{"id":117053,"date":"2018-03-11T10:46:27","date_gmt":"2018-03-11T10:46:27","guid":{"rendered":"https:\/\/www.deberes.net\/tesis\/sin-categoria\/reduccion-de-tipo-hopf-de-un-modelo-cuartico-aplicaciones-en-dinamica-rotacional-y-orbital\/"},"modified":"2018-03-11T10:46:27","modified_gmt":"2018-03-11T10:46:27","slug":"reduccion-de-tipo-hopf-de-un-modelo-cuartico-aplicaciones-en-dinamica-rotacional-y-orbital","status":"publish","type":"post","link":"https:\/\/www.deberes.net\/tesis\/matematicas\/reduccion-de-tipo-hopf-de-un-modelo-cuartico-aplicaciones-en-dinamica-rotacional-y-orbital\/","title":{"rendered":"Reducci\u00f3n de tipo hopf de un modelo cu\u00e1rtico. aplicaciones en din\u00e1mica rotacional y orbital."},"content":{"rendered":"<h2>Tesis doctoral de <strong> Francisco Crespo Cutillas <\/strong><\/h2>\n<p>Esta tesis aborda los sistemas m\u00e1s conocidos de la mec\u00e1nica cl\u00e1sica de forma unificada. Nuestro objetivo principal es desarrollar un marco de trabajo com\u00fan para el estudio de perturbaciones, dicha tarea se realiza desde un punto de vista geom\u00e9trico. Hemos estructurado esta memoria en tres partes:   parte i. Preliminares en mec\u00e1nica cl\u00e1sica y geometr\u00eda: en esta primera parte recogemos  herramientas que ser\u00e1n usadas a lo largo de nuestro estudio. En el primer cap\u00edtulo fijamos la notaci\u00f3n y se presentan algunos resultados b\u00e1sicos. En el segundo estudiamos el sistema extendido de euler, como un problema de valor inicial param\u00e9trico. Este enfoque permite derivar las principales propiedades de las funciones el\u00edpticas. En concreto, las conocidas relaciones cuadr\u00e1ticas entre funciones el\u00edpticas y la transformaci\u00f3n de jacobi para el m\u00f3dulo el\u00edptico se obtienen de nuestro an\u00e1lisis.  parte ii. Reducci\u00f3n de tipo hopf de un modelo cu\u00e1rtico: en el tercer cap\u00edtulo estudiamos una generalizaci\u00f3n de la fibraci\u00f3n de hopf cl\u00e1sica. Seguiremos la misma metodolog\u00eda que en la fibraci\u00f3n de hopf cl\u00e1sica, pero el cuerpo complejo ser\u00e1 reemplazado por cuaternios. En el cuarto cap\u00edtulo usamos las componentes de la representaci\u00f3n cuaterni\u00f3nica de la aplicaci\u00f3n de hopf para proponer una familia de hamiltonianos multiparam\u00e9trica. Para una elecci\u00f3n apropiada de los par\u00e1metros y considerando una regularizaci\u00f3n de la variable independiente, cuando sea necesario, algunos modelos destacados de la mec\u00e1nica cl\u00e1sica tales como el sistema de kepler, el flujo geod\u00e9sico, el oscilador isotr\u00f3pico de cuatro dimensiones y el s\u00f3lido r\u00edgido libre aparecen como casos particulares. El an\u00e1lisis del modelo cu\u00e1rtico se lleva a cabo a trav\u00e9s de una doble reducci\u00f3n. Por un lado, el sistema es geom\u00e9tricamente reducido, este modelo es un ejemplo detallado de reducci\u00f3n singular, en la cual la correspondiente reconstrucci\u00f3n es tambi\u00e9n proporcionada. Por otro lado, la reducci\u00f3n simpl\u00e9ctica llevada a cabo a trav\u00e9s del uso de nuevas coordenadas can\u00f3nicas es analizada. En concreto, se muestra la relaci\u00f3n entre la reducci\u00f3n geom\u00e9trica y simpl\u00e9ctica y se proporciona la formulaci\u00f3n expl\u00edcita para todos los cambios de variables que son usados.  parte iii. Aplicaciones a la din\u00e1mica roto-orbital: esta parte est\u00e1 dedicada al estudio de la din\u00e1mica de actitud y el movimiento orbital de modelos que aproximan un asteroide o un sat\u00e9lite con una triaxialidad gen\u00e9rica, bajo los efectos de una perturbaci\u00f3n gravitacional. Este problema, denominado problema completo de los dos cuerpos, es un sistema din\u00e1mico hamiltoniano no integrable, que requiere el uso de teor\u00edas de perturbaciones para su an\u00e1lisis. Dentro del contexto de poincar\u00e9 y arnold, una teor\u00eda de perturbaci\u00f3n deber\u00eda ser desarrollada a partir un orden cero integrable y no degenerado. Nosotros exploraremos nuevos candidatos para el orden cero llamados intermediarios.  la idea de los intermediarios consiste en definir un sistema integrable simplificado del problema en cuesti\u00f3n. En el quinto cap\u00edtulo recordamos el concepto de intermediario, presentamos cinco modelos y establecemos una metodolog\u00eda com\u00fan para su estudio. Es en este contexto donde el marco desarrollado para el modelo polin\u00f3mico cu\u00e1rtico es completamente explotado. El sistema simplificado incluye parte del potencial donde el acoplamiento roto-orbital esta presente de tal manera, que el sistema definido por el orden cero es integrable. Los cap\u00edtulos seis y siete aprovechan el marco de trabajo desarrollado en el estudio de dos intermediarios definidos en el cap\u00edtulo anterior. Se asume que estos intermediarios tienen orbitas circulares y el\u00edpticas respectivamente.  en el cap\u00edtulo seis estudiamos equilibrios relativos y bifurcaciones del intermediario circular. Este modelo de intermediario define un flujo poisson sobre espacio multiparam\u00e9trico. En el caso de un cuerpo de rotaci\u00f3n lenta, identificamos condiciones bajo las cuales aparecen bifurcaciones de las trayectorias inestables cl\u00e1sicas, siendo dichos escenarios de gran inter\u00e9s en relaci\u00f3n a la estabilizaci\u00f3n y control. Por otro lado, tambi\u00e9n se pone de manifiesto y se estudia en detalle el papel jugado por la triaxialidad del cuerpo.  en el \u00faltimo cap\u00edtulo la perturbaci\u00f3n contiene al radio y como consecuencia las \u00f3rbitas obtenidas ser\u00e1n de tipo roseta. Este modelo se asocia a dos tipos de aplicaciones, asteroides y sat\u00e9lites, es decir, en nuestra \u00faltima aplicaci\u00f3n consideramos \u00f3rbitas el\u00edpticas en general; tambi\u00e9n analizamos las condiciones para que este modelo admita circulares. El objetivo de este estudio es encontrar un modelo suficientemente simplificado para ser considerado un orden cero, pero que incorpore parte del efecto perturbativo gravitatorio.  conclusi\u00f3n  en esta tesis se aborda una generalizaci\u00f3n del sistema cl\u00e1sico de euler, la soluci\u00f3n general conecta con las doce funciones el\u00edpticas de jacobi. Usando esta generalizaci\u00f3n y la fibraci\u00f3n tipo hopf cuaterni\u00f3nica, se define y estudia en detalle una familia polin\u00f3mica param\u00e9trica de hamiltonianos. Sobre dicha familia se llevan a cabo reducciones de tipo geom\u00e9trico y simpl\u00e9ctico y se muestra que algunos modelos de la mec\u00e1nica cl\u00e1sica est\u00e1n incluidos para ciertas elecciones de los par\u00e1metros. En este sentido, la familia propuesta proporciona un marco de trabajo com\u00fan para abordar estos modelos cl\u00e1sicos.  en las aplicaciones nos centramos en la modelizaci\u00f3n de problemas roto-orbitales. El modelo completo requiere el desarrollo de teor\u00edas perturbativas para obtener soluciones aproximadas. En este trabajo consideramos algunos candidatos para el orden cero. Presentamos un detallado an\u00e1lisis para el caso en el que el sat\u00e9lite presenta rotaci\u00f3n lenta. Para cada misi\u00f3n concreta, el valor de los modelos depender\u00e1 de las comparaciones con experimentos num\u00e9ricos.  para el caso de radio no constante aparecen un buen n\u00famero de t\u00e9cnicas de la mec\u00e1nica cl\u00e1sica a investigar. En este sentido, el cap\u00edtulo siete es un primer paso que requiere m\u00e1s investigaci\u00f3n. Como ejemplo valga la comparaci\u00f3n de nuestro enfoque con la eliminaci\u00f3n de la paralaje como punto de partida. Un segundo aspecto es la producci\u00f3n de las correspondientes variables de \u00e1ngulo-acci\u00f3n.   summary  this thesis addresses some of the very well known systems in classical mechanics in a uniform manner. Our main target is to develop a common framework to deal with perturbations. As such, the structure of this memoir comprises three parts:  part i. Preliminaries on classical mechanics and geometry: in the first part of this memoir we gather some tools that will be used along our study. The first chapter sets notation and presents some basic results. In the second chapter we study the extended euler systems  as an initial value problem with parameters. Particular realizations of this system lead to several lie-poisson structures. The twelve jacobi elliptic functions are shown in a unified way.  part ii. Hopf reduction on a quartic polynomial model: in the third chapter we study a four dimensional generalization of the classical hopf fibration. We follow the same methodology as in the classical hopf fibration, but instead of complex numbers the generalization of the classic hopf map is defined in terms of quaternions. The fourth chapter uses the components of the quaternionic hopf map to propose a parametric hamiltonian function, which is an homogeneous quartic polynomial with six parameters, defining an integrable family of hamiltonian systems. For suitable choices of the parameters, adding an appropriate regularization when needed, some remarkable classical models such as the kepler, geodesic flow, 4-d isotropic oscillator and free rigid body systems appear as particular cases. The analysis of the quartic model is performed through a twofold reduction. On the one hand, the system is geometrically reduced. On the other hand, symplectic reduction is examined. Moreover, we show the relation between the geometric reduction and the reduction carried out by the projective andoyer variables.  part iii. Applications to roto-orbital dynamics: this part is devoted to the study of the attitude dynamics and the orbital motion of models approximating a generic triaxial spacecraft under gravity-gradient torque perturbation. The full problem is a non-integrable hamiltonian dynamical system. Within the context of poincar\u00e9 and arnold, a perturbation theory should be developed upon an integrable and non-degenerate zero order. We study alternative candidates for the zero order, the intermediaries. The idea of the intermediary is to define a simplified integrable system of the problem at stake. In the fifth chapter we recall the concept of intermediary, present five of them and we set a common methodology. Sixth and seventh chapters take advantage of the previous framework considering two intermediary models. Those intermediaries are assumed to be in circular and elliptic orbits respectively.  we study relative equilibria and bifurcations of the circular intermediary in chapter six. This intermediary model defines a poisson flow over a large parameter space. In the case of slow rotational motion we identify conditions under which different bifurcations of the classical unstable trajectories occur, being those scenarios of great interest in relation to stabilization and control purposes. The role played by the triaxiality is also shown.  the final chapter examines a body moving in a rosette-like orbit. More precisely we are thinking about two types of applications, namely to artificial satellites or asteroids around a planet. In other words, we consider perturbed elliptic orbits in general; we also investigate conditions for which this model admits the circular ones. This scenario leads to medium orbits rather than to the low type of orbits studied in the preceding chapter. The intention of this study is to analyze a model simply enough to be considered as an alternative zero order, but incorporating partially the effects of the gravity torque perturbation.  conclusions  the main conclusion of this memoire may be summarized as follows. A generalized study of the classical euler system is presented, connecting its solutions with the twelve jacobi elliptic functions. Using that and the quaternionic hopf fibration a quartic homogeneous polynomial parametric family is proposed and studied in detail. Geometric and symplectic reductions are performed in the family. It is shown that, for suitable choices of parameters, several classical mechanical systems arise as family realizations and we provide a common framework to study them. in the application we focus on modeling problems in the roto-orbital dynamics. The full model is a non-integrable problem which requires the development of perturbation theories in order to obtain approximate solutions. Several candidates for the zero order term, on which the whole theory relies, are considered in this context. We analyze the role played by the integrals and the relation with the physical parameters involved. In particular, we present a fairly complete analysis of the case when the satellite has slow rotation. For each mission, the relative value of each model will finally depend on numerical experiments. when the radius is not constant, there is a number of techniques of classical mechanics to be considered and the last chapter is just a preliminary step to do more research. As an example we mention the comparison of our approach with the elimination of the parallax as the starting point. A second aspect could be the production of the corresponding action-angle variables.<\/p>\n<p>&nbsp;<\/p>\n<h3>Datos acad\u00e9micos de la tesis doctoral \u00ab<strong>Reducci\u00f3n de tipo hopf de un modelo cu\u00e1rtico. aplicaciones en din\u00e1mica rotacional y orbital.<\/strong>\u00ab<\/h3>\n<ul>\n<li><strong>T\u00edtulo de la tesis:<\/strong>\u00a0 Reducci\u00f3n de tipo hopf de un modelo cu\u00e1rtico. aplicaciones en din\u00e1mica rotacional y orbital. <\/li>\n<li><strong>Autor:<\/strong>\u00a0 Francisco Crespo Cutillas <\/li>\n<li><strong>Universidad:<\/strong>\u00a0 Murcia<\/li>\n<li><strong>Fecha de lectura de la tesis:<\/strong>\u00a0 23\/01\/2015<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<h3>Direcci\u00f3n y tribunal<\/h3>\n<ul>\n<li><strong>Director de la tesis<\/strong>\n<ul>\n<li>Sebastian Ferrer Martinez<\/li>\n<\/ul>\n<\/li>\n<li><strong>Tribunal<\/strong>\n<ul>\n<li>Presidente del tribunal: Juan  pablo Ortega lahuerta <\/li>\n<li>Juan  f\u00e9lix San Juan  d\u00edaz (vocal)<\/li>\n<li>patricia Yanguas sayas (vocal)<\/li>\n<li>jean cees Van der meer (vocal)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Tesis doctoral de Francisco Crespo Cutillas Esta tesis aborda los sistemas m\u00e1s conocidos de la mec\u00e1nica cl\u00e1sica de forma unificada. [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""}},"footnotes":""},"categories":[3185,12585,126,8235],"tags":[230843,230844,214464,150508,98366,70529],"class_list":["post-117053","post","type-post","status-publish","format-standard","hentry","category-ecuaciones-diferenciales-en-derivadas-parciales","category-ecuaciones-diferenciales-ordinarias","category-matematicas","category-murcia","tag-francisco-crespo-cutillas","tag-jean-cees-van-der-meer","tag-juan-felix-san-juan-diaz","tag-juan-pablo-ortega-lahuerta","tag-patricia-yanguas-sayas","tag-sebastian-ferrer-Martinez"],"_links":{"self":[{"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/posts\/117053","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/comments?post=117053"}],"version-history":[{"count":0,"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/posts\/117053\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/media?parent=117053"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/categories?post=117053"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/tags?post=117053"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}