by Chan Kwan Leung. / Parallel title in Chinese characters. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves 168-169). / by Chan Kwan Leung. / Acknowledgement --- p.i / Abstract --- p.ii / Chapter 1. --- Introduction / Chapter 1.1 --- Objective of our variational method --- p.2 / Chapter 1.2 --- Outline of the content --- p.5 / Chapter 2. --- Formulation of the new variational method / Chapter 2.1 --- Formulation --- p.14 / Chapter 2.2 --- Motivation --- p.15 / Chapter 3. --- The variational method applied to the anharmonic oscillator problem / Chapter 3.1 --- Formalism --- p.18 / Chapter 3.2 --- Relationship with usual variational method --- p.32 / Chapter 3.3 --- Relationship with W.K.B. approximation --- p.37 / Chapter 3.4 --- Perturbative corrections --- p.45 / Chapter 3.5 --- Diagonalization of non-orthogonal basis --- p.57 / Chapter 3.6 --- Perturbative corrections using the non-orthogonal basis --- p.72 / Chapter 3.7 --- Some previous works on the anharmonic oscillator problem --- p.85 / Chapter 4. --- The variational method applied to the helium-like atomic problem / Chapter 4.1 --- Previous work on the problem --- p.90 / Chapter 4.2 --- Formulation of the variational method on the problem --- p.95 / Chapter 4.3 --- Zeroth order results for atomic helium --- p.103 / Chapter 4.4 --- Diagonalization using the non-orthogonal basis --- p.109 / Chapter 4.5 --- Results for some helium-like ions --- p.136 / Chapter 4.6 --- Possibility of generalization to systems with more electrons --- p.140 / Chapter 5 --- Concluding remarks / Chapter 5.1 --- Range of applicability of our variational method --- p.164 / Chapter 5.2 --- Ground state problem --- p.165 / Chapter 5.3 --- Completeness of our 'basis' --- p.166 / References --- p.168

Made available in DSpace on 2017-07-21T19:25:58Z (GMT). No. of bitstreams: 1 Samyr Ariel Abdulack.pdf: 1631733 bytes, checksum: 7d939ce2577fb7894eb9d9a200d53eb2 (MD5) Previous issue date: 2010-03-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Numerical solutions of a mathematical system presents noise due to the truncation and roundoff errors. If chaos cannot ruled out then these errors are amplified. The Hamiltonian dynamical systems may present chaos and periodicity in the same phase space for a given range of values of the control parameter. In particular the standard map is a Hamiltonian system widely investigated to be derived for many physical systems of interest. An answer for question of validity of numerical solutions is the shadowing of physical trajectories that ensures the existence of real orbits that stays near of noisy trajectories for long time. If the system present hyperbolic structure, then all conditions are fulfilled and shadowing can be done for every point of the set where the system is defined. On the other hand, most of systems are nonhyperbolic like standard map. This loss of hyperbolicity can ocurr in two ways: unstable dimension variability and tangencies between manifolds. This study aims the shadowing problem and investigate regions where tangencies can ocurr caracterizing periodic orbits structure in phase space. With the knowledge of unstable periodic orbits is possible to obtain manifolds and verify regions where shadowing is broken by tangencies. For this the Schmelcher-Diakonos method is employed for found periodic orbits. The manifolds are found by taking a ball of initial conditions in linear neighborhood of points of any period and by iteration foward in time of map to represent an aproximation of unstable manifold and by reverse iteration in time to represent stable manifold. As a result we found regions were possible tangencies ocurr and shadowing cannot be done. / Os cálculos numéricos envolvendo as soluções de um sistema matemático apresentam ruído em razão dos erros de truncamento e arredondamento efetuados a cada passo. Se o sistema dinâmico apresentar caos, então estes erros são amplificados. Os sistemas dinâmicos hamiltonianos podem apresentar regiões disjuntas onde há ocorrência de caos e periodicidade no mesmo espaço de fases para uma dada faixa de valores do parâmetro de controle. Em particular, o mapa padrão é um sistema hamiltoniano amplamente investigado por ser proveniente de vários sistemas físicos de interesse. Uma resposta à questão da validade das soluções numéricas é o sombreamento que garante a existência de órbitas reais próximas de órbitas ruidosas por longo tempo. Se o sistema apresentar estrutura hiperbólica, então o sombreamento é garantido inteiramente para o conjunto onde o sistema está definido. Por outro lado, a maioria dos sistemas não apresenta estrutura hiperbólica, a exemplo do que ocorre com o mapa padrão. Esta quebra de hiperbolicidade pode ocorrer de duas maneiras: pela variabilidade da dimensão instável ou por tangências entre as variedades. Este trabalho tem como objetivo estudar o problema do sombreamento e compreender as técnicas de contenção e refinamento bem como investigar as regiões onde ocorrem possíveis tangências buscando caracterizar a estrutura das órbitas periódicas instáveis no espaço de fases. De posse das órbitas periódicas instáveis é possível obter as variedades associadas e verificar regiões onde há quebra de sombreamento por tangências. Para tanto, emprega-se o método de Schmelcher- Diakonos para encontrar as órbitas periódicas. As variedades são encontradas tomando uma bola de condições iniciais na vizinhança linear dos pontos de algum período e iterando o mapa para representar aproximadamente a variedade instável e iterando a inversa do mapa para encontrar a variedade estável. Como resultado verificam-se regiões onde possivelmente ocorrem tangências e o sombreamento não pode ser efetuado.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Physics, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Includes bibliographical references. / by Jens Hoppe. / Ph.D.

Estudaremos uma fam lia de sistemas hamiltonianos no R 4 , H : R 4 R, satisfazendo certas condi c oes, dependendo de um parametro . Iremos ca- racterizar algumas condi c oes sobre n veis de energia desse sistema, que nos permitem concluir existencia e destrui c ao de toros invariantes, em tais n veis de energia. Al em disso, podemos concluir que o fluxo hamiltoniano, restrito a esses n veis de energia, possui entropia topol ogica positiva. / We will study a family of Hamiltonian Systems in R 4 , satisfying certain conditions, H : R 4 R, depending of a parameter . We will characterize some conditions about the energy levels of this system, which allow us to conclude existence and destruction of invariant torus, at such energy levels. Moreover, we can conclude that the hamiltonian flow, restricted to these energy level, has positive topological entropy.

A Hamiltonian system in potential form (formula in the original abstract) subject to smooth constraints on q can be viewed as a Hamiltonian system on a manifold, but numerical computations must be performed in Rn. In this paper methods which reduce "Hamiltonian differential algebraic equations" to ODEs in Euclidean space are examined. The authors study the construction of canonical parameterizations or local charts as well as methods based on the construction of ODE systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an invariant manifold. In each case, a Hamiltonian system of ordinary differential equations is produced. The stability of the constraint invariants and the behavior of the original Hamiltonian along solutions are investigated both numerically and analytically.

We prove the existence and linear stability of quasi-periodic breather solutions in a 1d Hamiltonian lattice of identical, weakly-coupled, anharmonic oscillators with general on-site potentials and under the effect of long-ranged interaction, via de KAM technique. We prove the persistence of finite-dimensional tori which correspond in the uncoupled limit to N arbitrary lattice sites initially excited. The frequencies of the invariant tori of the perturbed system are only slightly deformed from the frequencies of the unperturbed tori.

Tez (Yüksek Lisans) - Süleyman Demirel Üniversitesi, Fen Bilimleri Enstitüsü, Matematik Anabilim Dalı, 2008. / Kaynakça var.

Thesis (Ph. D.)--University of Washington, 2003. / Vita. Includes bibliographical references (p. 93-103).

The influence of matter described by the Vlasov equation, on the evolution of anisotropy in the spatially-homogeneous universes, called the Bianchi cosmologies, is studied. Due to the spatial-homogeneity, the Einstein equations for each Bianchi Type are reduced to a set of coupled ordinary differential equations, which has Hamiltonian form with the metric components being the canonical coordinates. In the vacuum Bianchi cosmologies, it is known that a curvature potential, which comes from the symmetries of the three-dimensional Lie groups, determines the basic properties of the evolution of anisotropy. In this work, matter potentials are constructed for Vlasov matter. They are obtained by first introducing a new matter action principle for the Vlasov equation, in terms of a conjugate pair of functions, and then enforcing the symmetry to obtain a reduction. This yields an expression for the matter potential in terms of the phase space density, which is further reduced by assuming cold streaming matter. Some vacuum Bianchi cosmologies and Type I with Vlasov matter are compared. It is shown that the Vlasov-matter potential for cold streaming matter results in qualitatively distinct dynamics from the well-known vacuum Bianchi cosmologies. / text

Η μελέτη συνήθων διαφορικών εξισώσεων συχνά χρησιμοποιεί μεθόδους γνωστές από την κλασική Μηχανική. Η πιο γνωστή από αυτές ϕέρει το όνομα του εμπνευστή της, του Ιρλανδού Sir William Rowan Hamilton (1805 - 1865), κι αποτελεί μία μαθηματικά πλήρη ϑεωρία για τα λεγόμενα συστήματα Hamilton. Πρόσφατα, όμως, δομές τύπου Hamilton άρχισαν να μελετώνται και σε συστήματα μερικών διαφορικών εξισώσεων, συγκεκριμένα εξισώσεων εξέλιξης. Σκοπός της παρούσας εργασίας είναι η ανάπτυξη της ϑεωρίας Hamilton για τα συστήματα αυτά και ιδιαίτερα για τις περιπτώσεις εκείνες που εμφανίζουν ολοκληρωσιμότητα. Η γραμμή που ϑα ακολουθήσουμε έχει ως κύριο οδηγό τις συμμετρίες των διαφορικών εξισώσεων, ένα πολύ χρήσιμο εργαλείο για την επίλυση οποιασδήποτε διαφορικής εξίσωσης, που πρώτος ανέδειξε ο Νορβηγός Marius Sophus Lie (1842 - 1899). Στο πρώτο κεφάλαιο λοιπόν γίνεται μία εισαγωγή στην ϑεωρία των (γεωμετρικών) συμμετριών, ενώ επίσης παρουσιάζονται τρόποι επίλυσης και γενικότερα αντιμετώπισης ξεχωριστά συνήθων και μερικών διαφορικών εξισώσεων με την χρήση των ομάδων συμμετρίας τους. Το δεύτερο κεφάλαιο ϕιλοδοξεί να αναδείξει την αντιστοιχία μεταξύ των συμμετριών ενός συστήματος διαφορικών εξισώσεων και των νόμων διατήρησης στους οποίους υπακούει το ϕυσικό σύστημα που περιγράφουν. Αυτό είναι και το περιεχόμενο του ϑεωρήματος που διατύπωσε η Γερμανίδα Amalie Emmy Noether (1882 - 1935), το οποίο ισχύει και στην ειδική περίπτωση των συστημάτων Hamilton. Το πρώτο, λοιπόν, ϐήμα προς αυτήν την κατεύθυνση είναι η επέκταση της έννοιας της συμμετρίας στις λεγόμενες γενικευμένες συμμετρίες, με ιδιαίτερη έμφαση στις εξισώσεις εξέλιξης. Το δεύτερο είναι ουσιαστικά μια μικρή εισαγωγή στην ϑεωρία μεταβολών, απαραίτητη όμως και για τα επόμενα κεφάλαια. Την γνωστή ϑεωρία Hamilton για πεπερασμένα συστήματα, συστήματα δηλαδή συνήθων διαϕορικών εξισώσεων πραγματεύεται το τρίτο κεφάλαιο. Σκοπός του κεφαλαίου αυτού δεν είναι η πλήρης περιγραφή της ϑεωρίας, αλλά η διατύπωση των εννοιών εκείνων που μπορούν να γενικευτούν και στην περίπτωση των απειροδιάστατων συστημάτων. Για τον λόγο αυτό έχει προτιμηθεί η κάπως πιο αφηρημένη και σίγουρα όχι τόσο συνηθισμένη περιγραφή στο πλαίσιο της γεωμετρίας Poisson. Αντιμετωπίζοντας τις συμπλεκτικές δομές, οι οποίες επικρατούν στην ϐιβλιογραφία, ως μια υποπερίπτωση των γενικότερων δομών Poisson, έχουμε ουσιαστικά αποφύγει τελείως την χρήση διαφορικών μορφών, στρέφοντας περισσότερο την προσοχή στις ομάδες συμμετρίας Hamilton, μία έννοια-κλειδί για την ολοκληρωσιμότητα των συστημάτων αυτών. Στο τέταρτο κεφάλαιο παρουσιάζουμε το κεντρικό ϑέμα αυτής της εργασίας, δηλαδή τη ϑεωρία Hamilton για απειροδιάστατα συστήματα εξισώσεων εξέλιξης, και ειδικότερα την ολοκληρωσιμότητα τους. Τα ϐασικά μας εργαλεία είναι αυτά που παρουσιάστηκαν νωρίτερα, δηλαδή οι (γενικευμένες) συμμετρίες και οι νόμοι διατήρησης από την μια, και τα διανυσματικά πεδία Hamilton από την άλλη που μας επιτρέπουν την μεταξύ τους αντιστοιχία. Με ϐάση αυτά τα εργαλεία ϐλέπουμε πως η μελέτη πολλών μερικών διαφορικών εξισώσεων ϑυμίζει εκείνων των κλασικών συστημάτων Hamilton της Μηχανικής. Στην παραπάνω αντιστοιχία ϐασίζεται και η έννοια των δι-Χαμιλτονικών συστημάτων, την οποία μελετάμε στο πέμπτο κεφάλαιο. Μέσα από το παράδειγμα της εξίσωσης Korteweg-de Vries αναδεικνύονται τα πλεονεκτήματα της εύρεσης δύο διαφορετικών, ανεξάρτητων εκφράσεων Hamilton, που οδηγούν στην κατασκευή άπειρων συμμετριών ή ακόμα και νόμων διατήρησης. Η διπλή αυτή δομή Hamilton των απειροδιάστατων συστημάτων συνδέεται, όπως ϑα δούμε, με την ολοκληρωσιμότητα είτε με την έννοια του Liouville, είτε με διάφορα άλλα κριτήρια. Γνωστά παραδείγματα παραθέτονται, πέρα από την KdV, όπως η εξίσωση Schroedinger, η modified KdV, κι άλλες μη γραμμικές κυματικές εξισώσεις. Στο έκτο και τελευταίο κεφάλαιο παρουσιάζουμε την περίπτωση, όπου ένα σύστημα επιδέχεται πολλαπλή δομή Hamilton. Τέτοιου είδους συστήματα μας επιτρέπουν να δούμε προϋπάρχουσες έννοιες από την ϑεωρία Hamilton, αλλά κι όχι μόνο, κάτω από μία άλλη σκοπιά. Γι΄ αυτό κι έχουν απασχολήσει την σύγχρονη ϐιβλιογραφία, πάνω στην οποία κάνουμε μία σύντομη επισκόπηση, τόσο στο κομμάτι εκείνο που ασχολείται με τις πρόσφατες εξελίξεις της ϑεωρίας Hamilton, όσο και με την μελέτη γενικότερα της ολοκληρωσιμότητας των μερικών διαφορικών εξισώσεων. / The study of ordinary differential equations has often borrowed well known methods from Classical Mechanics. The most popular one is due to Sir William Rowan Hamilton (1805-1865), which has become a complete mathematical theory for the so-called Hamiltonian systems. Recently, Hamiltonian structures have been developed for systems of partial differential equations, particularly evolution equations. The purpose of this master thesis is to present the Hamiltonian theory for this type of systems, and especially for integrable equations. Our description is based on Symmetries, a useful tool for solving any differential equation, first discovered by Marius Sophus Lie (1842-1899). Thus, an introduction to his theory of point or geometrical symmetries is given in the first chapter, along with some applications, such as integration of ordinary differential equations and group-invariant solutions of partial differential equations. In the second chapter we discuss the connection between the symmetries of a system of differential equations and the conservation laws of the physical problem that they describe. That is the content of Noether’s theorem, which also holds in the particular case of Hamiltonian systems. The first step towards this direction is the generalization of the basic symmetry concept, and the second one is a small introduction to variational problems, also necessary for the next chapters. The well known Hamilton’s theory for finite systems is presented in the third chapter. We do not wish to describe the whole theory in full detail but only focus on these points that will be needed to handle the infinite-dimensional case. Therefore, we introduce the general notion of a Poisson structure, instead of the more familiar symplectic one. Avoiding the use of differential forms almost entirely, we concentrate on the Hamiltonian symmetries and their key role in the reduction theory of these systems. In Chapter 4 lies the heart of the subject, the Hamiltonian approach to a system of evolution equations. We start off by drawing an analogy between first order ordinary differential equations and evolution equations, and then we establish the fundamental concepts of the Hamiltonian franework, i.e. the Poisson bracket and Hamiltonian vector fields. Through another version of Noether’s theorem, we are able to explore, once again, the correspondence between (generalized) symmetries and conservation laws. Thus, we see that the study of several partial differential equations is in some way very close to the one of classical mechanical Hamiltonian systems. Evolution equations possessing, not just one, but two Hamiltonian structures, called bi-Hamiltonian systems, are discussed in the next chapter. The advantages of finding two different, independent Hamiltonian expressions are pointed out through the example of the Korteweg-de Vries equation. We show that such systems have an infinite number of symmetries and, subject to a mild compatibility condition, they also have an infinite number of conservation laws. Therefore they are completely integrable in Liouville’s sense. Several examples are presented, besides the KdV equation, such as the nonlinear Schroedinger, the modified KdV and other nonlinear wave equations. The final chapter is devoted to some of the recent publications, regarding multi-Hamiltonian evolution equations. This type of systems puts the classical Hamiltonian theory of ordinary differential equations in a new perspective and at the same time allows us to draw some connections with other integrability criteria used in the field of partial differential equations.