Hamiltonian Formulation Research Articles

Lagrangian–Hamiltonian formalism for cocontact systems

<abstract><p>In this paper we present a unified Lagrangian–Hamiltonian geometric formalism to describe time-dependent contact mechanical systems, based on the one first introduced by K. Kamimura and later formalized by R. Skinner and R. Rusk. This formalism is especially interesting when dealing with systems described by singular Lagrangians, since the second-order condition is recovered from the constraint algorithm. In order to illustrate this formulation, some relevant examples are described in full detail: the Duffing equation, an ascending particle with time-dependent mass and quadratic drag, and a charged particle in a stationary electric field with a time-dependent constraint.</p></abstract>

Conservation laws and symmetry analysis of a generalized Drinfeld-Sokolov system

<abstract><p>The generalized Drinfeld-Sokolov system is a widely-used model that describes wave phenomena in various contexts. Many properties of this system, such as Hamiltonian formulations and integrability, have been extensively studied and exact solutions have been derived for specific cases. In this paper we applied the direct method of multipliers to obtain all low-order local conservation laws of the system. These conservation laws correspond to physical quantities that remain constant over time, such as energy and momentum, and we provided a physical interpretation for each of them. Additionally, we investigated the Lie point symmetries and first-order symmetries of the system. Through the point symmetries and constructing the optimal systems of one-dimensional subalgebras, we were able to reduce the system of partial differential equations to ordinary differential systems and obtain new solutions for the system.</p></abstract>

Hamilton formulation for the electrodynamics of generalized maxwell using fractional derivatives

A comparative analysis of the Hamiltonian and Lagrangian equations for the Maxwell field was conducted, and it was demonstrated that both methods are equivalent. Dirac’s and Euler’s techniques were employed to handle the Hamiltonian approach. Additionally, a novel fractional Hamilton formulation was developed for the Maxwell field using fractional derivatives. This formulation yielded a fractional Riemann-Liouville derivative operator and a fractional Hamilton function in terms of the variables Ai, Aj, and A0. The effectiveness of this approach was verified by employing it to examine Maxwell’s electrodynamic equation, and the results obtained were in perfect agreement, confirming the validity of the study

Dielectric response of high-κ hafnium oxide under finite electric field: nonlinearities from ab initio and experimental points of view.

Herein, we report on the dielectric-voltage nonlinearities under a constant electric field in metal insulator metal (MIM) capacitor-based hafnium oxide (HfO2) with respect to the frequency range. Via the Schottky emission mechanism obtained from the current-voltage characteristic (I-V), we calculated the optical dielectric constant εr,opt for different external DC bias values. The extracted εr,opt revealed a quadratic dependence on the applied external field. This confirmed that such dependence is a common feature of high-κ oxides in the low and high frequency ranges. The results were correlated with the ab initio calculations using the finite field (FF) method as implemented in the CRYSTAL 17 code. Good agreement between the results from the FF method, I-V plots, as well as the UV-visible spectrometry is observed. To assess any change in the dielectric constant upon the application of an external electric field, several parameters such as exchange-correlation functional, basis sets (BSs), as well as supercell expansion factor (N) were tested. The corresponding parameters have a great influence on the macroscopic electron density and voltage along the field direction and thus on the optical response. For N > 2 and rich basis sets, the hybrid functional B3LYP revealed good agreement with the experimental results as compared to other Hamiltonian's forms such as LDA, PW-GGA and HF.

On the integrability of twofold 1 : 2 Hamiltonian resonance

In a recent paper Mazrooei-Sebdani et al study a particular case of 1 : 2 : 1 : 2 Hamiltonian normal form, truncated to order 3. They find several cases of integrability. The main purpose of this work is to find other integrable cases. First, we obtain some integrable cases considering the known results about integrability the normal forms of 1 : 2 : 2 and 1 : 1 : 2 resonances. Then, we simplify this particular normal form, prove a non-integrability theorem and isolate a couple of non-trivial cases of integrability. These results can be applied, for example, in the study of normal forms of coupled elastic pendulums or coupled double pendulums, both in 1 : 2 resonance. On the other hand, it is definitely worth exploring the local and global properties of the geometry and the dynamics of all integrable cases, found here

Hamiltonian Formalism for Bianchi Type I Model for Perfect Fluid as Well as for the Fluid with Bulk and Shearing Viscosity

Hamiltonian Formalism for Bianchi Type I Model for Perfect Fluid as Well as for the Fluid with Bulk and Shearing Viscosity

Quantization of Hamiltonian and non-Hamiltonian systems

<abstract> <p>The quantization process was always tightly connected to the Hamiltonian formulation of classical mechanics. For non-Hamiltonian systems, traditional quantization algorithms turn out to be unsuitable. Numerous attempts to quantize non-Hamiltonian systems have shown that this problem is nontrivial and requires the development of new approaches. In this paper, we present the quantization methods that do not depend upon the Hamiltonian formulation of classical mechanics. Two approaches to the quantization of mechanical systems are considered: axiomatic and hydrodynamic. It is shown that the formal application of these approaches to the classical Hamilton-Jacobi theory allows obtaining the wave equation for the corresponding quantum system in natural way. Examples are considered that show the effectiveness of the proposed approaches, both for Hamiltonian and non-Hamiltonian systems. The spinor form of the relativistic Hamilton-Jacobi theory for classical particles is considered. It is shown that it naturally leads to the Dirac equation for the corresponding quantum particle and to its non-Hamiltonian generalization, the bispinor relativistic Kostin equation.</p> </abstract>

An efficient linearly-implicit energy-preserving scheme with fast solver for the fractional nonlinear wave equation

<abstract><p>The paper considers the Hamiltonian structure and develops efficient energy-preserving schemes for the nonlinear wave equation with a fractional Laplacian operator. To this end, we first derive the Hamiltonian form of the equation by using the fractional variational derivative and then applying the finite difference method to the original equation to obtain a semi-discrete Hamiltonian system. Furthermore, the scalar auxiliary variable method and extrapolation technique is used to approximate a semi-discrete system to construct an efficient linearly-implicit energy-preserving scheme. A fast solver for the proposed scheme is presented to reduce CPU consumption. Ample numerical results are given to finally confirm the efficiency and conservation of the developed scheme.</p></abstract>

A conservative exponential integrators method for fractional conservative differential equations

<abstract><p>The paper constructs a conservative Fourier pseudo-spectral scheme for some conservative fractional partial differential equations. The scheme is obtained by using the exponential time difference averaged vector field method to approximate the time direction and applying the Fourier pseudo-spectral method to discretize the fractional Laplacian operator so that the FFT technique can be used to reduce the computational complexity in long-time simulations. In addition, the developed scheme can be applied to solve fractional Hamiltonian differential equations because the scheme constructed is built upon the general Hamiltonian form of the equations. The conservation and accuracy of the scheme are demonstrated by solving the fractional Schrödinger equation.</p></abstract>

Hamiltonian form of extended cubic-quintic nonlinear Schrödinger equation in a nonlinear Klein-Gordon model.

We derive an extended cubic-quintic nonlinear Schrödinger equationwith Hamiltonian structure in a nonlinear Klein-Gordon model with cubic-quintic nonlinearity. We use the nonlinear dispersion relation to properly take into account the input of high-order nonlinear effects in the Hamiltonian perturbation approach to nonlinear modulation. We demonstrate that changing the balance between the cubic and quintic nonlinearities has a significant effect on the stability of unmodulated wave packets to long-wave modulations.

Mechanical properties of the particle associated with the Laguerre–Gauss beams via the quantum potential point of view

The aim of the present work is to show that the Laguerre–Gauss beams determine a Hamiltonian system with two degrees of freedom for a particle of mass m = 1 under the action of the quantum potential determined by these beams. We show that the integral curves of the Poynting vector constitute a particular subset of solutions to the corresponding Hamilton equations and that the geometrical light rays associated with these twisted beams turn out to be the tangent straight lines to the exact optics energy trajectories at the zeroes of the quantum potential. By using the Hamiltonian formulation, we determine the velocity, the linear momentum, the angular momentum, the torque, and the areal velocity characterizing the particle associated with the Laguerre–Gauss beams.

Hamiltonian formulation with reduced variables for flexible multibody systems under linear constraints: Theory and experiment

Mechanical structures, such as robot arms in space stations and rockets that are used to support futuristic activities and carry loads in modern society, tend to be flexible owing to their reduced weight. Flexible multibody dynamics is an effective method for analyzing the dynamic motion of flexible structures. Although the Hamiltonian formulation based on the canonical theory is commonly used in multibody dynamics, certain drawbacks exist, such as complexity and computational cost. In this study, a novel flexible multibody dynamics formulation based on canonical theory is proposed with reduced variables to improve computational efficiency. This method formulates the equation of motion using only independent generalized coordinates with the assumption of linear constraint equations and the introduction of the first derivative of the constraint condition. The proposed method is combined with absolute nodal coordinate formulation and the motions of flexible structures are simulated to verify its accuracy and effectiveness. In comparison with the conventional methods, the proposed method requires less calculation time while maintaining accuracy. In addition, a wind tunnel experiment was conducted to validate the proposed method. The concurrence between the simulation and experimental results further verifies the accuracy of the proposed method.

SU(N) fractional instantons and the Fibonacci sequence

We study, by means of numerical methods, new SU(N) self-dual instanton solutions on R × T3 with fractional topological charge Q = 1/N. They are obtained on a box with twisted boundary conditions with a very particular choice of twist: both the number of colours and the ’t Hooft ZN fluxes piercing the box are taken within the Fibonacci sequence, i.e. N = Fn (the nth number in the series) and left|overrightarrow{m}right| = left|overrightarrow{k}right| = Fn−2. Various arguments based on previous works and in particular on ref. [1], indicate that this choice of twist avoids the breakdown of volume independence in the large N limit. These solutions become relevant on a Hamiltonian formulation of the gauge theory, where they represent vacuum-to-vacuum tunneling events lifting the degeneracy between electric flux sectors present in perturbation theory. We discuss the large N scaling properties of the solutions and evaluate various gauge invariant quantities like the action density or Wilson and Polyakov loop operators.

O(n)-invariant Riemannian metrics on SPD matrices

Symmetric Positive Definite (SPD) matrices are ubiquitous in data analysis under the form of covariance matrices or correlation matrices. Several O(n)-invariant Riemannian metrics were defined on the SPD cone, in particular the kernel metrics introduced by Hiai and Petz. The class of kernel metrics interpolates between many classical O(n)-invariant metrics and it satisfies key results of stability and completeness. However, it does not contain all the classical O(n)-invariant metrics. Therefore in this work, we investigate super-classes of kernel metrics and we study which key results remain true. We also introduce an additional key result called cometric-stability, a crucial property to implement geodesics with a Hamiltonian formulation. Our method to build intermediate embedded classes between O(n)-invariant metrics and kernel metrics is to give a characterization of the whole class of O(n)-invariant metrics on SPD matrices and to specify requirements on metrics one by one until we reach kernel metrics. As a secondary contribution, we synthesize the literature on the main O(n)-invariant metrics, we provide the complete formula of the sectional curvature of the affine-invariant metric and the formula of the geodesic parallel transport between commuting matrices for the Bures-Wasserstein metric.

Solving hadron structures using the basis light-front quantization approach on quantum computers

Quantum computing has demonstrated the potential to revolutionize our understanding of nuclear, atomic, and molecular structure by obtaining forefront solutions in nonrelativistic quantum many-body theory. In this work, we show that quantum computing can be used to solve for the structure of hadrons, governed by strongly interacting relativistic quantum field theory. Following our previous work on light unflavored mesons as a relativistic bound-state problem within the nonperturbative Hamiltonian formalism, we present the numerical calculations on simulated quantum devices using the basis light-front quantization approach. We implement and compare the variational quantum eigensolver and the subspace-search variational quantum eigensolver to find the low-lying mass spectrum of the light meson system and its corresponding light-front wave functions as quantum states from ideal simulators, noisy simulators, and IBM quantum computers. Based on obtained quantum states, we evaluate the meson decay constants and parton distribution functions directly on the quantum circuits. Our calculations on the quantum computers and simulators are in reasonable agreement with accurate numerical solutions solved on classical computers when noises are moderately small, and our overall results are comparable with the available experimental data.

A symplectic Brezis-Ekeland-Nayroles principle for dynamic plasticity in finite strains

In a previous paper, the last author proposed with Buliga a symplectic version of Brezis–Ekeland–Nayroles principle based on the concepts of Hamiltonian inclusions and symplectic polar functions. It was illustrated by application to the standard plasticity in small deformations. The objective of this work is to generalize the previous formalism to dissipative media in finite strains. This aim is reached in three steps. Firstly, we develop a Lagrangian formalism for the reversible media based on the calculus of variation. Next, we propose a corresponding Hamiltonian formalism for such media. Finally, we deduce from it a symplectic minimum principle for dissipative media and we show how to get a minimum principle for plasticity in finite strains.

On the cosmological constant in the deformed Einstein–Cartan gauge gravity in De Donder–Weyl Hamiltonian formulation

AbstractA modification of the Einstein–Hilbert theory, the Covariant Canonical Gauge Gravity (CCGG), leads to a cosmological constant that represents the energy of the space–time continuum when deformed from its (A)dS ground state to a flat geometry. CCGG is based on the canonical transformation theory in the De Donder–Weyl (DW) Hamiltonian formulation. That framework modifies the Einstein–Hilbert Lagrangian of the free gravitational field by a quadratic Riemann–Cartan concomitant. The theory predicts a total energy‐momentum of the system of space–time and matter to vanish, in line with the conjecture of a “Zero‐Energy‐Universe” going back to Lorentz (1916) and Levi‐Civita (1917). Consequently, a flat geometry can only exist in presence of matter where the bulk vacuum energy of matter, regardless of its value, is eliminated by the vacuum energy of space–time. The observed cosmological constant Λobs is found to be merely a small correction attributable to deviations from a flat geometry and effects of complex dynamical geometry of space–time, namely torsion and possibly also vacuum fluctuations. That quadratic extension of General Relativity, anticipated already in 1918 by Einstein, thus provides a significant and natural contribution to resolving the “cosmological constant problem”.

Periodic perturbation of motion of an unbalanced circular foil in the presence of point vortices in an ideal fluid

The dynamics of a system governing the controlled motion of an unbalanced circular foil in the presence of point vortices is considered. The foil motion is controlled by periodically changing the position of the center of mass, the gyrostatic momentum, and the moment of inertia of the system. A derivation of the equations of motion based on Sedov's approach is proposed, the equations of motion are presented in the Hamiltonian form. A periodic perturbation of the known integrable case is considered.

A Puzzle Concerning Local Symmetries and Their Empirical Significance

In the last 5 years, the controversy about whether or not gauge transformations can be empirically significant has intensified. On the one hand, Greaves and Wallace developed a framework according to which, under some circumstances, gauge transformations can be empirically significant—and Teh further supported this result by using the constrained Hamiltonian formalism. On the other hand, Friederich claims to have proved that gauge transformation can never be empirically significant. In this article, I accomplish two tasks. First, I argue that there are strong reasons to resist Friederich’s proof because one of its assumptions is, at the very least, highly controversial. Second, I argue that despite criticism by Brading and Brown and Friederich, ‘t Hooft’s beam-splitter experiment is indeed a concrete example of a case where a local gauge symmetry has empirical significance. By shedding light on these two points, this article shows that recent arguments that claim gauge transformations cannot be empirically significant are not satisfactory.

On the role of geometry in statistical mechanics and thermodynamics. I. Geometric perspective

This paper contains a fully geometric formulation of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC). Although GENERIC, which is the sum of Hamiltonian mechanics and gradient dynamics, is a framework unifying a vast range of models in non-equilibrium thermodynamics, it has unclear geometric structure due to the diverse geometric origins of Hamiltonian mechanics and gradient dynamics. The difference can be overcome by cotangent lifts of the dynamics, which leads, for instance, to a Hamiltonian form of gradient dynamics. Moreover, the lifted vector fields can be split into their holonomic and vertical representatives, which provides a geometric method of dynamic reduction. The lifted dynamics can be also given physical meaning, here called the rate-GENERIC. Finally, the lifts can be formulated within contact geometry, where the second law of thermodynamics is explicitly contained within the evolution equations.