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The dynamics of a single underwater vehicle in an ideal irrotational fluid may be modeled by a Lagrangian system with configuration space the Euclidean group. If hydrodynamic coupling is ignored then two coupled vehicles may be modeled by the direct product of two single-vehicle systems. We consider this system in the case that the vehicles are coupled mechanically, with an ideal spherically symmetric joint, finding all of the relative equilibria. We demonstrate that there are relative equilibria in certain novel momentum-generator regimes identified by Patrick et.al. "<i>Stability of Poisson equilibria and Hamiltonian relative equilibria by energy methods</i>", Arch. Rational Mech. Anal., 174:301--344, 2004.

The dynamics of a single underwater vehicle in an ideal irrotational fluid may be modeled by a Lagrangian system with configuration space the Euclidean group. If hydrodynamic coupling is ignored then two coupled vehicles may be modeled by the direct product of two single-vehicle systems. We consider this system in the case that the vehicles are coupled mechanically, with an ideal spherically symmetric joint, finding all of the relative equilibria. We demonstrate that there are relative equilibria in certain novel momentum-generator regimes identified by Patrick et.al. "<i>Stability of Poisson equilibria and Hamiltonian relative equilibria by energy methods</i>", Arch. Rational Mech. Anal., 174:301--344, 2004.

<p>Pages 8, 38, 70, 116 and 120 have no body of text in the hardcopy. All are end pages of sections with a title at the top.</p> / <p>Many symbols could not be replicated using the Special Characters list. Please download thesis to read abstract.</p> / Doctor of Philosophy (PhD)

This work is concerned with the Almost Axisymmetric Flows with Forcing Terms which are derived from the inviscid Boussinesq equations. It is our hope that these flows will be useful in Meteorology to describe tropical cyclones. We show that these flows give rise to a collection of Monge-Ampere equations for which we prove an existence and uniqueness result. What makes these equations unusual is the boundary conditions they are expected to satisfy and the fact that the boundary is part of the unknown. Our study allows us to make inferences in a toy Almost Axisymmetric Flows with a forcing term model.

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Submitted by Célia Mayumi Kuwana (celiamkuwana@hotmail.com) on 2018-05-16T16:10:10Z No. of bitstreams: 1 kuwana_cm_me_rcla.pdf: 1196862 bytes, checksum: 37b452d62ccbc0a6e02de1a013df0849 (MD5) / Rejected by Adriana Aparecida Puerta null (dripuerta@rc.unesp.br), reason: Prezada Célia, O documento enviado para a coleção Instituto de Biociências Rio Claro foi recusado pelo(s) seguinte(s) motivo(s): - Falta a capa, elemento obrigatório, que deve ser inserida antes da folha de rosto no arquivo pdf. - Falta a informação de Aprovada na folha de aprovação, sendo que a folha, deve ser solicitada à Seção de Pós-Graduação e inserida após a ficha catalográfica. O documento enviado não foi excluído. Para revisá-lo e realizar uma nova tentativa de envio, acesse: https://repositorio.unesp.br/mydspace Em caso de dúvidas entre em contato pelo email repositoriounesp@reitoria.unesp.br. Agradecemos a compreensão e aguardamos o envio do novo arquivo. Atenciosamente, Biblioteca Campus Rio Claro Repositório Institucional UNESP https://repositorio.unesp.br on 2018-05-16T17:44:23Z (GMT) / Submitted by Célia Mayumi Kuwana (celiamkuwana@hotmail.com) on 2018-05-17T18:36:34Z No. of bitstreams: 2 kuwana_cm_me_rcla.pdf: 1196862 bytes, checksum: 37b452d62ccbc0a6e02de1a013df0849 (MD5) kuwana_cm_me_rcla.pdf: 1484457 bytes, checksum: 49f6c72467f2a1cd318e79d6f53b0ec8 (MD5) / Approved for entry into archive by Adriana Aparecida Puerta null (dripuerta@rc.unesp.br) on 2018-05-18T16:28:17Z (GMT) No. of bitstreams: 1 kuwana_cm_me_rcla.pdf: 1334658 bytes, checksum: f623f773fd644ffaefb15c97d13db854 (MD5) / Made available in DSpace on 2018-05-18T16:28:17Z (GMT). No. of bitstreams: 1 kuwana_cm_me_rcla.pdf: 1334658 bytes, checksum: f623f773fd644ffaefb15c97d13db854 (MD5) Previous issue date: 2018-02-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho apresentaremos e discutiremos algumas propriedades dinâmicas para uma família de mapeamentos discretos que preservam a área no espaço de fases nas variáveis momentum, I, e coordenada generalizada, θ. O mapeamento é descrito por dois parâmetros de controle, sendo eles ε, ajustando a intensidade da não linearidade, e γ, um parâmetro que fornece a forma da divergência da variável “θ”no limite em que I → 0. O parâmetro ε controla a transição de integrabilidade, quando ε = 0, para não integrabilidade, no limite em que ε ≠ 0. O objetivo principal deste trabalho é descrever o comportamento das curvas do momentum médio, I_RMS(ε,n), em função de n, a partir de uma função de probabilidade, P(I(n)), de observar um determinado momentum I em um instante n. Para tanto, resolveremos a Equação da Difusão analiticamente, considerando os casos: (i) o momentum inicial nulo, I_0 = 0, e (ii) o momentum inicial não nulo, I_0 ≠ 0. Nossos resultados descrevem bem os resultados fenomenológicos conhecidos na literatura (Physics Letters A, 379: 1808 (2015)). / In this work we will present and discuss some dynamical properties of a family of mappings that preserves area in the phase space for two variables momentum, I, and generalized coordinate, θ. The mapping is controled by two parameters: ε, tunning the intensity of nonlinearity, and γ, that describes the form of divergence of θ when I → 0. The parameter ε defines a transition from integrability, when ε = 0, to nonintegrability, when ε ≠ 0. The main goal of this work is to describe the curves of average momentum, I_RMS(ε,n), in terms of n, from a probability function, P(I(n)), to observe a determined momentum I at an instant n. Therefore, we will solve the Diffusion equation analitically considering the cases: (i) the initial momentum is null, I_0 = 0, and (ii) the initial momentum is nonzero, I_0 ≠ 0. Our results describe well the known phenomenological results in literature (Physics Letters A, 379: 1808 (2015)). / CAPES-DS: 3300413-7.

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

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.

Cette thèse porte sur l’étude des solutions périodiques du problème des N-tourbillons à vorticité positive. Ce problème, formulé par Helmholtz il y a plus de 160 ans, possède une histoire très riche et reste un domaine de recherche très actif. Pour un nombre quelconque de tourbillons et sans contrainte sur les vorticités, ce système n’est pas intégrable au sens de Liouville : on ne peut trouver de solution périodique non triviale par des méthodes explicites. Dans cette thèse, à l’aide de méthodes variationnelles, nous prouvons l’existence d’une infinité de solutions périodiques non triviales pour un système de N tourbillons à vorticités positives. De plus, lorsque les vorticités sont des nombres rationnels positifs, nous montrons qu’il n’existe qu’un nombre fini de niveaux d’énergie sur lesquels un équilibre relatif pourrait exister. Enfin, pour un système de N-tourbillons identiques, nous montrons qu’il existe une infinité de chorégraphies simples. / This thesis focuses on the study of the periodic solutions of the N-vortex problem of positive vorticity. This problem was formulated by Helmholtz more than 160 years ago and remains an active research field. For an undetermined number of vortices and general vorticities the system is not Liouville integrable and periodic solutions cannot be determined explicitly, except for relative equilibria. By using variational methods, we prove the existence of infinitely many non-trivial periodic solutions for arbitrary N and arbitrary positive vorticities. Moreover, when the vorticities are positive rational numbers, we show that there exists only finitely many energy levels on which there might exist a relative equilibrium. Finally, for the identical N-vortex problem, we show that there exists infinitely many simple choreographies.