An overview of the historical developments of the paradigms of classical mechanics, the free particle, oscillator and the Kepler problem, is given ito (in terms of) their conserved quantities. Next, the orbits of the three paradigms are found from quadratic forms. The quadratic forms are constructed using first integrals found by the application of Poisson's theorem. The orbits are presented ito expanding surfaces defined by the quadratic forms. The Lie and Noether symmetries of the paradigms are investigated. The free particle is discussed in detail and an overview of the work done on the oscillator and Kepler problem is given. The Lie and Noether theories are compared from various aspects. A technical description of Lie groups and algebras is given. This provides a basis for a discussion of the historical development of the paradigms of mechanics ito their group properties. Lastly the paradigms are discussed ito of Quantum Mechanics. / Thesis (M.Sc.)-University of Natal, 1996.

This thesis is devoted to creating a systematic way of determining all inequivalent orthogonal coordinate systems which separate the Hamilton-Jacobi equation for a given natural Hamiltonian defined on three-dimensional spaces of constant, non-zero curvature. To achieve this, we represent the problem with Killing tensors and employ the recently developed invariant theory of Killing tensors. Killing tensors on the model spaces of spherical and hyperbolic space enjoy a remarkably simple form; even more striking is the fact that their parameter tensors admit the same symmetries as the Riemann curvature tensor, and thus can be considered algebraic curvature tensors. Using this property to obtain invariants and covariants of Killing tensors, together with the web symmetries of the associated orthogonal coordinate webs, we establish an equivalence criterion for each space. In the case of three-dimensional spherical space, we demonstrate the surprising result that these webs can be distinguished purely by the symmetries of the web. In the case of three-dimensional hyperbolic space, we use a combination of web symmetries, invariants and covariants to achieve an equivalence criterion. To completely solve the equivalence problem in each case, we develop a method for determining the moving frame map for an arbitrary Killing tensor of the space. This is achieved by defining an algebraic Ricci tensor. Solutions to equivalence problems of Killing tensors are particularly useful in the areas of multiseparability and superintegrability. This is evidenced by our analysis of symmetric potentials defined on three-dimensional spherical and hyperbolic space. Using the most general Killing tensor of a symmetry subspace, we derive the most general potential “compatible” with this Killing tensor. As a further example, we introduce the notion of a joint invariant in the vector space of Killing tensors and use them to characterize a well-known superintegrable potential in the plane. xiii

We pursue topics in optics that follow three major themes; time averaged dynamics with the associated Effective Hamiltonian theory, quantification and transformation of polarization, and periodicity within quantum circuits. Within the first theme, we develop a technique for finding the dynamical evolution in time of a time averaged density matrix. The result is an equation of evolution that includes an Effective Hamiltonian, as well as decoherence terms that sometimes manifest in a Lindblad-like form. We also apply the theory to examples of the AC Stark Shift and Three-Level Raman Transitions. In the theme of polarization, the most general physical transformation on the polarization state has been represented as an ensemble of Jones matrix transformations, equivalent to a completely positive map on the polarization matrix. This has been directly assumed without proof by most authors. We follow a novel approach to derive this expression from simple physical principles, basic coherence optics and the matrix theory of positive maps. Addressing polarization measurement, we first establish the equivalence of classical polarization and quantum purity, which leads to the identical structure of the Poincar\' and Bloch spheres. We analyze and compare various measures of polarization / purity for general dimensionality proposed in the literature, with a focus on the three dimensional case. % entanglement? In pursuit of the final theme of periodic quantum circuits, we introduce a procedure that synthesizes the circuit for the simplest periodic function that is one-to-one within a single period, of a given period p. Applying this procedure, we synthesize these circuits for p up to five bits. We conjecture that such a circuit will need at most n Toffoli gates, where p is an n-bit number. Moreover, we apply our circuit synthesis to compiled versions of Shor's algorithm, showing that it can create more efficient circuits than ones previously proposed. We provide some new compiled circuits for experimentalists to use in the near future. A layer of "classical compilation" is pointed out as a method to further simplify circuits. Periodic and compiled circuits should be helpful for creating experimental milestones, and for the purposes of validation.

We pursue topics in optics that follow three major themes; time averaged dynamics with the associated Effective Hamiltonian theory, quantification and transformation of polarization, and periodicity within quantum circuits. Within the first theme, we develop a technique for finding the dynamical evolution in time of a time averaged density matrix. The result is an equation of evolution that includes an Effective Hamiltonian, as well as decoherence terms that sometimes manifest in a Lindblad-like form. We also apply the theory to examples of the AC Stark Shift and Three-Level Raman Transitions. In the theme of polarization, the most general physical transformation on the polarization state has been represented as an ensemble of Jones matrix transformations, equivalent to a completely positive map on the polarization matrix. This has been directly assumed without proof by most authors. We follow a novel approach to derive this expression from simple physical principles, basic coherence optics and the matrix theory of positive maps. Addressing polarization measurement, we first establish the equivalence of classical polarization and quantum purity, which leads to the identical structure of the Poincar\' and Bloch spheres. We analyze and compare various measures of polarization / purity for general dimensionality proposed in the literature, with a focus on the three dimensional case. % entanglement? In pursuit of the final theme of periodic quantum circuits, we introduce a procedure that synthesizes the circuit for the simplest periodic function that is one-to-one within a single period, of a given period p. Applying this procedure, we synthesize these circuits for p up to five bits. We conjecture that such a circuit will need at most n Toffoli gates, where p is an n-bit number. Moreover, we apply our circuit synthesis to compiled versions of Shor's algorithm, showing that it can create more efficient circuits than ones previously proposed. We provide some new compiled circuits for experimentalists to use in the near future. A layer of "classical compilation" is pointed out as a method to further simplify circuits. Periodic and compiled circuits should be helpful for creating experimental milestones, and for the purposes of validation.

Un système intégrable semi-torique sur une variété symplectique de dimension 2n est un système intégrable dont le flot de n − 1 composantes de l'application moment est 2 -périodique. On obtient donc une action hamiltonienne du tore Tn−1. En outre, on demande que tous les points critiques du système soient non-dégénérés et sans composante hyperbolique. En dimension 4, San V˜u Ngo.c et Álvaro Pelayo ont étendu à ces systèmes semi-toriques les résultats célèbres d'Atiyah, Guillemin, Sternberg et Delzant concernant la classification des systèmes toriques. Dans cette thèse nous proposons une extension de ces résultats en dimension quelconque, à commencer par la dimension 6. Les techniques utilisées relèvent de l'analyse comme de la géométrie symplectique, ainsi que de la théorie de Morse dans des espaces différentiels stratifiés. Nous donnons d'abord une description de l'image de l'application moment d'un point de vue local, en étudiant les asymptotiques des coordonnées actionangle au voisinage d'une singularité foyer-foyer, avec le phénomène de monodromie du feuilletage qui en résulte. Nous passons ensuite à une description plus globale dans la veine des polytopes d'Atiyah, Guillemin et Sternberg. Ces résultats sont basés sur une étude systématique de la stratification donnée par les fibres de l'application moment. Avec ces résultats, nous établissons la connexité des fibres des systèmes intégrables semi-toriques de dimension 6 et indiquons comment nous comptons démontrer ce résultat en dimension quelconque.

Exact solution of the Schr&ouml / dinger equation with some potentials is obtained. The normal and supersymmetric cases are considered. Deformed ring-shaped potential is solved in the parabolic and spherical coordinates. By taking appropriate values for the parameter q, similar results are obtained for Hulth&eacute / n and exponential type screened potentials. Similarly, Morse, P&ouml / schl-Teller and Hulth&eacute / n potentials are solved for the supersymmetric case. Supersymmetric solution of PT-/non-PT-symmetric and non-Hermitian Morse potential is also studied. The Nikiforov-Uvarov and Hamiltonian Hierarchy methods are used in the calculations. Eigenfunctions and corresponding energy eigenvalues are calculated analytically. Results are in good agreement with ones obtained before.

Cette thèse aborde les problèmes de la modélisation et de la commande d'un micro-réseau courant continu (CC) en vue de la gestion énergétique optimale, sous contraintes et incertitudes. Le micro-réseau étudie contient des dispositifs de stockage électrique (batteries ou super-capacités), des sources renouvelables (panneaux photovoltaïques) et des charges (un système d'ascenseur motorise par une machine synchrone a aimant permanent réversible). Ces composants, ainsi que le réseau triphasé, sont relies a un bus commun en courant continu, par des convertisseurs dédies. Le problème de gestion énergétique est formule comme un problème de commande optimale qui prend en compte la dynamique du système, des contraintes sur les variables, des prédictions sur les prix, la consommation ou la production et des profils de référence.Le micro-réseau considère est un système complexe, de par l'hétérogénéité de ses composants, sa nature distribuée, la non-linéarité de certaines dynamiques, son caractère multi-physiques (électromécanique, électrochimique, électromagnétique), ainsi que la présence de contraintes et d'incertitudes. La représentation consistante des puissances échangées et des énergies stockées, dissipées ou fournies au sein de ce système est nécessaire pour assurer son opération optimale et fiable.Le problème pose est abordé via l'usage combine de la formulation hamiltonienne a port, de la platitude et de la commande prédictive économique base sur le modelé. Le formalisme hamiltonien a port permet de décrire les conservations de la puissance et de l'énergie au sein du micro-réseau explicitement et de relier les composants hétérogènes dans un même cadre théorique. Les non linéarités sont gérées par l'introduction de la notion de platitude démentielle et la sélection de sorties plates associées au modèle hamiltonien a ports. Les profils de référence sont génères a l'aide d'une para métrisation des sorties plates de telle sorte que l'énergie dissipée soit minimisée et les contraintes physiques satisfaites. Les systèmes hamiltoniens sur graphes sont ensuite introduits pour permettre la formulation et la résolution du problème de commande prédictive _économique a l'échelle de l'ensemble du micro-réseau CC. Les stratégies de commande proposées sont validées par des résultats de simulation pour un système d'ascenseur multi-sources utilisant des données réelles, identifiées sur base de mesures effectuées sur une machine synchrone. / The goals of this thesis is to propose modelling and control solutions for the optimal energy management of a DC microgrid under constraints. The studied microgrid system includes electrical storage units (e.g., batteries, supercapacitors), renewable sources (e.g., solar panels) and loads (e.g., an electro-mechanical elevator system). These interconnected components are linked to a three phase electrical grid through a DC bus and associated DC/AC converters. The optimal energy management is usually formulated as an optimal control problem which takes into account the system dynamics, cost, constraints and reference profiles.An optimal energy management for the microgrid is challenging with respect to classical control theories. Needless to say, a DC microgrid is a complex system due to its heterogeneity, distributed nature (both spatial and in sampling time), nonlinearity of dynamics, multi-physic characteristics, the presence of constraints and uncertainties. Moreover, the power-preserving structure and the energy conservation of a microgrid are essential for ensuring a reliable operation.This challenges are tackled through the combined use of port-Hamiltonian formulations, differential flatness, and economic Model Predictive Control.The Port-Hamiltonian formalism allows to explicitly describe the power-preserving structure and the energy conservation of the microgrid and to connect different components of different physical natures through the same formalism. The strongly non-linear system is then translated into a flat representation. Taking into account differential flatness properties, reference profiles are generated such that the dissipated energy and various physical constraints are taken into account. Lastly, we minimize the purchasing/selling electricity cost within the microgrid using the economic Model Predictive Control with the Port-Hamiltonian formalism on graphs.The proposed control designs are validated through simulation results.

Orientador: Prof. Dr. Maurício Firmino Silva Lima / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática , 2015. / Neste trabalho estudamos generalizações do Método de Melnikov para sistemas descontínuos no plano. Neste sentido, inicialmente abordamos esse problema como uma variação do estudo [1] onde um campo Hamiltoniano que admite um ciclo heteroclínico, cujo interior é folheado de órbitas periódicas, é perturbado por um campo Hamiltoniano não autonomo. Neste trabalho estendemos esse resultado para perturbações mais gerais (não conservativas) e apresentamos funções de Melnikov nesse novo contexto. Finalmente, abordamos o problema mais geral, relativo à perturbação de campos não conservativos, onde a função de Melnikov, associada a órbita heteroclínica, é obtida. / In this work we study generalizations of Melnikov's method to planar discontinuous dynamical system. Initially we study this problem as a variation of the work [1] where a Hamiltonian vector field that admits an heteroclinic cycle with its interior foliated by a family of periodic orbits is perturbed by a Hamiltonian perturbation. In this work we extended the results to more general perturbation (non conservative) and we show the Melnikov's functions in this new context. Finally, we approach a more general problem related to a perturbation of the non-conservative vector field where we obtained the Melnikov's function that is associated with a heteroclínic orbit.

Made available in DSpace on 2016-06-02T20:15:18Z (GMT). No. of bitstreams: 1 1995.pdf: 1566210 bytes, checksum: 7c5a0c3ef27c706d0ac9145573694cdf (MD5) Previous issue date: 2008-05-17 / Financiadora de Estudos e Projetos / In this work we first present a protocol to build effective interactions between two cavity modes, considering a two-level atom under the action of classical fields. Bilinear Hamiltonians associated with parametric up- and down-conversion processes are derived, apart from nonlinear interactions associated with the degenerate parametric down-conversion process, resulting in the squeezing operation of a cavity mode. We also demonstrate how to construct nonlinear Hamiltonians related with a Kerr-type process for one or two cavity modes. In particular, we show how to implement, in the bimodal cavity, the Hamiltonian describing a two-specieis Bose-Einstein condensate in the two-mode approximation. Next, considering a two-level ion trapped in a cavity, under the action of classical amplification field, we show how to build an artificial reservoir for the electronic states of the ion. This reservoir is suited to protect nonstationary superpositions of the electronic levels, enabling us to measure the geometric phase acquired by these states under nonadiabatic evolutions of the system. Finally, we show how to construct squeezed reservoirs, either for a cavity mode or two-level atoms, by previously engineering an effective interaction between the atom(s) and the cavity mode which comprehends the simultaneous implementation of the Jaynes-Cummings and anti-Jaynes-Cummings Hamiltonians. / Nesta tese, apresentamos primeiramente um protocolo para a construção de interações efetivas entre dois modos de uma cavidade, através de um átomo de dois níveis sob a ação de campos clássicos. hamiltonianos bilineares associados à processos de conversões paramétricas ascendente e descendente de frequências foram obtidos, bem como hamiltonianos não-lineares associados à compressão paramétrica de um modo da cavidade. Mostramos também como construir hamiltonianos associados a processos não-lineares do tipo Kerr para um ou dois modos da cavidade. Em especial, mostramos como implementar, na cavidade bi-modal, o hamiltoniano que descreve um condensado de Bose-Einstein de duas espécies atômicas na aproximação de dois modos. Em seguida, considerando um íon de dois níveis aprisionado no interior de uma cavidade e submetido à ação de campos clássicos, mostramos como construir um reservatório artificial para os estados eletrônicos do íon. Este reservatório permite a proteção de superposições não estacionárias dos níveis eletrônicos, possibilitando a medida de fases geométricas por elas adquiridas mediante evoluções não adiabáticas do sistema. Por fim, mostramos como construir reservatórios comprimidos tanto para um modo da cavidade como para átomos de dois níveis, mediante a construção prévia de uma interação efetiva entre átomo(s) e modo que compreende a realização simultânea dos hamiltonianos de Jaynes-Cummings e anti-Jaynes-Cummings. Para tanto, recorremos a átomo(s) de três níveis sob a ação de campos clássicos.

Le cube de Fibonacci est un sous-graphe isométrique de l'hyper- cube ayant un nombre de Fibonacci de sommets. Le cube de Fibonacci a été initialement introduit par W-J. Hsu comme un réseau d'interconnexion et, comme l'hypercube, il a des propriétés topologiques très attractives, mais avec une croissance plus modérée. Parmi ces propriétés, nous discutons de l'hamiltonicité dans le cube de Fibonacci et aussi dans le cube de Lucas qui est obtenu à partir du cube de Fibonacci en supprimant toutes les chaînes qui commencent et nissent avec 1. Nous trouvons également le nombre de som- mets des cubes de Fibonacci et Lucas ayant une certaine excentricité. En n, nous présentons une étude de deux cubes du point de vue de la domination et du 2-packing. / The Fibonacci cube is an isometric subgraph of the hypercube having a Fibonacci number of vertices. The Fibonacci cube was originally proposed by W-J. Hsu as an interconnection network and like the hypercube it has very attractive topological properties but with a more moderated growth. Among these properties, we discuss the hamiltonicity in the Fibonacci cube and also in the Lucas cube which is obtained by removing all the strings that begin and end with 1 from the Fibonacci cube. We give also the eccentricity sequences of the Fibonacci and the Lucas cubes. Finally, we present a study of both cubes from the domination and the 2-packing points of view.