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.

Orientador: Marco Antonio Teixeira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T12:10:27Z (GMT). No. of bitstreams: 1 Roberto_LuciAnyFrancisco_D.pdf: 627926 bytes, checksum: 0c8cb4e26df805282fa716847859d82f (MD5) Previous issue date: 2008 / Resumo: Neste trabalho estudamos sistemas dinâmicos possuindo estruturas Hamiltonianas e reversíveis( / Abstract: In this work we study dynamical systems possessing Hamiltonian and time-reversible structures. The reversibility concept is de¯ned in terms of an involution. Initially we discuss the dynamics of Hamiltonian vector ¯elds with 2 and 3 degrees of freedom around an elliptic equilibrium in the presence of an involution which preserves the symplectic structure. The main results discuss the existence of one-parameter families of reversible periodic solutions terminating at the equilibrium. The main techniques that are used in the proofs are Belitskii and Birkho® normal forms and the Liapunov-Schmidt Reduction. Next we consider a case of the 3-body restricted problem in rotating coordinates. In this case the two primaries are oving in an elliptic collision orbit. By the continuation method of Poincare we characterize that the periodic circular orbits and the symmetric periodic elliptic orbits from the Kepler problem which can be prolonged to pseudo periodic orbits of the planar restricted 3{body problem in rotating coordinates with the two primaries moving in an elliptic collision orbit / Doutorado / Topologia e Geometria / Doutor em Matemática

Orientador: Paulo Regis Caron Ruffino / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-13T23:01:19Z (GMT). No. of bitstreams: 1 Costa_PauloHenriquePereirada_M.pdf: 875708 bytes, checksum: 8862a1813f1bb85b5d0269462a80501e (MD5) Previous issue date: 2009 / Resumo: O objetivo desse trabalho é estudar as equações de Hamilton no contexto estocástico. Sendo necessário para tal um pouco de conhecimento a cerca dos seguintes assuntos: cálculo estocástico, geometria de segunda ordem, estruturas simpléticas e de Poisson. Abordamos importantes resultados, dentre eles o teorema de Darboux (coordenadas locais) em variedades simpléticas, teorema de Lie-Weinstein que de certa forma generaliza o teorema de Darboux em variedades de Poisson. Veremos que apesar de o ambiente natural para se estudar sistemas hamiltonianos ser variedades simpléticas, no caso estocástico esses sistemas se adaptam bem em variedades de Poisson. Além disso, para atingir a nossa meta, estudaremos equações diferenciais estocásticas em variedades de dimensão finita usando o operador de Stratonovich / Abstract: This dissertation deals with transfering Hamilton's equations in stochastic context. This requires some knowledge about the following: stochastic calculus, second order geometry and Poisson and simplectic structures. Important results that will be discussed in this theory are Darboux's theorem (local coordinates) for simplectic manifolds, and Lie-Weintein's theorem that is in a certain way of Darboux's theorem on Poisson manifolds. We will see that although the natural environment for studying hamiltonian systems is symplectic manifolds, if we have a Poisson structure we will still be able to study them. Moreover, to achieve our goal, we will study stochastic differential equations on finite dimensional manifolds using the Stratonovich operator / Mestrado / Geometria Estocastica / Mestre em Matemática

Orientador: Kyoko Furuya / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataghin / Made available in DSpace on 2018-08-04T01:53:26Z (GMT). No. of bitstreams: 1 SousaJunior_DelcidesFlaviode_M.pdf: 32680218 bytes, checksum: aa259799e554166260b37c235e19a803 (MD5) Previous issue date: 1998 / Resumo:Apresentamos nesta dissertação um estudo da conexão entre a Mecânica Clássica e a Mecânica Quântica através dos diagramas de energia vs. período para as principais famílias de órbitas periódicas de um dado sistema dinâmico. O diagrama quântico é definido através do espectro do sistema quântico correspondente, que mostra cicatrizes dessas famílias no regime semiclássico. Dois sistemas hamiltonianos, com dois graus de liberdade e apresentando comportamento misto ( caótico e regular ) , são estudados. O primeiro é o pêndulo elástico, usado como paradigma de caos clássico. Aspectos essenciais da sua dinâmica são estudados e o diagrama clássico de energia vs. período com as principais famílias de órbitas periódicas é construido. O segundo sistema é o Hamiltoniano Spin-Bóson, um sistema quântico para o qual trabalhos anteriores definiram um análogo clássico, para o qual estudou-se o comportamento caótico e famílias de órbitas periódicas. Uma versão quântica deste diagrama de energia vs. período é mostrada para este modelo. As duas versões são comparadas no regime de caos misto e o ajuste no limite semiclássico discutido. Uma concordância qualitativa é obtida, com indicações de que as cicatrizes são mais acentuadas nas regiões onde ocorrem bifurcações de órbitas periódicas / Abstract:We study the connection between Classical and Quantum Mechanics using the plots of Energy VS. Period for the main families of periodic orbits of certain dynamical system .The quantum E-t plot is defined through the spectrum of the corresponding quantum system, which shows scars of the classical families in the semiclagsical regime. Two Hamiltonian systems with two degrees of freedom both displaying mixed (chaotic and regular) behaviour are analized. The first one is the elastic pendulum, its behaviour ususally presented as a paradigm of classical chaos. Essential aspects of its dinamics are studied to some extent and the classical (E, t ) plot is shown. The second system is the Spin- Boson Hamiltonian, a quantum system for which previous works have defined a classical analogue with chaotic behaviour and compiled the main families of periodic orbits. A quantum version of the (E, t ) plot for this model is shown, and the classical and quantum plots are compared in the regime of soft chaos. The fitting in the semiclassical limit is discussed with a qualitative agreement that indicates enhancements of the scars in the regions where bifurcations of period orbits occur / Mestrado / Física / Mestre em Física

L'objectif de cette thèse est d'investiguer les relations entre les propriétés topologiques d'une variété de contact et la dynamique des flots de Reeb dans la variété de contact en question. Dans la première partie de la thèse, nous établissons une relation entre la croissance de l’homologie de contact cylindrique d'une variété de contact et l'entropie topologique des flots de Reeb dans cette variété de contact. Nous utilisons ce résultat dans les chapitres 8 et 9 pour montrer l'existence d'un grand nombre des nouvelles variétés de contact de dimension 3 dans lesquelles tous les flots de Reeb ont entropie topologique positive. Dans le chapitre 10, nous prouvons un résultat obtenu en collaboration avec Chris Wendl qui donne une obstruction dynamique pour qu'une variété de contact de dimension 3 soit planaire. Cette obstruction est utilisée pour montrer que, si une variété de contact de dimension 3 possède un flot de Reeb qui est uniformément hyperbolique (Anosov) avec variétés invariantes traversalement orientables, alors cette variété de contact n'est pas planaire. Dans le chapitre 11, nous étudions l'entropie topologique des flots de Reeb dans les fibrés unitaires des surfaces de genre plus grand que 1. Nous montrons que la restriction de chaque flot de Reeb en au ensemble limite de presque toute fibre unitaire a une entropie topologique positive. / In this thesis we study the relations between the contact topological properties of contact manifolds and the dynamics of Reeb flows. On the first part of the thesis, we establish a relation between the growth of the cylindrical contact homology of a contact manifold and the topological entropy of Reeb flows on this manifold. We build on this to show in Chapter 6 that if a contact manifold M admits a hypertight contact form A for which the cylindrical contact homology has exponential homotopical growth rate, then the Reeb flow of every contact form on M has positive topological entropy. Using this result, we exhibit in Chapter 8 and 9 numerous new examples of contact 3-manifolds on which every Reeb flow has positive topological entropy. On Chapter 10 we present a joint result with Chris Wendl that gives a dynamical obstruction for contact 3-manifold to be planar. We then use the obstruction to show that a contact 3-manifold that possesses a Reeb flow that is a transversely orientable Anosov flow, cannot be planar. On Chapter 11 we study the topological entropy for Reeb flows on spherizations. The result we obtain is a refinement of a result of Macarini and Schlenk, that states that every Reeb flow on the unit tangent bundle U of a high genus surface S has positive topological entropy. We show that for any Reeb flow on U, the omega-limit of almost every Legendrian fiber is a compact invariant set on which the dynamics has positive topological entropy.

Indiana University-Purdue University Indianapolis (IUPUI) / We derive the differential identities for isomonodromic tau functions, describing their monodromy dependence. For Painlev´e equations we obtain them from the relation of tau function to classical action which is a consequence of quasihomogeneity of corresponding Hamiltonians. We use these identities to solve the connection problem for generic solution of Painlev´e-III(D8) equation, and homogeneous Painlev´e-II equation. We formulate conjectures on Hamiltonian and symplectic structure of general isomonodromic deformations we obtained during our studies and check them for Painlev´e equations.

Phonon anharmonicity is critical for describing various phenomena in crystals, including lattice thermal conductivity, thermal expansion, structural phase transitions, and many others. Including anharmonicity in the calculation of condensed matter observables developed rapidly in the past decade. First-principles computation of cubic phonon interactions have been performed in many systems, and the quartic interactions have begun to receive more attention. In this study, reliable Hamiltonians are constructed purely in terms of quadratic, cubic, and quartic irreducible derivatives, which are calculated efficiently and precisely using the lone and bundled irreducible derivative approaches (LID and BID). The resulting Hamiltonians give rise to a nontrivial many-phonon problem which requires some approximation in order to compute observables. We implemented self-consistent diagrammatic approaches to evaluate the phonon self-energy, including the Hartree-Fock approximation for phonons and quasiparticle perturbation theory, where both the 4-phonon loop and the real part of the 3-phonon bubble are employed during self-consistency. Additionally, we implemented molecular dynamics in order to yield the numerically exact solution in the classical limit. The molecular dynamics solution is robust for directly comparing to experimental results at sufficiently high temperatures, and for assessing our diagrammatic approaches in the classical limit. Anharmonic vibrational Hamiltonians were constructed for CaF₂, ThO₂, and UO₂. Diagrammatic approaches were used to evaluate the phonon self-energy, yielding the phonon lineshifts and linewidths and the thermal conductivity within the relaxation time approximation. Our systematic results allowed us to resolve the paradox of why first-principles phonon linewidths strongly disagree with results extracted from inelastic neutron scattering (INS). We demonstrated that the finite region in reciprocal space required in INS data analysis, the 𝑞-voxel, must be explicitly accounted for within the calculation in order to draw a meaningful comparison. We also demonstrated that the 𝑞-voxel is important to properly compare the spectrum measured in inelastic X-ray scattering (IXS), despite the fact that the ?-voxel is much smaller. Accounting for the 𝑞-voxel, we obtained good agreement for the scattering function linewidths up to intermediate temperatures. Additionally, good agreement was obtained for the thermal conductivity. Another topic we addressed is translation symmetry breaking caused by factors such as defects, chemical disorders, and magnetic order. These phenomena will lead to shifts and a broadening of the phonon spectrum, and formally the single-particle Green’s function encodes these effects. However, it is often desirable to obtain an approximate non-interacting spectrum that contains the effective shifts of the phonon frequencies, allowing straightforward comparison with experimentally measured scattering peak locations. Such an effective phonon dispersion can be obtained using a band unfolding technique, and in this study, we formulated unfolding in the context of irreducible derivatives. We showcased the unfolding of phonons in UZr₂, where chemical disorder is present, and compared the results with experimental IXS data. Additionally, we extended the unfolding technique to anharmonic terms and demonstrated this using 3rd and 4th order terms in the antiferromagnetic phase of UO₂.

L'objectif de ce thèse est l'étude des systèmes dynamiques avec interaction à longue portée. La complexité de leur dynamique met en évidence des propriétés contre-intuitives et inattendues, comme l'existence d'états stationnaires hors-équilibre (QSS). Dans le QSS on peut observer des propriétés particulières: chaleur spécifique négative, inéquivalence des ensembles statistiques et phénomènes d'auto-organisation. Les théories des interactions LR ont été appliquées pour décrire la dynamique des systèmes auto-gravitants, de tourbillons bidimensionnels, de systèmes avec interactions onde-particule et des plasmas chargés. Mon travail s'est tout d'abord consacré à l'extension de la solution de Lynden-Bell pour le modèle HMF, en généralisant l'analyse à des conditions initiales de «water-bag" à plusieurs niveaux, qui approchent des conditions initiales continues. En suite je me suis intéressé à la caractérisation formelle de la thermodynamique des QSS dans l'ensemble statistique canonique. En appliquant la théorie standard, il est possible de mesurer une chaleur spécifique "cinétique'' négative. Cette propriété inattendue amène à la violation du second principe de la thermodynamique. Un tel résultat nous pousse à reconsidérer l'applicabilité de la théorie thermodynamique actuelle aux systèmes LR. En suite j'ai étudié, pour le modèle α-HMF, la persistance des caractéristiques typiques du régime LR, dans le limite dynamique à courte portée. Les résultats suggèrent une généralisation de la définition des systèmes LR. Le dernier chapitre est consacré à la caractérisation d'un nouveau modèle LR, extension naturelle du précédent α-HMF et d'intérêt potentiel applicatif. / The scope of this thesis is the study of systems with long-range interactions (LR). The complexity of their dynamics evidences counter-intuitive and unexpected properties, as for instance the existence of out-of-equilibrium stationary states (QSS). Considering a system in the QSS, one may observe peculiar properties, as negative specific heat, statistical ensemble inequivalence and phenomena of self-organizations. The main theories of long-range interactions have been applied to describing self-gravitating systems, two-dimensional vortices, systems with wave-particle interactions and charged plasmas. My work has been initially dedicated to extending the Lynden-Bell solution for the HMF model, generalizing the analysis to multi-level water-bag initial condition that could approximate continuous distributions. Then I concentrated to the formal characterization of the thermodynamics of QSS in the canonical statistical ensemble. By applying the standard theory, it is possible to measure negative “kinetic” specific heat. This latter unexpected property leads to a violation of the second principle of thermodynamics. Such result forces us to reconsider the applicability of the accepted thermodynamic theory to LR systems. Afterwards I studied, in the context of the α-HMF model, the persistence of the typical characteristics of the LR regime in the limit of short-range dynamics. The results obtained suggests a generalization of the definition of LR systems. The last chapter is dedicated to the characterization of a novel LR model, a natural extension of α-HMF and of potential applicability.

In this work we study the existence of nontrivial weak solutions for some Hamiltonian elliptic systems in dimension two, involving a potential function and nonlinearities which possess maximal growth with respect to a critical curve (hyperbola). We consider four different cases. First, we study Hamiltonian systems in bounded domains with potential function identically zero. The second case deals with systems of equations on the whole space, the potential function is bounded from below for some positive constant and satisfies some integrability conditions, while the nonlinearities involve weight functions containing a singulatity at the origin. In the third case, we consider systems with coercivity potential functions and nonlinearities with weight functions which may have singularity at the origin or decay at infinity. In the last case, we study Hamiltonian systems, where the potential can be unbounded or can vanish at infinity. To establish the existence of solutions, we use variational methods combined with Trudinger-Moser type inequalities for Lorentz-Sobolev spaces and a finite-dimensional approximation. / Neste trabalho estudamos a existência de soluções fracas não triviais para sistemas hamiltonianos do tipo elíptico, em dimensão dois, envolvendo uma função potencial e não linearidades tendo crescimento exponencial máximo com respeito a uma curva (hipérbole) crítica. Consideramos quatro casos diferentes. Primeiramente estudamos sistemas de equações em domínios limitados com potencial nulo. No segundo caso, consideramos sistemas de equações em domínio ilimitado, sendo a função potencial limitada inferiormente por alguma constante positiva e satisfazendo algumas de integrabilidade, enquanto as não linearidades contêm funções-peso tendo uma singularidade na origem. A classe seguinte envolve potenciais coercivos e não linearidades com funções peso que podem ter singularidade na origem ou decaimento no infinito. O quarto caso é dedicado ao estudo de sistemas em que o potencial pode ser ilimitado ou decair a zero no infinito. Para estabelecer a existência de soluções, utilizamos métodos variacionais combinados com desigualdades do tipo Trudinger-Moser em espaços de Lorentz-Sobolev e a técnica de aproximação em dimensão finita.

Cette thèse est motivée par un problème aéronautique: le ballottement du carburantdans des réservoirs d’ailes d’avion très flexibles. Les vibrations induites par le couplagedu fluide avec la structure peuvent conduire à des problèmes tels que l’inconfort des passagers,une manoeuvrabilité réduite, voire même provoquer un comportement instable. Cette thèse apour objectif de développer de nouveaux modèles d’interaction fluide-structure, en mettant enoeuvre la théorie des systèmes Hamiltoniens à ports d’interaction (pHs). Le formalisme pHsfournit d’une part un cadre unifié pour la description des systèmes multi-physiques complexeset d’autre part une approche modulaire pour l’interconnexion des sous-systèmes grâce auxports d’interaction. Cette thèse s’intéresse aussi à la conception de contrôleurs à partir desmodèles pHs. Des modèles pHs sont proposés pour les équations de ballottement du liquide en partantdes équations de Saint Venant en 1D et 2D. L’originalité du travail est de donner des modèlespHs pour le ballottement dans des réservoirs en mouvement. Les ports d’interaction sont utiliséspour coupler la dynamique du ballottement à la dynamique d’une poutre contrôlée par desactionneurs piézo-électriques, celle-ci étant préalablement modélisée sous forme pHs. Aprèsl’écriture des équations aux dérivées partielles dans le formalisme pHs, une approximation endimension finie est obtenue en utilisant une méthode pseudo-spectrale géométrique qui conservela structure pHs du modèle continu au niveau discret. La thèse propose plusieurs extensionsde la méthode pseudo-spectrale géométrique, permettant la discrétisation des systèmesavec des opérateurs différentiels du second ordre d’une part et avec un opérateur d’entrée nonborné d’autre part. Des essais expérimentaux ont été effectués sur une structure constituéed’une poutre liée à un réservoir afin d’assurer la validité du modèle pHs du ballottementdu liquide couplé à la poutre flexible, et de valider la méthode pseudo-spectrale de semi-discrétisation.Le modèle pHs a finalement été utilisé pour concevoir un contrôleur basé surla passivité pour réduire les vibrations du système couplé. / This thesis is motivated by an aeronautical issue: the fuel sloshing in tanksof very flexible wings. The vibrations due to these coupled phenomena can lead to problemslike reduced passenger comfort and maneuverability, and even unstable behavior. Thisthesis aims at developing new models of fluid-structure interaction based on the theory ofport-Hamiltonian systems (pHs). The pHs formalism provides a unified framework for thedescription of complex multi-physics systems and a modular approach for the coupling ofsubsystems thanks to interconnection ports. Furthermore, the design of controllers using pHsmodels is also addressed. PHs models are proposed for the equations of liquid sloshing based on 1D and 2D SaintVenant equations and for the equations of structural dynamics. The originality of the workis to give pHs models of sloshing in moving containers. The interconnection ports are used tocouple the sloshing dynamics to the structural dynamics of a beam controlled by piezoelectricactuators. After writing the partial differential equations of the coupled system using thepHs formalism, a finite-dimensional approximation is obtained by using a geometric pseudospectralmethod that preserves the pHs structure of the infinite-dimensional model at thediscrete level. The thesis proposes several extensions of the geometric pseudo-spectral method,allowing the discretization of systems with second-order differential operators and with anunbounded input operator. Experimental tests on a structure made of a beam connected to atank were carried out to validate both the pHs model of liquid sloshing in moving containersand the pseudo-spectral semi-discretization method. The pHs model was finally used to designa passivity-based controller for reducing the vibrations of the coupled system.