The main asteroid belt spans ~2-4 AU in heliocentric distance and is sparsely populated by rocky debris. The dynamical structure of the main belt records clues to past events in solar system history. Evidence from the structure of the Kuiper belt, an icy debris belt beyond Neptune, suggests that the giant planets were born in a more compact configuration and later experienced planetesimal-driven planet migration. Giant planet migration caused both mean motion and secular resonances to sweep across the main asteroid belt, raising the eccentricity of asteroids into planet-crossing orbits and depleting the belt. I show that the present-day semimajor axis and eccentricity distributions of large main belt asteroids are consistent with excitation and depletion due to resonance sweeping during the epoch of giant planet migration. I also use an analytical model of the sweeping of the &nu;₆ secular resonance, to set limits on the migration speed of Saturn.After planet migration, dynamical chaos became the dominant loss mechanism for asteroids with diameters D>10 km in the current asteroid belt. I find that the dynamical loss history of test particles from this region is well described with a logarithmic decay law. My model suggests that the rate of impacts from large asteroids may have declined by a factor of three over the last ~3 Gy, and that the present-day impact flux of D>10 km objects on the terrestrial planets is roughly an order of magnitude less than estimates used in crater chronologies and impact hazard risk assessments.Finally, I have quantified the change in the solar wind <super>6</super>Li/<super>7</super>Li ratio due to the estimated in-fall of chondritic material and enhanced dust production during the epoch of planetesimal-driven giant planet migration. The solar photosphere is currently highly depleted in lithium relative to chondrites, and <super>6</super>Li is expected to be far less abundant in the sun than <super>7</super>Li due to the different nuclear reaction rates of the two isotopes. Evidence for a short-lived impact cataclysm that affected the entire inner solar system may be found in the composition of implanted solar wind particles in lunar regolith.

Waveguides in Euclidian space are piecewise path connected subsets of R^n that can be written as the union of a compact domain with boundary and their cylindrical ends. The compact and non-compact parts share a common boundary. This boundary is assumed to be Lipschitz, piecewise smooth and piecewise path connected. The ends can be thought of as the cartesian product of the boundary with the positive real half-line. A notable feature of Euclidian waveguides is that the scattering matrix admits a meromorphic continuation to a certain Riemann surface with a countably infinite number of leaves [2], which we will describe in detail and deal with. In order to construct this meromorphic continuation, one usually first constructs a meromorphic continuation of the resolvent for the Laplace operator. In order to do this, we will use a well known glueing construction (see for example [5]), which we adapt to waveguides. The construction makes use of the meromorphic Fredholm theorem and the fact that the resolvent for the Neumann Laplace operator on the ends of the waveguide can be easily computed as an integral kernel. The resolvent can then be used to construct generalised eigenfunctions and, from them, the scattering matrix. Being in possession of the scattering matrix allows us to calculate resonances; poles of the scattering matrix. We are able to do this using a combination of numerical contour integration and Newton s method.

The nonlinear nonplanar steady-state responses of cantilever beams to direct and parametric harmonic excitations are investigated using perturbation techniques. Modal interactions between the bending-bending and bending-bending-twisting motions are studied. Using a variational formulation, we obtained the governing equations of motion and associated boundary conditions for monoclinic composite and isotropic metallic inextensional beams. The method of multiple scales is applied either to the governing system of equations and associated boundary conditions or to the Lagrangian and virtual-work term to determine the modulation equations that govern the slow dynamics of the responses. These equations are shown to exhibit symmetry properties, reflecting the conservative nature of the beams in the absence of damping. It is popular to first discretize the partial-differential equations of motion and then apply a perturbation technique to the resulting ordinary-differential equations to determine the modulation equations. Due to the presence of quadratic as well as cubic nonlinearities in the governing system for the bending-bending-twisting oscillations of beams, it is shown that this approach leads to erroneous results. Furthermore, the symmetries are lost in the resulting equations. Nontrivial fixed points of the modulation equations correspond, generally, to periodic responses of the beams, whereas limit-cycle solutions of the modulation equations correspond to aperiodic responses of the beams. A pseudo-arclength scheme is used to determine the fixed points and their stability. In some cases, they are found to undergo Hopf bifurcations, which result in limit cycles. A combination of a long-time integration, a two-point boundary-value continuation scheme, and Floquet theory is used to determine in detail branches of periodic and chaotic solutions and assess their stability. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations. The chaotic attractors undergo attractor-merging and boundary crises as well as explosive bifurcations. For certain cases, it is determined that the response of a beam to a high-frequency excitation is not necessarily a high-frequency low-amplitude oscillation. In fact, low-frequency high-amplitude components that dominate the responses may be activated by resonant and nonresonant mechanisms. In such cases, the overall oscillations of the beam may be significantly large and cannot be neglected. / Ph. D.

Much effort in recent times has been devoted to the study of low energy transport in multibody gravitational systems. Despite continuing advancements in computational abilities, such studies can often be demanding or time consuming in the three-body and four-body settings. In this work, the Hamiltonian describing the planar circular restricted three-body problem is rewritten for systems having small mass parameters, resulting in a 2D symplectic twist map describing the evolution of a particle's Keplerian motion following successive close approaches with the secondary. This map, like the true dynamics, admits resonances and other invariant structures in its phase space to be analyzed. Particularly, the map contains rotational invariant circles reminiscent of McGehee's invariant tori blocking transport in the true phase space, adding a new quantitative description to existing chaotic zone estimates about the secondary. Used in a patched three-body setting, the map also serves as a tool for investigating transfer trajectories connecting loose captures about one secondary to the other without any propulsion systems. Any identified initial conditions resulting in such a transfer could then serve as initial guesses to be iterated upon in the continuous system. In this work, the projection of the McGehee torus within the interior realm is identified and quantified, and a transfer from Earth to Venus is exemplified. / Master of Science / The transport of a particle between celestial bodies, such as planets and moons, is an important phenomenon in astrodynamics. There are multiple ways to mediate this objective; commonly, the motion can be influenced directly via propulsion systems or, more exotically, by utilizing the passive dynamics admitted by the system (such as gravitational assists). Gravitational assists are traditionally modelled using two-body dynamics. That is, a space- craft or particle performs a flyby within that body's sphere of influence where momentum is exchanged in the process. Doing so provides accurate and reliable results, but the design space effecting the desired outcome is limited when considering the space of all possibilities. Utilizing three-body dynamics, however, provides a significant improvement in the fidelity and variety of trajectories over the two-body approach, and thus a broader space through which to search. Through a series of approximations from the three-body problem, a discrete map describing the evolution of nearly Keplerian orbits through successive close encounters with the body is formed. These encounters occur outside of the body's sphere of influence and are thus uniquely formed from three-body dynamics. The map enables computation of a trajectory's fate (in terms of transit) over numerical integration and also provides a boundary for which transit is no longer possible. Both of these features are explored to develop an algorithm able to rapidly supply guesses of initial conditions for a transfer in higher fidelity models and further develop the existing literature on the chaotic zone surrounding the body.

This Master’s dissertation focuses on engineering artificial nanostructures, namely, arrays of metamolecules on a substrate (metasurfaces), with the goal to achieve the desired linear and nonlinear optical responses. Specifically, a simple analytical model capable of predicting optical nonlinearity of an individual metamolecule has been developed. The model allows one to estimate the nonlinear optical response (linear polarizability and nonlinear hyperpolarizabilities) of a metamolecule based on the knowledge of its shape, dimensions, and material. In addition, a new experimental approach to measure hyperpolarizability has also been investigated. As another research effort, a 2D plasmonic metasurface with the collective behaviour of the metamolecules known as hybrid plasmonic-Fabry-Perot cavity and surface lattice resonances was designed, fabricated and optically characterized. We experimentally discovered a novel way of coupling the microcavity resonances and the diffraction orders of the plasmonic metamolecule arrays with the low-quality plasmon resonance to generate multiple sharp resonances with the higher quality factors. Finally, we experimentally observed and demonstrated a record ultra-high-Q surface lattice resonance from a plasmonic metasurface. These novel results can be used to render highly efficient nonlinear optical responses relying on high optical field localization, and can serve as the stepping stone towards achieving practical artificial nanophotonic devices with tailored linear and nonlinear optical responses.

<p>When studying the structure formation and fragmentation of complex atomic and nuclear systems it is preferable to start with simple systems where all details can be explored. Some of the knowledge gained from studies of atomic dimers can be generalised to more complex systems. Adding a third atom to an atomic dimer gives a first chance to study how the binding between two atoms is affected by a third. Few-body physics is an intermediate area which helps us to understand some but not all phenomena in many-body physics.</p><p>Very weakly bound, spatially very extended quantum systems with a wave function reaching far beyond the classical forbidden region and with low angular momentum are characterized as halo systems. These unusual quantum systems, first discovered in nuclear physics may also exist in systems of neutral atoms.</p><p>Since the first clear theoretical prediction in 1977, of a halo system possessing an Efimov state, manifested in the excited state of the bosonic van der Waals helium trimer ⁴₂He₃, small helium and different spin-polarised halo hydrogen clusters and their corresponding isotopologues have been intensively studied the last three decades.</p><p>In the work presented here, the existence of the spin-polarized tritium trimer ground state, ³₁H₃, is demonstrated, verifying earlier predictions, and the system's properties elucidated. Detailed analysis has found no found evidence for other bound states and shape resonances in this system. The properties of the halo helium trimers, ⁴₂He₃ and ⁴₂He₂-³₂He have been investigated. Earlier predictions concerning the ground state energies and structural properties of these systems are validated using our three-dimensional finite element method. In the last part of this work we present results on the bound states and structural properties of the van der Waals bosonic atomic trimers Ne₃ and Ar₃. We believe to be the first to find evidence of a possible shape resonance just above the three-body dissociation limit of the neon trimer.</p>

K+Λ photoproduction provides the best possibility for a model independent extraction of the photoproduction process and contributing resonances. To do this, it is vital that cross section measurements are well understood. This thesis presents pγ K+Λ differential cross sections from the reaction threshold, to an invariant centre of mass energy of 1.87 GeV. The data was taken at MAMI-C electron microtron facility in Mainz, Germany, during July 2007 and April 2009. The 1.5 GeV MAMI-C electron beam was used to produce an energy tagged bremsstrahlung photon beam with a maximum energy of 1.4 GeV and an intensity of 105γs-1MeV-1. The beam impinged upon a liquid hydrogen target, with reaction products detected in two segmented calorimeter arrays; the Crystal Ball detector and TAPS. This work pioneers a new method of K+ detection in segmented calorimeters, in which the K+ was identified from the signature of its weak decay inside the crystals of the calorimeter. This proved to be an excellent method of isolating K+ and accessing strangeness photoproduction channels, with good agreement between experimental and Geant4 simulated data. A novel method in seperating K+Λ and K+Σ0 final states was also developed by identifying the photon from the decay: Σ0 → Λγ. The intense photon beam at the MAMI-C facility enabled differential cross section data with greater invariant mass resolution than previous measurements. The new measurement near threshold imposes important constraints to effective field theories based on the approximate chiral symmetry of QCD. At higher centre of mass energies it also addresses the current problem of discrepant data sets and will form an important constraint on partial wave analysis for the nucleon excitation spectrum. As such, this work contributes to a major world wide programme aiming to extract the excitation spectrum of the nucleon and to understand the dynamics and interactions of its constituents. The greater statistics near threshold, and particularly at backwards K+ centre of mass angles will give new valuable constraints to contributions from meson and hyperonic resonances on the reaction mechanism. The high resolution of the photon beam (approximately 2 MeV) also allows the first search for narrow resonances coupling to KΛ final states. The differential cross sections give good agreement with Kaon-MAID partial wave solutions, apart from at backward angles close to threshold, where the data is lower. Near threshold, the data agrees with calculations from the chiral unitary framework of Borasoy et al, tending to be in better accordance with the model than previous data. No strong structure from potential narrow resonance states was observed over the centre of mass energy region of 1650-1700 MeV, where narrow structure has been observed in recent η photoproduction of the neutron.

In the thesis we study a method for determining resonance energies - gen- eralization of the method of analytical continuation in the coupling constant, which is based on continuation of the coupling constant λ as a function of the momentum k. A formula for λ(k) is derived for spherically symmetric potential consisting of finite number of δ-functions and its Taylor series is studied. Taylor series of λ(k) and its asymptotic behavior is studied for sep- arable potential. Proper choice of added potential parameters is studied on examples. A method for determining λ(k) poles is described for spherically symmetric potential with added δ-function. It is tested whether the knowl- edge of λ(k) poles can be useful to improve the accuracy of the determination of the resonance parameters of the original potential.

A existência de um méson escalar-isoescalar leve, conhecido como ?, foi proposta pela primeira vez na década de 60. A partícula tinha então um papel importante na construção teórica das interações ?? mas, apesar dos esforços experimentais, ela não foi detectada nos anos que se seguiram. Essa situação foi radicalmente alterada em 2001, quando uma ressonância escalar foi descoberta nos canais ?+?- do decaimento D+ -> ?+?-?+ e recebeu o rótulo ?(500). Sua existência é bem estabelecida hoje em dia. Contudo, no tratamento dos dados dos vários grupos experimentais são empregadas expressões com pouca base teórica e, por isso, os valores de sua massa e largura ainda são mal conhecidos. Neste tipo de decaimento, a formação da ressonância pode se dar no vértice fraco. Em sua subseqüente propagação, ocorrem as chamadas interações de estado final, cuja descrição não é trivial. Normalmente, essas interações não são levadas em conta de maneira criteriosa na análise de dados experimentais. Neste trabalho introduzimos uma função _(s) que descreve a propagação e decaimento da ressonância em presença das interações de estado final. No regime elástico, a fase de _(s) é determinada pelo chamado teorema de Watson, segundo o qual ela deve ser a mesma do espalhamento. Conseguimos estabelecer, sem ambigüidades, como a informação do espalhamento deve ser usada de forma a determinar não somente a fase de _(s), mas também seu módulo. Nosso principal resultado é uma expressão para _(s) em termos da fase elástica e de uma outra fase relacionada a uma integral de loop bem controlada. Três casos particulares foram explorados numericamente: os modelos sigma linear e não linear e ainda um modelo fenomenológico que leva em conta o acoplamento de canais p´?on-p´?on e k´aon-k´aon. Em consonância com a teoria quântica de campos, nosso resultado incorpora a unitariedade, considera a ressonância como grau de liberdade explícito e representa, ainda, uma generalização do procedimento usual de unitarizacao pela matriz K. Por permitir uma ligação clara entre espalhamento e produção, a função _(s) pode ser útil na análise de dados experimentais e ajudar na determinação da posição do pólo do ? e de outras ressonâncias escalares. / The existence of a light scalar-isoscalar meson, known as ?, was suggested in the 60\'s. This particle played an important role in the theoretical construction of ?? interactions but, in spite of all experimental effort, it failed to be detected. This scenario changed radically in 2001, when a scalar-isoscalar resonance was discovered in the ?+?- channel of the D+ -> ?+?-?+ decay and was called ?(500). Nowadays, its existence is rather well established. However, in the analysis of experimental data, expressions loosely based on theory are employed and therefore its mass and width are still not well known. In this kind of decay, the production of the resonance may occur at the weak vertex. When it propagates, final state interactions take place. Usually these interactions are not properly taken into account in data analysis. In this work, we introduce a function _(s), which describes the propagation and decay of the resonance in the presence of the final state interactions. In the elastic regime, the phase of _(s) is determined by the Watson\'s theorem, which states that it must be the same as the scattering phase. We were able to establish, unambiguously, how the information from scattering should be used to determine not only the phase of _(s) but also its modulus. Our main result is an expression for _(s) in terms of the elastic phase and another one related to a well controlled loop integral. Three special cases are explored numerically, namely: the linear and non linear sigma models and a phenomenological model that takes into account the coupling between pion-pion and kaon-kaon channels. In agreement with quantum field theory, our result encompasses unitarity, treats the resonance as an explicit degree of freedom and, moreover, corresponds to a generalisation of the usual K-matrix unitarization procedure. Since it represents a clear way to relate scattering and production, our function _(s) can be useful in data analysis and may be instrumental in the determination of the pole position of the ? as well as other scalar resonances.

When studying the structure formation and fragmentation of complex atomic and nuclear systems it is preferable to start with simple systems where all details can be explored. Some of the knowledge gained from studies of atomic dimers can be generalised to more complex systems. Adding a third atom to an atomic dimer gives a first chance to study how the binding between two atoms is affected by a third. Few-body physics is an intermediate area which helps us to understand some but not all phenomena in many-body physics. Very weakly bound, spatially very extended quantum systems with a wave function reaching far beyond the classical forbidden region and with low angular momentum are characterized as halo systems. These unusual quantum systems, first discovered in nuclear physics may also exist in systems of neutral atoms. Since the first clear theoretical prediction in 1977, of a halo system possessing an Efimov state, manifested in the excited state of the bosonic van der Waals helium trimer 42He3, small helium and different spin-polarised halo hydrogen clusters and their corresponding isotopologues have been intensively studied the last three decades. In the work presented here, the existence of the spin-polarized tritium trimer ground state, 31H3, is demonstrated, verifying earlier predictions, and the system's properties elucidated. Detailed analysis has found no found evidence for other bound states and shape resonances in this system. The properties of the halo helium trimers, 42He3 and 42He2-32He have been investigated. Earlier predictions concerning the ground state energies and structural properties of these systems are validated using our three-dimensional finite element method. In the last part of this work we present results on the bound states and structural properties of the van der Waals bosonic atomic trimers Ne3 and Ar3. We believe to be the first to find evidence of a possible shape resonance just above the three-body dissociation limit of the neon trimer.