Hamiltonian dynamics Research Papers - Academia.edu (original) (raw)

We develop a regularization for binary collisions in some restricted 3-body problems moving in planar one-dimensional spaces with constant curvature. The main characteristic of the regularization is that it preserves the Hamiltonian... more

We develop a regularization for binary collisions in some restricted 3-body problems moving in planar one-dimensional spaces with constant curvature. The main characteristic of the regularization is that it preserves the Hamiltonian structure of the equations and it regularizes all the binary collisions with just one transformation. We apply this global symplectic regularization to the 2-body problem on the unit circle and we show the global dynamics. Also, we tackle the restricted 3-body problem with one fixed center in the unit circle and we give the global dynamics for the case when it has two fixed centers.

This paper presents port-Hamiltonian models for describing flow dynamics of incompressible fluids in rigid pipelines with faults. Two types of faults are addressed in this paper: leaks and partial blockages. In order to facilitate the... more

This paper presents port-Hamiltonian models for describing flow dynamics of incompressible fluids in rigid pipelines with faults. Two types of faults are addressed in this paper: leaks and partial blockages. In order to facilitate the understanding of the modeling, the proposed formulation is introduced starting from the analogy between electrical and hydraulic circuits. Thanks to the port-Hamiltonian formalism the models proposed here have a particular structure that makes them plug-in and modular, so that they can be interconnected for building holistic models for faulty water distribution networks.

Hamilton’s principle is one of the great achievements of analytical mechanics. It offers a methodical manner of deriving equations motion for many systems, with the additional benefit that appropriate and correct boundary conditions are... more

Hamilton’s principle is one of the great achievements of analytical mechanics. It offers a methodical manner of deriving equations motion for many systems, with the additional benefit that appropriate and correct boundary conditions are automatically produced as part of the derivation. It allows insight into the manner that the system is modeled, as any modelling assumptions are clear and the effects of changing basic system properties become apparent and are accounted for in a consistent manner. Simplifications may also be made and Hamilton’s principle can be used as the basis for an approximate solution. Classical mechanics dictates that Hamilton’s principle can only be used for systems that are always composed of the same particles. This has been more recently extended to include systems whose constitutent particles change with time, including open systems of changing mass. In this chapter, we review the principle and its extended version and show through application to examples how it can lead to insightful observations about the system being modelled.

O formalismo Hamiltoniano é uma importante ferramenta no estudo de problemas físicos e matemáticos. Sistemas físicos que envolvem pêndulos e molas são muito empregados em cursos de mecânica clássica como exemplos de aplicação dos... more

O formalismo Hamiltoniano é uma importante ferramenta no estudo de problemas físicos e matemáticos. Sistemas físicos que envolvem pêndulos e molas são muito empregados em cursos de mecânica clássica como exemplos de aplicação dos formalismos estudados. Este trabalho tem por objetivos fazer uma breve introdução ao formalismo Hamiltoniano e mostrar de maneira mais detalhada a resolução, dentro deste formalismo, do problema de dois pêndulos acoplados por uma mola, encontrando seus modos normais de oscilação. São investigados também os invariantes adiabáticos deste sistema, quando diminuímos lentamente o comprimento de um dos fios de pêndulo, no limite do acoplamento fraco.
Palavras-Chave: Formalismo Hamiltoniano, Pêndulos Acoplados, Mola, Modos Normais de Oscilação, Invariantes Adiabáticos.

II EDIZIONE:

Risposte alle domande di teoria del corso di fisica matematica (FISICA, UNIPD, II ANNO)

In this 607 page book, in Spanish, are described in clear and complete way several problems of statics, mechanics, kinematics, dynamics and analytical dynamics. Includes non conventional subjects like perturbation theory, Kepler problem... more

In this 607 page book, in Spanish, are described in clear and complete way several problems of statics, mechanics, kinematics, dynamics and analytical dynamics. Includes non conventional subjects like perturbation theory, Kepler problem in parabolic coordinates, and connection with quantum mechanics.

It is my great honour to welcome you on behalf of the Bureau of IUTAM to this Symposium on Hamiltonian dynamics, vortex structures and turbulence. The Symposium has been in preparation for two years, and I congratulate our hosts here at... more

It is my great honour to welcome you on behalf of the Bureau of IUTAM to this Symposium on Hamiltonian dynamics, vortex structures and turbulence. The Symposium has been in preparation for two years, and I congratulate our hosts here at the Steklov Institute of the Russian Academy of Sciences for having prepared an excellent and wide-ranging programme, and for having succeeded in attracting such a distinguished gathering to debate problems in fluid dynamics many of which have a long history, yet still today present many challenges of a fundamental nature. The letters IUTAM, as you all know, stand for the International Union of Theoretical and Applied Mechanics. This Union is one of the International Scientific Union members of ICSU, the International Council for Science, which this year celebrates its 75th anniversary. The roots of IUTAM itself go back to the early Congresses in Mechanics, the first of which was held in Delft in the Netherlands, in 1924. IUTAM was formally established as an International Union at the 7th Congress, which was held in London in 1948. The 13th Congress of Theoretical and Applied Mechanics was held here in Moscow in 1972, under the Presidency of the great Mushkhelishvili. The most recent 21st Congress was held in Warsaw in 2004, and the next will be held in Adelaide, South Australia, in 2008.
Professor Keith Moffatt, Vice-President, IUTAM

We use the global stochastic analysis tools introduced by P. A. Meyer and L. Schwartz to write down a stochastic generalization of the Hamilton equations on a Poisson manifold that, for exact symplectic manifolds, satisfy a natural... more

We use the global stochastic analysis tools introduced by P. A. Meyer and L. Schwartz to write down a stochastic generalization of the Hamilton equations on a Poisson manifold that, for exact symplectic manifolds, satisfy a natural critical action principle similar to the one encountered in classical mechanics. Several features and examples in relation with the solution semimartingales of these equations are presented.

In the 1830s, W. R. Hamilton established a formal analogy between optics and mechanics by constructing a mathematical equivalence between the extremum principles of ray optics (Fermat's principle) and corpuscular mechanics (Maupertuis's... more

In the 1830s, W. R. Hamilton established a formal analogy between optics and mechanics by constructing a mathematical equivalence between the extremum principles of ray optics (Fermat's principle) and corpuscular mechanics (Maupertuis's principle). Almost a century later, this optical-mechanical analogy played a central role in the development of wave mechanics. Schrödinger was well acquainted with Hamilton's analogy through earlier studies. From Schrödinger's research notebooks, we show how he used the analogy as a heuristic tool to develop de Broglie's ideas about matter waves and how the role of the analogy in his thinking changed from a heuristic tool into a formal constraint on possible wave equations. We argue that Schrödinger only understood the full impact of the optical-mechanical analogy during the preparation of his second communication on wave mechanics: Classical mechanics is an approximation to the new undulatory mechanics, just as ray optics is an approximation to wave optics. This completion of the analogy convinced Schrödinger to stick to a realist interpretation of the wave function, in opposition to the emerging mainstream. The transformations in Schrödinger's use of the optical-mechanical analogy can be traced in his research notebooks, which offer a much more complete picture of the development of wave mechanics than has been previously thought possible.

Bu pdf fizik (veya astronomi, matematik) lisans öğrencilerine kuantum mekaniğinin kullandığı matematiğin göründüğü kadar karmaşık olmadığını göstermek amacıyla Cohen ve Shankar'ı kaynak alarak oluşturduğum yaklaşık 80 sayfalık bir... more

Bu pdf fizik (veya astronomi, matematik) lisans öğrencilerine kuantum mekaniğinin kullandığı matematiğin göründüğü kadar karmaşık olmadığını göstermek amacıyla Cohen ve Shankar'ı kaynak alarak oluşturduğum yaklaşık 80 sayfalık bir nottur. Parçacıkları artık doğrudan üç konum ve üç momentum ile tanımlamak (ara ara klasik mekaniğe de değiniyorum) yerine bir dalga fonksiyonu ile tanımlayarak başlayıp küresel harmonikler ile bitiriyorum. Spin operatörü için kuantum mekaniği II'de devam edeceğim.

With this paper we will try to introduce the foundations and the formalism of relativistic mean field theory and its applications. We begin by discussing the formulation of the theory of special relativity. Then we derive the Lagrangian... more

With this paper we will try to introduce the foundations and the formalism of relativistic mean field theory and its applications. We begin by discussing the formulation of the theory of special relativity. Then we derive the Lagrangian formulation of a field from the continuous limit of a discrete system. Afterwards, we formulate a relativistically invariant Lagrangian for a field and use the previous formalism to investigate several problems of continuous systems. Finally, reference is made to the application of the mean field approximation to the nuclear model of Quantum Hadrodynamics (QHD).

"François Beets, Michel Dupuis et Michel Weber (éditeurs), Alfred North Whitehead. De l’algèbre universelle à la théologie naturelle. Actes des Journées d’étude internationales tenues à l’Université de Liège les 11-12-13 octobre 2001.... more

"François Beets, Michel Dupuis et Michel Weber (éditeurs), Alfred North Whitehead. De l’algèbre universelle à la théologie naturelle. Actes des Journées d’étude internationales tenues à l’Université de Liège les 11-12-13 octobre 2001. Publiés avec le concours du FNRS, Frankfurt / Paris / Lancaster, ontos verlag, Chromatiques whiteheadiennes II, 2004. (377 p. ; ISBN 3-937202-64-1 ; 79 €)
Les premières journées « Chromatiques » se sont donné pour objectif de faciliter une réflexion globale sur la trajectoire conceptuelle du philosophe et mathématicien britannique Alfred North Whitehead (1861–1947). Afin de mener le lecteur au cœur de l’ontologie organique de l’époque de Harvard, il est en effet urgent d’élucider Whitehead à partir de lui-même, de montrer — sans être victime d’une « illusion rétrospective » — la continuité qui s’atteste dans un développement idéel qui exploite cependant quelques notables « changements d’amure ». Les contributions au colloque, placées sous le signe du commerce avec les textes eux-mêmes, furent traversées par une double tension : d’une part, l’éclairement systématique d’un aspect technique d’une des époques spéculatives de l’auteur ; d’autre part, la mise en horizon de ce questionnement ponctuel à l’aide d’une perspective globale sur le cheminement spéculatif whiteheadien.
"

Inspired by the Hilbert–Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly nontrivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which... more

Inspired by the Hilbert–Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly nontrivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the asymptotic (very large [Formula: see text]) region. The ordinates [Formula: see text] are the positive imaginary parts of the nontrivial zeta zeroes in the critical line :[Formula: see text]. The latter results are consistent with the validity of the Bohr–Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches not only the first two zeroes but the other consecutive zeroes. The highly nontrivial functional form of the potential is found via the Bohr–Sommerfeld quantization formula using the full-fledged Riemann–von Mangoldt count...

In this article we derive the equations for a rotating stratified fluid governed by inviscid Euler–Boussinesq and primitive equations that account for the effects of the perturbations upon the mean. Our method is based on the concept of... more

In this article we derive the equations for a rotating stratified fluid governed by inviscid Euler–Boussinesq and primitive equations that account for the effects of the perturbations upon the mean. Our method is based on the concept of the geometric generalized Lagrangian mean recently introduced by Gilbert and Vanneste, combined with generalized Taylor and horizontal isotropy of fluctuations as turbulent closure hypotheses. The models we obtain arise as Euler–Poincaré equations and inherit from their parent systems conservation laws for energy and potential vorticity. They are structurally and geometrically similar to Euler–Boussinesq-α and primitive equations-α models, however feature a different regularizing second order operator.

Classical mechanics, relativity, electrodynamics and quantum mechanics are often depicted as separate realms of physics, each with its own formalism and notion. This remains unsatisfactory with respect to the unity of nature and to the... more

Classical mechanics, relativity, electrodynamics and quantum mechanics are often depicted as separate realms of physics, each with its own formalism and notion. This remains unsatisfactory with respect to the unity of nature and to the necessary number of postulates. We uncover the intrinsic connection of these areas of physics and describe them using a common symplectic Hamiltonian formalism. Our approach is based on a proper distinction between variables and constants, i.e. on a basic but rigorous ontology of time and on a simple analysis of the conditions for measurements in physics. The result put the measurement problem of quantum mechanics and the Copenhagen interpretation of the quantum mechanical wavefunction into perspective. Based on this (onto-) logic spacetime can not be fundamental and we show how a geometric interpretation of symplectic dynamics emerges from the isomorphism between corresponding Lie algebra and the representation of a Clifford algebra. We derive the di...

We study the dynamical and statistical behavior of the Hamiltonian mean field (HMF) model in order to investigate the relation between microscopic chaos and phase transitions. HMF is a simple toy model of N fully coupled rotators which... more

We study the dynamical and statistical behavior of the Hamiltonian mean field (HMF) model in order to investigate the relation between microscopic chaos and phase transitions. HMF is a simple toy model of N fully coupled rotators which shows a second-order phase transition. The solution in the canonical ensemble is briefly recalled and its predictions are tested numerically at finite N. The Vlasov stationary solution is shown to give the same consistency equation of the canonical solution and its predictions for rotator angle and momenta distribution functions agree very well with numerical simulations, A link is established between the behavior of the maximal Lyapunov exponent and that of thermodynamical fluctuations, expressed by kinetic energy fluctuations or specific heat. The extensivity of chaos in the N → ∞ limit is tested through the scaling properties of Lyapunov spectra and of the Kolmogorov-Sinai entropy. Chaotic dynamics provides the mixing property in phase space necessary for obtaining equilibration; however, the relaxation time to equilibrium grows with N, at least near the critical point. Our results constitute an interesting bridge between Hamiltonian chaos in many degrees of freedom systems and equilibrium thermodynamics.

The goal of the present account is to review our efforts to obtain and apply a “collective” Hamiltonian for a few, approximately decoupled, adiabatic degrees of freedom, starting from a Hamiltonian system with more or many more degrees of... more

The goal of the present account is to review our efforts to obtain and apply a “collective” Hamiltonian for a few, approximately decoupled, adiabatic degrees of freedom, starting from a Hamiltonian system with more or many more degrees of freedom. The approach is based on an analysis of the classical limit of quantum-mechanical problems. Initially, we study the classical problem within the framework of Hamiltonian dynamics and derive a fully self-consistent theory of large-amplitude collective motion with small velocities. We derive a measure for the quality of decoupling of the collective degree of freedom. We show for several simple examples, where the classical limit is obvious, that when decoupling is good, a quantization of the collective Hamiltonian leads to accurate descriptions of the low energy properties of the systems studied. In nuclear physics problems we construct the classical Hamiltonian by means of time-dependent mean-field theory, and we transcribe our formalism to this case. We report studies of a model for monopole vibrations, of 28Si with a realistic interaction, several qualitative models of heavier nuclei, and preliminary results for a more realistic approach to heavy nuclei. Other topics included are a nuclear Born–Oppenheimer approximation for an ab initio quantum theory and a theory of the transfer of energy between collective and noncollective degrees of freedom when the decoupling is not exact. The explicit account is based on the work of the authors, but a thorough survey of other work is included.

The magnetic hysteresis of a two-dimensional lattice of rotors with four-way anisotropy interaction and a Heisenberg exchange interaction is studied. The Hamiltonian dynamics of the lattice is thermostated using the Nosé thermostat,... more

The magnetic hysteresis of a two-dimensional lattice of rotors with four-way anisotropy interaction and a Heisenberg exchange interaction is studied. The Hamiltonian dynamics of the lattice is thermostated using the Nosé thermostat, resulting in a system that approaches thermal equilibrium and which under certain conditions can remain in metastable states. Using physically realistic values for the interactions in a nanoparticle of monolayer thickness, we locate the Curie temperature of our lattice by determining the peak of the heat capacity curve. We then compare the coercive field of our two-dimensional lattice below this Curie temperature to the coercive field of an elliptical cobalt nanoparticle measured in experiment. We find an order of magnitude agreement between our lattice model and the experimental results, even though the value of the anisotropy used is more appropriate for a monolayer film than for the nanoparticle.

This article presents a multiphase interleaved boost converter supplied by a fuel-cell (FC)/reformer power source for highly dynamic transportation applications. A control theory based on the Hamiltonian function approach is considered.... more

This article presents a multiphase interleaved boost converter supplied by a fuel-cell (FC)/reformer power source for highly dynamic transportation applications. A control theory based on the Hamiltonian function approach is considered. Using the port-controlled Hamiltonian system, we propose simple solutions to the dynamic performance and convergence problems Manuscript when an interaction occurs between the power sources and constant power loads. To corroborate the proposed control law, an FC boost converter (2.5-kW two-phase interleaved converter) is used and investigated in the laboratory. The methanol FC system is composed of a fuel reformer reactor that transforms water and methanol liquid fuel into hydrogen gas to a polymer electrolyte membrane FC stack (2.5 kW, 50 V). The studied control approach is realized by digital calculation using a Micro-LabBox controller board (dSPACE platform). The simulation using the MATLAB/Simulink program and the experimental results validate that our proposed solution is an excellent control algorithm for highly dynamic power-load cycles. Index Terms-Constant power load (CPL), DC microgrid, fuel cell (FC), interleaved boost converter, passivity-based controller (PBC), port-controlled Hamiltonian (PCH), voltage regulation.

Reconstruction of equations of motion from incomplete or noisy data and dimension reduction are two fundamental problems in the study of dynamical systems with many degrees of freedom. For the latter, extensive efforts have been made, but... more

Reconstruction of equations of motion from incomplete or noisy data and dimension reduction are two fundamental problems in the study of dynamical systems with many degrees of freedom. For the latter, extensive efforts have been made, but with limited success, to generalize the Zwanzig–Mori projection formalism, originally developed for Hamiltonian systems close to thermodynamic equilibrium, to general non-Hamiltonian systems lacking detailed balance. One difficulty introduced by such systems is the lack of an invariant measure, needed to define a statistical distribution. Based on a recent discovery that a non-Hamiltonian system defined by a set of stochastic differential
equations can be mapped to a Hamiltonian system, we develop such general projection formalism. In the resulting generalized Langevin equations, a set of generalized fluctuation–dissipation relations connect the memory kernel and the random noise terms, analogous to Hamiltonian systems obeying detailed balance. Lacking of these relations restricts previous application of the generalized Langevin formalism. Result of this work may serve as the theoretical basis for further technical developments
on model reconstruction with reduced degrees of freedom. We first use an analytically solvable example to illustrate the formalism and the fluctuation–dissipation relation. Our numerical test on a chemical network with end-product inhibition further demonstrates the validity of the formalism. We suggest that the formalism can find wide applications in scientific modeling. Specifically, we discuss potential applications to biological networks. In particular, the method provides a suitable framework for gaining insights into network properties such as robustness and parameter transferability

In the framework of covariant theory of gravitation the Euler-Lagrange equations are written and equations of motion are determined by using the Lagrange function, in the case of small test particle and in the case of continuously... more

In the framework of covariant theory of gravitation the Euler-Lagrange equations are written and equations of motion are determined by using the Lagrange function, in the case of small test particle and in the case of continuously distributed matter. From the Lagrangian transition to the Hamiltonian was done, which is expressed through three-dimensional generalized momentum in explicit form, and also is defined by the 4-velocity, scalar potentials and strengths of gravitational and electromagnetic fields, taking into account the metric. The definition of generalized 4-velocity, and the description of its application to the principle of least action and to Hamiltonian is done. The existence of a 4-vector of the Hamiltonian is assumed and the problem of mass is investigated. To characterize the properties of mass we introduce three different masses, one of which is connected with the rest energy, another is the observed mass, and the third mass is determined without taking into account the energy of macroscopic fields. It is shown that the action function has the physical meaning of the function describing the change of such intrinsic properties as the rate of proper time and rate of rise of phase angle in periodic processes.

RESUMO: A noção de sustentabilidade já traz em si a questão do tempo: o que quer que se busque sustentar, busca-se fazê-lo por um período (determinado ou indeterminado). A sustentabilidade seria, portanto, a capacidade de mantermos o grau... more

RESUMO: A noção de sustentabilidade já traz em si a questão do tempo: o que quer que se busque sustentar, busca-se fazê-lo por um período (determinado ou indeterminado). A sustentabilidade seria, portanto, a capacidade de mantermos o grau de organização de nossa sociedade por meio de um padrão de funcionamento que não a coloque em conflito com o ambiente que a contém, que denominamos padrão de metabolismo social. A obtenção de tal padrão "saudável" de metabolismo social está relacionada à questão dos diferenciais de ritmo entre diferentes processos intrínsecos à dinâmica do subsistema e dos processos pertencentes a dinâmica do sistema. O objetivo deste artigo é tratar a questão dos diferenciais de ritmo entre processos ocorridos nos subsistemas e no sistema maior que os contém como ponto fundamental para obter a sustentabilidade de um sistema, seja qual for a dimensão de análise. ABSTRACT: The notion of sustainability brings a temporal problem: anything that is sustainable, it is for a period (finite or indefinite). Sustainability is, therefore, the ability to keep our society's degree of organization through a pattern of social metabolism. To reach a "healthy" pattern of social metabolism it is necessary to consider the time differentials among different process that occurs among a system and its subsystems. This paper's goal is to treat the question of time differentials among process in a holarchical structure as an essential point to reach a system's sustainability, in any of sustainability's dimensions.

Abstract: We develop the argument that the Gibbs-von Neumann entropy is the appropriate statistical mechanical generalisation of the thermodynamic entropy, for macroscopic and microscopic systems, whether in thermal equilibrium or not, as... more

Abstract: We develop the argument that the Gibbs-von Neumann entropy is the appropriate statistical mechanical generalisation of the thermodynamic entropy, for macroscopic and microscopic systems, whether in thermal equilibrium or not, as a consequence of Hamiltonian dynamics. The mathematical treatment utilises well known results [Gib02, Tol38, Weh78, Par89], but most importantly, incorporates a variety of arguments on the phenomenological properties of thermal states [Szi25, TQ63, HK65, GB91] and of ...