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.

This thesis is devoted to two aspects of nonlinear PDEs which are fundamental for the understanding of the order and coherence observed in the underlying physical systems. These are symmetry properties and soliton solutions. We analyse these fundamental aspects for a number of models arising in various branches of theoretical physics and appli ed mathematics. We start with a fluid model of a plasma in the case of a general polytropic process. We propose a method of the analysis of unmagnetized travelling structures, alternative to the conventional formalism of Sagdeev 's pseudopotential. This method is then utilized to obtain the existence domain for compressive solitons and to establish the absence of rarefactive solitons and monotonic double layers in a two-component plasma. The second class of models under consideration arises in (2+1)-dimensional condensed matter physics. These are the Abelian gauge theories with Chern-Simons term, which are currently considered as candidates for the description of high-Te superconductivity and fra ctional quantum Hall effect. The emphasis here is on nonrelativistic theories. The standard model of a self-gravitating gas of nonrelativistic bosons coupled to the Chern-Simons gauge field is capable of describing asymptotically vanishing field configurations , such as lump-like solitons. We formulate an alternative model, which describes systems of repulsive particles with a background electric charge and allows to incorporate asymptotically nonvanishing configurations, such as condensate and its topological excitations. We demonstrate the absence of the condensate state in the standard nonrelativistic gauge theory and relate this fact to the inadequate Lagrangian formulation of its nongauged precursor. Using an appropriate modification of this Lagrangian as a basis for the gauge theory naturally leads to the new model. Reformulating it as a constrained Hamiltonian system allows us to find two self-duality limit s and construct a large variety of self-dual solutions. We demonstrate the equivalence of the model with the background charge and the standard model in the external magnetic field. Finally we discuss nontopological bubble solutions in Chem-Simons-Maxwell theories and demonstrate their absence in nonrelativistic theories. Finally, we consider a model of a nonhomogeneous nonlinear string. We continue the group theoretical classification of the string equations initiated by Ibragimov et al. and present their preliminary group classification with respect to a countable dimensional subalgebra of their equivalence algebra. This subalgebra is an extension of the 10-dimensional subalgebra considered by Ibragimov et al. Our main result here is a table of non-equivalent equations possessing an additional symmetry. / Thesis (Ph.D.)-University of Natal, 1994.

Classical mechanics provides a phase space representation of mechanical systems in terms of position and momentum state variables. The Hamiltonian system, a set of partial differential equations, defines a vector field in phase space and uniquely determines the evolutionary process of the system given its initial state.A closed form solution describing system trajectories in phase space is only possible if the system of differential equations defining the Hamiltonian is linear. For nonlinear cases approximate and qualitative methods are required.Generalized phase space methods do not confine state variables to position and momentum, allowing other observables to describe the system. Such a generalization adjusts the description of the system to the required information and provides a method for studying physical systems that are not strictly mechanical.This thesis presents and uses the methods of generalized phase space to compare linear to nonlinear systems.Ball State UniversityMuncie, IN 47306 / Department of Physics and Astronomy

In this thesis we consider mathematical problems related to different aspects of hard sphere systems. In the first part we study planar billiards, which arise in the context of hard sphere systems when only one or two spheres are present. In particular we investigate the possibility of elliptic periodic orbits in the general construction of hyperbolic billiards. We show that if non-absolutely focusing components are present there can be elliptic periodic orbits with arbitrarily long free paths. Furthermore, we show that smooth stadium like billiards have elliptic periodic orbits for a large range of separation distances. In the second part we consider hard sphere systems with a large number of particles, which we model by the Boltzmann equation. We develop a new approach to derive hydrodynamic limits, which is based on classical methods of geometric singular perturbation theory of ordinary differential equations. This method provides new geometric and dynamical interpretations of hydrodynamic limits, in particular, for the of the dissipative Boltzmann equation.

Covariance assignment theory, introduced in the late 1980s, provided the only means to directly control the steady-state error properties of a linear system subject to Gaussian white noise and parameter uncertainty. This theory, however, does not extend to control of the transient uncertainties and to date there exist no practical engineering solutions to the problem of directly and optimally controlling the uncertainty in a linear system from one Gaussian distribution to another. In this thesis I design a dual-mode Receding Horizon Controller (RHC) that takes a controllable, deterministic linear system from an arbitrary initial covariance to near a desired stationary covariance in finite time. The RHC solves a sequence of free-time Optimal Control Problems (OCP) that directly controls the fundamental solution matrices of the linear system; each problem is a right-invariant OCP on the matrix Lie group GLn of invertible matrices. A terminal constraint ensures that each OCP takes the system to the desired covariance. I show that, by reducing the Hamiltonian system of each OCP from T?GLn to gln? x GLn, the transversality condition corresponding to the terminal constraint simplifies the two-point Boundary Value Problem (BVP) to a single unknown in the initial or final value of the costate in gln?. These results are applied in the design of a dual-mode RHC. The first mode repeatedly solves the OCPs until the optimal time for the system to reach the de- sired covariance is less than the RHC update time. This triggers the second mode, which applies covariance assignment theory to stabilize the system near the desired covariance. The dual-mode controller is illustrated on a planar system. The BVPs are solved using an indirect shooting method that numerically integrates the fundamental solutions on R4 using an adaptive Runge-Kutta method. I contend that extension of the results of this thesis to higher-dimensional systems using either in- direct or direct methods will require numerical integrators that account for the Lie group structure. I conclude with some remarks on the possible extension of a classic result called Lie?s method of reduction to receding horizon control.