We extend the concept of central configurations to the N-body problem in spaces of nonzero constant curvature. Based on the work of Florin Diacu on relative equilib- ria of the curved N-body problem and the work of Smale on general relative equilibria, we find a natural way to define the concept of central configurations with the effective potentials. We characterize the ordinary central configurations as constrained critical points of the cotangent potential, which helps us to establish the existence of ordi- nary central configurations for any given masses. After these fundamental results, we study central configurations on H2, ordinary central configurations in S3, and special central configurations in S3 in three separate chapters. For central configurations on H2, we generalize the theorem of Moulton on geodesic central configurations, the theorem of Shub on the compactness of central configurations, the theorem of Conley on the index of geodesic central configurations, and the theorem of Palmore on the lower bound for the number of central configurations. We show that all three-body central configurations that form equilateral triangles must have three equal masses. For ordinary central configurations in S3, we construct a class of S3 ordinary central configurations. We study the geodesic central configurations of two and three bodies. Three-body non-geodesic ordinary central configurations that form equilateral trian- gles must have three equal masses. We also put into the evidence some other classes of central configurations. For special central configurations, we show that for any N ≥ 3, there are masses that admit at least one special central configuration. We then consider the Dziobek special central configurations and obtain the central con- figuration equation in terms of mutual distances and volumes formed by the position vectors. We end the thesis with results concerning the stability of relative equilibria associated with 3-body special central configurations. We find that these relative equilibria are Lyapunov stable when confined to S1, and that they are linearly stable on S2 if and only if the angular momentum is bigger than a certain value determined by the configuration. / Graduate

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Les matériaux actifs sont des matériaux qui réagissent quand on leur applique un champ extérieur comme la température, la lumière, un champ magnétique ou un champ électrique. Ces champs changent les propriétés du matériau comme la longueur, la susceptibilité magnétique ou la permittivité électrique. Ces changements peuvent être utilisé pour faire du travail. Quelques exemples sont les matériaux piézoélectriques, qui changent de longueur quand on applique un champ électrique, les alliages à mémoire de forme qui changent leur longueur sous l’action de la température. Un matériau plus récent qu’on appelle les alliages mémoire de forme magnétique se de forme sous l’action d’un champ magnétique. Dans cette thèse, on utilise ce matériau pour Confectionner un actionneur. Pour ce faire, on utilise la thermodynamique des procédés irréversibles pour modéliser le matériau. La thermodynamique s’avère très versatile pour ce type de matériau car il permet de quantifier l’ échange et la transformation d’ énergie dans le matériau. Aussi, étant donné que le matériau se comporte d’une façon non-linéaire et hystérique, le cadre énergétique nous permets justement de prendre en compte ces non- linearités. Cette thèse utilise l’approche énergétique notamment les Hamiltonien à ports pour modéliser un actionneur à base d’alliage à mémoire de forme. Cette méthode nous permets aussi de concevoir des lois de commande pour contrôler le matériau. / Active materials are a class of material which react to an external stimulus such as temperature,photons, magnetic field or electric field. These stimuli cause some properties of the material tochange usually their length. Some examples are piezoelectric material which change their lengthunder the action of an electric field, Shape Memory alloys which alter their shape on applicationof heat, and more recently Magnetic Shape Memory Alloys (MSMA) which undergo a deformationon application of a magnetic field. Harnessing this property of MSMAs, we hereby present anactuator using this novel material. We extensively make use of an energy framework, namely thethermodynamics of irreversible processes to model the material. This framework has been provento be very versatile in modelling energy exchange and transformation as it occurs in the materialand also to incorporate hysteresis which arises naturally in such materials. Another advantage of thismethod is its ability to give us constitutive laws based on simple assumptions. Furthermore, usingan energy framework allows us to apply some energy based control. Port Hamiltonian Control is onesuch method and it is not limited only to linear models. This latter characteristic has proven veryuseful since MSMAs are very non-linear in nature.

In this thesis Moore and Yeo's proposed mapping of the structural glass to the Ising spin glass in a random field is presented. In contrast to Random First Order Theory and Mode Coupling Theory, this mapping predicts that there should be no glass transition at finite temperature. However, a growing correlation length is predicted from the size of rearranging regions in the supercooled liquid, and from this a growing structural relaxation time is predicted. Also presented is a study of the propensity of binary fluids (i.e. fluids containing particles of two sizes) to phase separate into regions dominated by one type of particle only. Binary fluids like this are commonly used as model glass formers and the study shows that this phase separation behaviour is something that must be taken into account.The mapping relies on the use of replica theory and is therefore very opaque. Here a model is presented that may be mapped directly to a system of spins, and also prevents the process of phase separation from occurring in binary fluids. The system of spins produced in the mapping is then analysed through the use of an effective Hamiltonian, which is in the universality class of the Ising spin glass in a random field. The behaviour of the correlation length depends on the spin-spin coupling J and the strength of the random field h. The variation of these with packing fraction and temperature T is studied for a simple model, and the results extended to the full system. Finally a prediction is made for the critical exponents governing the correlation length and structural relaxation time.

Orientador: Fabiano Borges da Silva / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-18T12:59:08Z (GMT). No. of bitstreams: 1 Fernandes_AristonLopes_M.pdf: 1807643 bytes, checksum: 1b8f6422f12372fd95aefa1b55d12fc7 (MD5) Previous issue date: 2011 / Resumo: Este trabalho tem como propósito estudar trajetórias geradas por sistemas hamiltonianos no plano. Para isso, são analisados os diversos tipos de retratos de fase dos sistemas lineares planares e a classificação destes. Sistemas hamiltonianos surgiram na mecânica clássica e seus pontos de equilíbrio são classificados em selas ou centros, conforme os sinais dos autovalores da matriz do sistema linearizado. Além disso, é apresentada a relação entre campos de vetores hamiltonianos e espaços vetoriais simpléticos / Abstract: This work has the objective of studing trajectories generated by Hamiltonian systems on the plane. For this, we analyse the various types of phase portraits of planar and linear systems. Hamiltonian systems have emerged in the mechanical and their classical equilibrium points are classified into saddles or centers, as the signs of the eigenvalues of linearized system matrix. We have also illustrated the connection between Hamiltonian vector fields and symplectic spaces / Mestrado / Matemática Universitária / Mestre em Matemática Universitária

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1979. / Bibliography: leaves 58-59. / by Boris A. Kupershmidt. / Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1979.

La structure dynamique de la région transneptunienne est encore loin d'être entièrement comprise, surtout concernant les objets ayant un périhélie très éloigné. Dans cette région, les perturbations orbitales sont très faibles, autant de l'intérieur (les planètes) que de l'extérieur (les étoiles de passage et les marées galactiques). Pourtant, de nombreux objets ont des orbites très excentriques, ce qui indique qu'ils ne se sont pas formés tels qu'on les observe actuellement. De plus, certaines accumulations dans la distribution de leurs éléments orbitaux ont attiré l'attention de la communauté scientifique, conduisant à de nombreuses conjectures sur l'origine et l'évolution du Système Solaire externe.Avant d'envisager des théories plus "exotiques", une analyse exhaustive doit être menée sur les différents mécanismes qui peuvent reproduire les trajectoires observées à partir de ce qui est jugé "certain" dans la dynamique du Système Solaire, à savoir les perturbations par les planètes connues et par les marées galactiques. Cependant, nous ne pouvons pas nous fier uniquement aux simulations numériques pour explorer efficacement l'espace des comportements possibles. Dans ce contexte, notre objectif est de dégager une vision globale de la dynamique entre Neptune et le nuage de Oort, y compris les orbites les plus extrêmes (même si elles sont improbables ?).Les orbites entièrement extérieures à la région planétaire peuvent être divisées en deux classes générales : d'un côté, les objets soumis à une diffusion du demi grand-axe (ce qui empêche toute variation importante du périhélie) ; de l'autre côté les objets qui présentent une dynamique intégrable à court terme (ou quasi-intégrable). La dynamique de ces derniers peut être décrite par des modèles séculaires. Il existe deux sortes d'orbites régulières : les orbites non résonnantes (demi grand-axe fixe) et celles piégées dans une résonance de moyen mouvement avec une planète (demi grand-axe oscillant).La majeur partie de ce travail de thèse se concentre sur le développement de modèles séculaires pour les objets transneptuniens, dans les cas non résonnant et résonnant. Des systèmes à un degré de liberté peuvent être obtenus, ce qui permet de représenter chaque trajectoire par une courbe de niveau du hamiltonien. Ce type de formalisme est très efficace pour explorer l'espace des paramètres. Il révèle des trajectoires menant à des périhélies éloignés, de même que des "mécanismes de captures", capables de maintenir les objets sur des orbites très distantes pendant des milliards d'années. L'application du modèle séculaire résonnant aux objets connus est également très instructive, car elle montre graphiquement quelles orbites observées nécessitent un scénario complexe (comme la migration planétaire ou un perturbateur extérieur), et lesquelles peuvent être expliquées par l'influence des planètes connues. Dans ce dernier cas, l'histoire dynamique des petits corps peut être retracée depuis leur capture en résonance.La dernière partie de ce travail est consacrée à l'extension du modèle séculaire non résonnant au cas d'un perturbateur extérieur massif. S'il est doté d'une excentricité et/ou d'une inclinaison non négligeable, cela introduit un, voire deux degrés de liberté supplémentaires dans le système, d'où une dynamique en général non intégrable. Dans ce cas, l'analyse peut être réalisée à l'aide de sections de Poincaré, qui permettent de distinguer les régions chaotiques et régulières de l'espace des phases. Pour des demi grands-axes croissants, le chaos se propage très rapidement. Les structures les plus persistantes sont des résonances séculaires produisant des trajectoires alignées ou anti-alignées avec la planète distante. / The dynamical structure of the transneptunian region is still far from being fully understood, especially concerning high-perihelion objects. In that region, the orbital perturbations are very weak, both from inside (the planets) and from outside (passing stars and galactic tides). However, numerous objects have very eccentric orbits, which indicates that they did not form in their current orbital state. Furthermore, some intriguing clusters in the distribution of their orbital elements have attracted attention of the scientific community, leading to numerous conjectures about the origin and evolution of the external Solar System.Before thinking of "exotic" theories, an exhaustive survey has to be conducted on the different mechanisms that could produce the observed trajectories involving only what we take for granted about the Solar System dynamics, that is the orbital perturbations by the known planets and/or by galactic tides. However, we cannot rely only on numerical integrations to efficiently explore the space of possible behaviours. In that context, we aim at developing a general picture of the dynamics between Neptune and the Oort Cloud, including the most extreme (even if improbable?) orbits.The orbits entirely exterior to the planetary region can be divided into two broad classes: on the one hand, the objects undergoing a diffusion of semi-major axis (which prevents from large variation of the perihelion distance); on the other hand, the objects which present an integrable (or quasi-integrable) dynamics on a short time-scale. The dynamics of the latter can be described by secular models. There are two kinds of regular orbits: the non-resonant ones (fixed semi-major axis) and those trapped in a mean-motion resonance with a planet (oscillating semi-major axis).The major part of this Ph.D. work is focussed on the development of secular models for transneptunian objects, both in the non-resonant and resonant cases. One-degree-of-freedom systems can be obtained, which allows to represent any trajectory by a level curve of the Hamiltonian. Such a formalism is pretty efficient to explore the parameter space. It reveals pathways to high perihelion distances, as well as "trapping mechanisms", able to maintain the objects on very distant orbits for billion years. The application of the resonant secular model to the known objects is also very informative, since it shows graphically which observed orbits require a complex scenario (as the planetary migration or an external perturber), and which ones can be explained by the influence of the known planets. In this last case, the dynamical history of the small bodies can be tracked back to the resonance capture.The last part of this work is devoted to the extension of the non-resonant secular model to the case of an external massive perturber. If it has a substantial eccentricity and/or inclination, it introduces one or two more degrees of freedom in the system, so the secular dynamics is non integrable in general. In that case, the analysis can be realised by Poincaré sections, which allow to distinguish the chaotic regions of the phase space from the regular ones. For increasing semi-major axes, the chaos spreads very fast. The most persistent structures are secular resonances producing trajectories aligned or anti-aligned with the orbit of the distant planet.