Hamiltonian dynamics

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 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 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 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.

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.

This are some notes on mathematical physics.

Contents:
- Introduction and basics in differential geometry
-Symplectic geometry
          -Symplectic vector space
          -Symplectic mfd
          -Symplectic structure of cotangent bundle
          -Symplectomorphism and canonical transformation
          -Darboux Theorem                                                               
-Newtonian Mechanics
          -Poisson brackets
          -Physical interpretation
        -Integrability                                             
        -Co-adjoint orbits                                         
-Poisson Manifold
        -Poisson fields
        -Poisson cohomology
        -Classical R-matrix
        -Lax Representation
-Deformation quantization (with Kontsevich's thm)

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.

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 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).

Acta Mechanica, 2017, DOI: 10.1007/s00707-016-1775-2

This paper presents an adaptive power harvester using a shunted piezoelectric control system with segmented electrodes. This technique has spurred new capability for widening the three simultaneous resonance frequency peaks using only a single piezoelectric laminated beam where normally previous works only provide a single peak for the resonance at the first mode. The benefit of the proposed techniques is that it provides effective and robust broadband power generation for application in self-powered wireless sensor devices. The smart structure beam with proof mass offset is considered to have simultaneous combination between vibration-based power harvesting and shunt circuit control-based electrode segments. As a result, the system spurs new development of the two mathematical methods using electromechanical closed-boundary value techniques and Ritz method-based weak-form analytical approach. The two methods have been used for comparison giving accurate results. For different electrode lengths using certain parametric tuning and harvesting circuit systems, the technique enables the prediction of the power harvesting that can be further proved to identify the performance of the system using the effect of varying circuit parameters so as to visualize the frequency and time waveform responses.

Acta Mechanica, vol. 228 (2), pp 631–650, 2017

This paper discusses, compares and contrasts two important techniques for formulating the electromechanical piezoelectric equations for power harvesting system applications. It presents important additions to existing literature by providing intrinsic formulation techniques of the harvesting system for the two different electromechanical dynamic equation-based voltage and charge-type systems associated with the standard AC–DC circuit interface developed using the extended Hamiltonian principle. The derivations of the two analytical methods rely on the fundamental continuum thermopiezoelectricity concepts of the electrical enthalpy energy and Helmholtz free energy. The benefit of using analytical charge-type modelling is that the technique shows more compact formulation for developing simultaneous derivations by coupling the mechanical and electromechanical systems of the piezoelectric devices and electronic system so that the frequency response functions (FRFs) and time wave form systems can be formulated. On the other hand, the analytical voltage-type modelling is obviously convenient but can show tedious derivation process for joining with the electronic circuit part. To tackle this situation, the analytical voltage type with mechanical and electromechanical forms of the piezoelectric structure can be derived separately from the electronic system where they can be combined together after applying further derivations to formulate the FRFs. In this paper, the two analytical techniques also show particular benefit and even further development of how to model the power harvesting scheme with the combinations of piezoelectric structure and electronic system. Moreover, validations of the two analytical methods show good agreement with previous authors’ electromechanical finite element analysis and experimental works. Further parametric electromechanical energy harvesting behaviours have been explored to study the system responses.

A lesser-known property of Hamiltonian dynamics is that it can be formally mapped to the Riemannian geometry of classical gravitation. Taking advantage of this property, we explore here the possibility that the onset of Hamiltonian chaos in the ultraviolet (UV) sector of field theory generates the cosmological and Fermi scales. This finding supports the conjecture that both Standard and the Lambda-CDM  models emerge as non-trivial attractors of the UV to infrared (IR) flow.

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...

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 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 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, 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.

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 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.

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 ...