Search

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2021, Director: Arturo Vieiro Yanes, [en] The $n$-body problem is a classical problem in celestial mechanics which attempts to describe the motion of $n$ bodies under their mutual gravitational attraction. The problem is only solvable for two masses, and not much is known for the general case of three or more bodies. This work deals with some particular solutions of the $n$-body problem. First, using its underlying Hamiltonian structure, we state the main properties of the problem, its symmetries and first integrals. Next, we study central configurations and their relation with homothetic and relative equilibria solutions. For three bodies, the well-known Lagrange configuration provides a relative equilibria in which three shifted particles in an equilateral triangle move along a periodic orbit, known as a choreography. In the last chapter we consider the figure eight solution which is another choreography of three bodies with some particular geometrical and dynamical properties. Using an ad-hoc implementation of the Taylor method developed for the numerical integration of the $n$-body problem we illustrate the orbits and the properties of the particular solutions discussed in this work as well as a numerical check of the remarkable linear stability property of the figure eight.

This paper deals with direct transfers from the Earth to Halo orbits related to the translunar point. The gravitational influence of the Sun as a fourth body is taken under consideration by means of the Bicircular Problem (BCP), which is a periodic time dependent perturbation of the Restricted Three Body Problem (RTBP) that includes the direct effect of the Sun on the spacecraft. In this model, the Halo family is quasi-periodic. Here we show how the effect of the Sun bends the stable manifolds of the quasi-periodic Halo orbits in a way that allows for direct transfers.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2020, Director: Àngel Jorba i Monte, [en] The mechanics of celestial bodies has been studied since the origin of human reasoning, specially the planets. We have seen examples of this in the counting of lunar cycles of the primitive homo sapiens, the stars dome of the greek, or in the geocentric and heliocentric visions of our Solar system. Nowadays, with all the avances in the fields of Maths, modern Physics and computers, the study of celestial bodies mechanics has advanced a lot. This paper aims to explain the behaviour of big mass particles attracted to each other in space, according to the laws of Keppler and Newton. This is carried out from a modern point of view and with mathematical rigor.

We consider planar central configurations of the Newtonian kn-body problem consisting in k groups of regular n-gons of equal masses, called (k , n)-crown. We derive the equations of central configurations for a general ( k, n)-crown. When k = 2 we prove the existence of a twisted (2, n)-crown for any value of the mass ratio. Moreover, for n = 3, 4 and any value of the mass ratio, we give the exact number of twisted (2, n)-crowns, and describe their location. Finally, we conjecture that for any value of the mass ratio there exist exactly three (2; n)-crowns for n>=5 The authors are supported by MINECO grants MTM2016-80117-P and MTM2016-77278-P (FEDER) and Catalan (AGAUR) grants 2017 SGR 1374 and 2017 SGR 1617

Made available in DSpace on 2018-12-11T17:26:44Z (GMT). No. of bitstreams: 0 Previous issue date: 2015-01-01 In this work, we are interested in avoiding large variations in the mutual distances among multiple satellites and also in controlling their geometric configuration around an Earth-Moon triangular point. Previous studies about triangular libration points have determined the existence of zero drift regions with respect to the nominal trajectory, in which the expansion or contraction of the formation never take place. Our goal is to carry out two different control strategies for a formation near a given nominal trajectory around L4: a bang-off-bang control and a minimum weighted total ΔV consumption. A linearization relative to the reference trajectory around the triangular libration point is carried out, and different geometrical possibilities in the zero drift regions are studied. To investigate the influence of the gravitational force of the Sun, the BiCircular Four Body Problem is considered here. According to the results obtained, some meaningful insights to allow a proper design of the geometric configuration of the formation are drawn. UNESP-Grupo de Dinâmica Orbital e Planetologia Instituto Nacional de Pesquisas Espaciais, Av. dos Astronautas 1758 Departament de Matemàtica Aplicada i ETSEIB-UPC, Avda. Diagonal 647 Departament de Matemàtica Aplicada i Anàlisi UB, Gran Vía 585 UNESP-Grupo de Dinâmica Orbital e Planetologia

[cat] El problema quàntic de molts cossos ha estat sempre un repte de la física teòrica. L'objectiu al qual s'intenta arribar es la descripció, des d'un punt de vista quàntic, dels observables d'un sistema de moltes partícules en interacció. En el camp específic dels sistemes nuclears, dels més petits com els nuclis als més grans com la matèria nuclear infinita dins de les estrelles, el problema nuclear de molts cossos s'ha intentat resoldre amb la construcció de diferents formalismes. L'objectiu ha sigut descriure aproximadament el comportament de molts nucleons, obtenen resultats els més propers als experimentals i empírics. En la present tesis ens hem centrat sobre l'estudi de la matèria nuclear infinita. En particular sobre els dos sistemes infinits que es situen en els extrems en termes d'asimetria d'isospín: la matèria nuclear simètrica (SNM), el sistema format d'un igual nombre de protons i nucleons, i la matèria purament neutrónica (PNM), el sistema format solament de neutrons. Per dur a terme aquests càlculs hem expandit, de manera consistent, el formalisme de funcions de Green auto-consistents per incloure forces a tres cossos. Hem treballat amb un potencial basat en la teoria efectiva de baixes energies aplicada a la Cromodinàmica Quàntica (QCD), i.e. la teoria efectiva quiral. En aquesta teoria, les 2NFs i 3NFs apareixen consistentment. La inclusió de les forces a tres cossos en els càlculs s'ha demostrat ser fonamental per obtenir el mecanisme de saturació en SNM al voltant de valors acceptables d'energia/densitat i per proporcionar una equació d'estat més rígida per al sistema de PNM. Això demostra, un cop més, la necessitat de considerar les forces de molts cossos per fel càlculs teòrics consistents amb els resultats empírics i experimentals., [eng] The quantum many-body problem is an everlasting challenge for theoretical physics. The aim sought is to analyze, at the quantum level, the observables arising from a group formed by a certain number of interacting particles N. In the specific case of nuclear physics, the study and interpretation of interacting nuclear systems has spanned from the finite nuclei to the infinite nuclear matter case. In order to characterize such systems, the forces with which nucleons interact in their interior need to be sorted out. Once the Hamiltonian is chosen, the many-body problem for the group of interacting fermions, the nucleons, has to be solved. In the present thesis we focus our study on infinite nuclear matter, analyzing the properties of the two systems which stand at the extremes in terms of isospin asymmetry: symmetric nuclear matter (SNM), with an equal number on neutrons and protons, and pure neutron matter (PNM), with only neutrons. We base our approach in terms of realistic microscopic potentials, hence we need to choose a many-body technique in order to treat the potential. We choose to perform our calculations exploiting the formalism provided by the self-consistent Green's function's (SCGF) theory. This approach is based on a diagrammatic expansion of the single-particle (SP) Green's function, or propagator. This method was devised to treat the correlated, i.e. beyond mean-field, behavior of strongly interacting systems, such as nuclear matter. Unfortunately, it is a well established fact that whatever microscopic two-nucleon forces (2NFs) are used in the many-body calculation, empirical saturation properties of nuclear matter fail to be reproduced. Saturation densities appear at high values, presenting energies which are inconsistently too attractive, overbinding nuclear matter. This inconsistency bears some similarities to the case of light nuclei where, on the contrary, the 2NF-only based theory underbinds experimental data. The inclusion of three-nucleon forces (3NFs) has been the indispensable factor to cure this deficiency. Hence, microscopic 3NFs have been mostly devised to provide attraction in light finite systems, small densities, and repulsion in infinite systems, high densities. However, in modeling these 3NFs, the inclusion of phenomenological ingredients has often been the way followed. To bypass the need to adjust the potential with ad hoc contributions, an alternative has been provided by chiral effective field theory (EFT). Featuring the consistent Hamiltonian at the 2N and 3N level provided by EFT, in this thesis we present calculations including both chiral 2NFs and 3NFs. This is dealt within an extended SCGF formalism, specifically formulated to include consistently 3B forces by means of effective interactions. We observe the striking effect of the inclusion of 3N forces for the total energy of the many-body ground state. Exploiting the extended SCGF approach presented in this thesis, we see how the modifications induced by the 3B force are larger as the density increases. This increasing with density is what provides the saturation mechanism for nuclear matter. In pure neutron matter, 3NFs are the main cause for the stiffening of the equation of state. In view of astrophysical studies for neutron star masses, this stiffening is a major ingredient for the achievement of theoretical results which can better match recent astrophysical observations. The idea, on which the development of this thesis was based, has been all the way the quest to introduce consistently three-body forces in the formalism of the SCGF theory. While the main motivation of this study has been nuclear systems, the extended SCGF formalism can be easily applied to other many-body systems, of either atomic or molecular nature.

THESIS SUMMARYTEXT:This thesis is made of two parts. In the first one, the issue of entanglement in many body systems is addressed. The concept of entanglement and some of the recent progress on the study of entropy of entanglement in many body quantum systems are reviewed. Emphasis is placed on the scaling properties of entropy for one-dimensional models at quantum phase transitions. Then, we focus on the area-law scaling of the entanglement entropy. An explicit computation in arbitrary dimensions of the entanglement entropy of the ground state of a discretized scalar free field theory that shows the expected area law result is also presented. For this system, it is shown that area law scaling is a manifestation of a deeper reordering of the vacuum produced by majorization relations.To finish this first part, the issue of how simple can a quantum system be such as to give a highly entangled ground state is addressed. In particular, we propose a Hamiltonian of a XX model with a ground state whose entropy scales linearly with the size of the block. It provides a simple example of a one dimensional system of spin-1/2 particles with nearest neighbour interactions that violates area-law for the entanglement entropy.The second part of this thesis deals with the problem of simulating quantum mechanics for highly entangled systems. Two different approaches to this issue are considered. One consists of using ultra-cold atoms systems as quantum simulators. With this aim, some experimental techniques related to cold atoms that allow to simulate strongly correlated many body quantum systems are reviewed an explicit example of simulation is presented. In particular, we analyze how to achieve a Mott state of Laughlin wave functions in an optical lattice and study the consequences of considering anharmonic corrections to each single site potential expansion that were not taken into account until now.Finally, a different approach to simulate strongly correlated systems is considered: to use small quantum computers to simulate them. An explicit quantum algorithm that creates the Laughlin state for an arbitrary number of particles n in the case of falling fraction equal to one is presented. We further prove the optimality of the circuit using permutation theory arguments and we compute exactly how entanglement develops along the action of each gate. We also discuss its experimental feasibility decomposing the qudits and the gates in terms of qubits and two qubit-gates as well as the generalization to arbitrary falling fraction.KEYWORDS: Entanglement, Many body quantum systems, Quantum Information Condensed Matter, Cold atoms, Spin chains, Quantum simulator, Quantum computation., "Entrellaçament quàntic en sistemes de molts cossos"TEXT:Aquesta tesi està composada per dues parts. En la primera, adrecem la qüestió de l'entrellaçament quàntic en els sistemes de molts cossos. Així, introduïm primer el concepte d'entrellaçament i revisem els progressos recents sobre aquest camp. A continuació, ens centrem la llei d'àrea per l'entropia d'entrellaçament i presentem un càlcul explícit d'aquesta entropia per a l'estat fonamental d'un camp escalar no interactuant obtenint la llei d'àrea esperada. Finalment, acabem aquesta part presentant un sistema molt senzill 1-dimensional que tot i tenir interaccions locals mostra una llei de volum per l'entropia.En la segona part de la tesi tractem el problema de la simulació de sistemes quàntics altament entrellaçats. Considerem dos possibles vies per tractar aquest problema. Una d'elles consisteix en la utilització d'àtoms ultra-freds com a simuladors quàntics. En particular, analitzem un mètode per obtenir un estat producte de funcions d'ona de Laughlin en un xarxa òptica i estudiem les conseqüències de considerar la correcció anharmònica de l'expansió del potencial a cada pou de la xarxa. Finalment, considerem una altra aproximació a la simulació de sistemes fortament correlacionats: utilitzar petits ordinadors quàntics per a simular-los. Per il.lustrar aquest tipus de simulació, presentem un algoritme quàntic que crea un estat de Laughlin per un nombre arbitrari de partícules i en el cas de fracció d'ocupació 1.

En aquesta tesi presentem un conjunt d'estudis sobre sistemes atòmics on els efectes quàntics són especialment destacats. Aquests estudis s'han dut a terme aplicant diverses tècniques de la física teòrica de molts cossos.En primer lloc hem estudiat la possible existència d'una transició de fase superfluida en un gas ultrafred d'àtoms fermiònics, mitjançant una generalització de la teoria BCS de la superconductivitat que dóna especial rellevància al paper jugat per l'asimetria de densitat entre les dues espècies, i permet que l'estat fonamental presenti un trencament espontani de simetria.En una segona part, hem estudiat la dinàmica d'un condensat de Bose-Einstein el grau de llibertat d'espí del qual pot evolucionar dins d'una trampa òptica quasi-unidimensional, tant a temperatura zero com finita, mitjançant una formulació de camp mitjà.Finalment, hem dut a terme un estudi detallat de l'estat fonamental i la tensió lineal de sistemes bidimensional d'heli-4, primerament mitjançant les tècniques de Monte Carlo, i posteriorment amb un funcional de la densitat construit amb aquest objectiu., EN CASTELLÀ: En esta tesis se presenta un conjunto de estudios sobre sistemas atómicos donde los efectos cuánticos son especialmente destacados. Dichos estudios se han llevado a cabo aplicando varias técnicas de la física teórica de muchos cuerpos.En primer lugar, se ha estudiado la posible existencia de una transición superfluida en un gas ultrafrío de átomos fermiónicos mediante una generalización de la teoría BCS de la superconductividad que presta especial atención al papel jugado por la asimetría de densidad entre las dos especies, y permite que el estado fundamental presente una rotura espontánea de simetría.En una segunda parte, se ha estudiado la dinámica de un condensado de Bose-Einstein cuyo grado de libertad de espín puede evolucionar en una trampa óptica cuasi-unidimensional, tanto a temperatura cero como finita, mediante una formulación de campo medio.Finalmente, se ha llevado a cabo un estudio detallado del estado fundamental y la tensión lineal de sistemas bidimensionales de helio-4, primeramente mediante las técnicas de Monte Carlo, y posteriormente con un funcional de la densidad construido al efecto.PALABRAS CLAVE: Átomos fríos, Aparejamiento, Condensado espinorial, Helio, Dos dimensiones, SUMMARY: This thesis presents a set of studies on atomic systems where quantum efects are particularly relevant. These studies have been developed by applying a variety of tools from many-body physics.First of all, we have studied the prospects for the existance of a superfluid transition in an ultracold gas of fermionic atoms, by generalizing the BCS theory of superconductivity to the case when the two species that pair have different densities and the ground state may spontaneously break one or more symmetries.In a second part, we have studied the dynamics of a Bose-Einstein condensate whose spin degree of freedom is free to evolve inside a quasi-onedimensional optical trap. We have used a mean-field formulation to address both the zero temperature case and the finite temperature one.Finally, we have performed a careful study of the ground state and the line tension of two-dimensional systems of helium-4. First, we have used Monte Carlo techniques, then with a Density Functional built on-purpose.KEYWORDS: Cold gases, Pairing, Spinor condensate, Helium, Two dimensions