les modèles en amas microscopiques tiennent compte de tous les nucléons et le principe de Pauli est traité de manière exacte à l’aide de l’opérateur d’antisymmétrisation. Les modèles microscopiques, plus réalistes, requièrent de longs temps de calculs et sont difficiles à implémenter.<p><p>Dans le présent travail, nous avons développé une méthode de calcul semi-analytique plus rapide et plus précise qu’une méthode purement numérique. Notre modèle est basé sur la méthode des coordonnées génératrices dans le formalisme hypersphérique. Les états du continu sont étudiés à l’aide de la méthode des déphasages. Ces derniers sont obtenus à partir de la méthode de la matrice R microscopique.<p><p>Avec ce modèle, nous avons étudié les effets d’antisymmétrisation dans les noyaux d’6He, 6Be, 9Be, 9B et 12C. Nous avons montré que ces effets restent non négligeables mêmes à grandes distances dans le cas des noyaux 9Be, 9B et du 12C. <p><p>Nous avons ensuite étudié les états du continu du 6He et du 6Be. Nous avons déterminé des valeurs théoriques pour les propriétés des résonances connues expérimentalement. Nos résultats sont en bon accord avec l’expérience. Nous avons également observé des structures larges dans les déphases associés aux états 0+ et 1- de l’6He et 1- du 6Be.<p><p>Finalement, nous avons étudié les états du 12C à travers la collision 8Be+α. Dans notre calcul, le 8Be est considéré comme un système comportant deux amas α. Nous avons montré que les états 2+ et 4+ du 8Be devaient être pris en compte pour une étude appropriée des états 0+, 2+ et 4+ du 12C. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Le premier objectif de cette thèse est de bien comprendre les propriétés de la dynamique du problème circulaire restreint des trois corps et de les utiliser pour calculer des missions pour satellites pourvus de moteurs à faible poussée. Une propriété fondamentale est l'existence de variétés invariantes associées à des orbites périodiques autour des points de \bsc{Lagrange}. En suivant l'idée de l'\emph{Interplanetary Transport Network}, la connaissance et le calcul des variétés invariantes, comme courants gravitationnels, sont cruciaux pour le \emph{design} de missions spatiales. Une grande partie de ce travail de thèse est consacrée au développement de méthodes numériques pour calculer le transfert entre variétés invariantes de façon optimale. Le coût que l'on cherche alors à minimiser est la norme $L^{1}$ du contrôle car elle est équivalente à minimiser la consommation des moteurs. On considère aussi la norme $L^{2}$ du contrôle car elle est, numériquement, plus facile à minimiser. Les méthodes numériques que nous utilisons sont des méthodes indirectes rendues plus robustes par des méthodes de continuation sur le coût, sur la poussée, et sur l'état final. La mise en œuvre de ces méthodes repose sur l'application du Principe du Maximum de Pontryagin. Les algorithmes développés dans ce travail permettent de calculer des missions réelles telles que des missions entre des voisinages des points de \bsc{Lagrange}. L'idée principale est d'initialiser un tir multiple avec une trajectoire admissible composée de parties contrôlées (des transferts locaux) et de parties non-contrôlées suivant la dynamique libre (les variétés invariantes). Les méthodes mises au point ici, sont efficaces et rapides puisqu'il suffit de quelques minutes pour obtenir la trajectoire optimale complète. Enfin, on développe une méthode hybride, avec à la fois des méthodes directes et indirectes, qui permettent d'ajuster la positions des points de raccord sur les variétés invariantes pour les missions à grandes variations d'énergie. Le gradient de la fonction valeur est donné par les valeurs des états adjoints aux points de raccord et donc ne nécessite pas de calculs supplémentaire. Ainsi, l'implémentation de algorithme du gradient est aisée. / The first objective of this work is to understand the dynamical properties of the circular restricted three body problem in order to use them to design low consumption missions for spacecrafts with a low thrust engine. A fundamental property is the existence of invariant manifolds associated with periodic orbits around Lagrange points. Following the Interplanetary Transport Network concept, invariant manifolds are very useful to design spacecraft missions because they are gravitational currents. A large part of this work is devoted to designing a numerical method that performs an optimal transfer between invariant manifolds. The cost we want to minimize is the $L^{1}$-norm of the control which is equivalent to minimizing the consumption of the engines. We also consider the $L^{2}$-norm of the control which is easier to minimize numerically. The numerical methods are indirect ones coupled with different continuations on the thrust, on the cost, and on the final state, to provide robustness. These methods are based on the application of the Pontryagin Maximum Principal. The algorithms developed in this work allow for the design of real life missions such as missions between the realms of libration points. The basic idea is to initialize a multiple shooting method with an admissible trajectory that contains controlled parts (local transfers) and uncontrolled parts following the natural dynamics (invariant manifolds). The methods developed here are efficient and fast (less than a few minutes to obtain the whole optimal trajectory). Finally, we develop a hybrid method, with both direct and indirect methods, to adjust the position of the matching points on the invariant manifolds for missions with large energy gaps. The gradient of the value function is given by the values of the costates at the matching points and does not require any additional computation. Hence, the implementation of the gradient descent is easy.

The wear process in three-body contact causes problems of abrasion such as volume loss and changes of geometry of the triboelements. The wear problem leads to increased failure and high costs for repairing or replacing equipment. To understand the nature of the wear behaviour and to predict the wear rate in advance, experimental investigation and numerical simulation of the wear process are required. In this work, the wear process is analysed and the influencing parameters governing the wear behaviour are investigated experimentally to develop a new wear model. Main influential factors are considered such as kinematics of abrasive particles, contact stiffness of the particle layer, friction characteristics, and wear factors. The experiments to study kinematics of particle layers are performed on a new observation tester. To define the contact stiffness of abrasive particles, experiments are conducted by the uniaxial spindle compression tester. Moreover, a tribometer test rig with applied load up to 200 N and velocity up to 1000 mm/s is used to investigate the friction characteristics and the wear behaviour of three-body tribosystem. Analyses of influential factors on the wear behaviour in dependency of predefined process parameter are carried out. Additionally, based on the results of the experimental investigations, approximation equations representing the relation of the influential factors and the process parameters are determined. A three-body wear model is build up to represent the wear behaviour by physical wear laws. Furthermore, these approximation equations and the relevant parameters obtained by experimental investigations are included in the Fleischer’s wear equation to simulate the wear process. With the coupled model the wear process of the sample can be simulated twodimensional over the sliding distance. It is possible to predict the wear depth and the wear intensity, which can be used to estimate the wear rate. Additionally, from the results of the wear simulation the worn surface and the local contact pressure in the contact region are determined which provide a deeper insight into the wear process. With this simulation the understanding of the wear behaviour can be improved which is important to solve wear problems.

<p>An increasing interest in lunar exploration calls for low-cost techniques of reaching the Moon. Ballistic lunar transfers are long duration trajectories that leverage solar perturbations to reduce the multi-body energy of a spacecraft upon arrival into cislunar space. An investigation is conducted to explore methods of constructing ballistic lunar transfers. The techniques employ dynamical systems theory to leverage the underlying dynamical flow of the multi-body regime. Ballistic lunar transfers are governed by the gravitational influence of the Earth-Moon-Sun system; thus, multi-body gravity models are employed, i.e., the circular restricted three-body problem (CR3BP) and the bicircular restricted four-body problem (BCR4BP). The Sun-Earth CR3BP provides insight into the Sun’s effect on transfers near the Earth. The BCR4BP offers a coherent model for constructing end-to-end ballistic lunar transfers. Multiple techniques are employed to uncover ballistic transfers to conic and multi-body orbits in cislunar space. Initial conditions to deliver the spacecraft into various orbits emerge from Periapse Poincaré maps. From a chosen geometry, families of transfers from the Earth to conic orbits about the Moon are developed. Instantaneous equilibrium solutions in the BCR4BP provide an approximate for the theoretical minimum lunar orbit insertion costs, and are leveraged to create low-cost solutions. Trajectories to the <i>L</i>2 2:1 synodic resonant Lyapunov orbit, <i>L</i>2 2:1 synodic resonant Halo orbit, and the 3:1 synodic resonant Distant Retrograde Orbit (DRO) are investigated.</p>