Equations Of Motion Research Articles

Noether’s currents for conformable fractional scalar field theories

The construction of fractional derivatives with the right properties for use in field theory is reputed to be a difficult task, essentially because of the absence of a unique definition and uniform properties. The conformable fractional derivative introduced in 2014 by Khalil et al. in their seminal paper is a novel and well-behaved definition of fractional derivative for a function that is derivable in the usual sense. In this paper, we investigate the consistency of the Euler–Lagrange formalism for a field theory defined on such a fractional space–time. We especially focus on the relation between symmetries and conservation laws (Noether’s currents), about the symmetry group introduced to construct the Lagrangian of the field. In particular, we show that the use of the conformable derivative induces additional terms in the calculation of the action variation. We also investigate the conservation of the Noether current and show that this property only takes place on condition that the equations of motion are verified with a new definition of the conserved law.

Charged scalar bosons in a Bonnor–Melvin-Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varLambda $$\\end{document} universe at conical approximation

The quantum dynamics of charged scalar bosons in a Bonnor–Melvin-Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varLambda $$\\end{document} universe is considered. In this study, the behavior of charged scalar bosons is explored within the framework of the Duffin–Kemmer–Petiau (DKP) formalism. Adopting a conical approximation (Λ≪1),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\varLambda \\ll 1),$$\\end{document} we are considered two scenarios for the vector potential: a linear and quadratic vector potentials. In particular, the effects of this background in the equation of motion, phase shift, S-matrix, energy spectrum and DKP spinor are analyzed and discussed.

RANSE simulations for unmanned sailboat performance prediction: Coupling aerodynamics and hydrodynamics in self-sailing

During the design phase of sailboats, naval architects typically employ hydrodynamic and aerodynamic models of the sailboat, utilizing a Velocity Prediction Program (VPP) to solve the equations of motion, thus determining the sailboat's speed and performance. Traditional methods for solving sailboat motion equations often employ simplified empirical formulas or costly experimental procedures, which yield less accurate results. This study presents the RANSE-based CFD software STAR-CCM + to comprehensively model the sailboat's hull-sail-rudder assembly. By implementing overset grid technology and the Volume of Fluid (VOF) model, the paper aims to couple the solution of aerodynamic and hydrodynamic forces under wind influence and introduces PID rudder control technology to maintain a balance of forces and convergence of motion state. This approach enables the prediction of the sailboat's speed and performance, such as yaw angle, heel, and rudder angle. Results demonstrate minimal variations in sailboat speed (<0.05 m/s) and heading (<0.1°) in upwind, downwind and reaching sailing conditions, underscoring the method's feasibility and advocating for further research into self-sailing performance.

High accuracy exponential decomposition of bath correlation functions for arbitrary and structured spectral densities: Emerging methodologies and new approaches.

This study investigates the decomposition of bath correlation functions (BCFs) in terms of complex exponential functions, with an eye on the realistic modeling of open quantum systems based on the hierarchical equations of motion. We introduce the theoretical background of various decomposition schemes in both time and frequency domains and assess their efficiency and accuracy by demonstrating the decomposition of various BCFs. We further develop a new procedure for the decomposition of BCFs originating from highly structured spectral densities with a high accuracy and compare it with existing fitting techniques. Advantages and disadvantages of each methodology are discussed in detail with special attention to their application to the corresponding quantum dynamical problem. This work provides fundamental tools for choosing and using a variety of decomposition techniques of BCFs for the study of open quantum systems in structured environments.

Thermodynamics of Irreversible Processes: Fundamental Constraints, Representations, and Formulation of Boundary Conditions

Starting from the analysis of the lack of positivity of the Cattaneo heat equation, this work addresses the thermodynamic relevance of the positivity constraint in irreversible thermodynamics, that is at least as significant as the entropic constraints. The fulfillment of this condition in hyperbolic models leads to the parametrization of the concentration fields with respect to internal variables associated with the microscopic dynamics. Using Brownian motion theory as a landmark example for deriving macroscopic transport equations from the equations of motion at the particle/molecular level, we discuss two typical problems involving hydrodynamic interactions at the microscale: surface chemical reactions at a solid interface of a diffusing reactant, and mass-balance equations in a complex viscoelastic fluid, in which the physics of the interaction leads either to overcoming the parabolic diffusion model or to considering the parametrization of the concentration with respect to the degrees of freedom associated with the relaxation dynamics of the solvent fluid.

Overturning resistant capacities of rocking frames with supplemental vibration control devices under pulse-type excitations

Rocking structures are vulnerable to overturn under earthquake shaking. It is necessary to reduce the overturning probability of rocking structures. Recently, inerters were proposed to control the vibrations of rocking columns, and it was found that the structure equipped with inerter has reduced probability of overturning. However, inerter does not have capacity to dissipate energy, its control performance is believed able to be enhanced once being used in conjunction with dampers. Therefore, this paper approaches rocking structural vibration control from two directions: by applying supplemental inertia and applying supplemental damping to the structure. To this end, inerter, clutched-inerter and viscous damper are selected as control devices, and two combined scenarios are also considered, with configuration Ⅰ consisting of an inerter and a viscous damper in parallel and configuration Ⅱ consisting of a clutched-inerter and a viscous damper in parallel. The analytical models of a rocking frame with supplemental control devices in variable positions are presented and the equations of motion for each model are derived. In addition, the maximum coefficient of restitution of the rocking frame with supplemental control device is derived based on the angular momentum-impulse theorem. The effects and applicable conditions of the considered devices are evaluated by means of assessing the overturning resistances, maximum responses and fragility curves of rocking frames under different conditions. The results show that, with appropriate damping coefficient and apparent mass of the inerter, the combined configurations Ⅰ and Ⅱ provide better control effectiveness than the cases with inerter, clutched-inerter or viscous damper applied individually.

Free flexural vibrations of standard wide-flange H-beams with consideration of the shear effect

The subject of the paper is a simply supported standard wide-flange H-beam. Cross sections of this beam is analytically described as a three-layer structure. The shear effect in its succesive layers is taking into account with consideration of the classical shear stress formula called Zhuravsky shear stress. Based on Hamilton’s principle, two differential equations of motion are obtained. These equations are analytically solved and the fundamental natural frequency of flexural vibration for this beam is derived. Examplary calculations are carried out for selected five I-beams.

A novel dynamic stiffness matrix for the nonlocal vibration characteristics of porous functionally graded nanoplates on elastic foundation with small-scale effects

The present paper deals with the development of a dynamic stiffness matrix to evaluate the free vibration response of functionally graded nanoplate (FG-nP) resting on the Winkler-Pasternak elastic foundation. The complete mathematical modeling of the dynamic stiffness matrix for nanostructures is given for the first time. The equation of motion for rectangular FG-nP plates supported on an elastic foundation is derived using Hamilton’s principle in conjunction with nonlocal elasticity theory. The non-local theory is incorporated to account for the size effect in the small-scale plate. The effective material property of the porous FG-nP has been calculated using three recently developed models of porosity. The developed dynamic stiffness matrix is solved using the Wittrick-Williams algorithm to extract the natural frequencies of the FG-nP. The variation of natural frequencies with the change of numerical values, such as nonlocal parameter, aspect ratio, elastic foundation parameters, and porosity volume fraction is analyzed. The validity and accuracy of the results are confirmed through comparison with the available literature. The use of non-local theory in dynamic stiffness analysis is shown to be effective in predicting the natural frequency of the FG-nP on a Winkler-Pasternak elastic foundation, providing new insights into the dynamic behavior of small-scale structures.

On wave-driven propulsion

A theory is presented for wave-driven propulsion of floating bodies driven into oscillation at the fluid interface. By coupling the equations of motion of the body to a quasipotential flow model of the fluid, we derive expressions for the drift speed and propulsive thrust of the body which in turn are shown to be consistent with global momentum conservation. We explore the efficacy of our model in describing the motion of SurferBot (Rhee et al., Bioinspir. Biomim., vol. 17, issue 5, 2022), demonstrating close agreement with the experimentally determined drift speed and oscillatory dynamics. The efficiency of wave-driven propulsion is then computed as a function of driving oscillation frequency and the forcing location, revealing optimal values for both of these parameters which await confirmation in experiments. A comparison with other modes of locomotion and applications of our model with competitive water sports is discussed in conclusion.

Massless Lifshitz field theory for arbitrary z

By using the notion of fractional derivatives, we introduce a class of massless Lifshitz scalar field theory in (1+1)-dimension with an arbitrary anisotropy index z. The Lifshitz scale invariant ground state of the theory is constructed explicitly and takes the form of Rokhsar-Kivelson (RK). We show that there is a continuous family of ground states with degeneracy parameterized by the choice of solution to the equation of motion of an auxiliary classical system. The quantum mechanical path integral establishes a 2d/1d correspondence with the equal time correlation functions of the Lifshitz scalar field theory. We study the entanglement properties of the Lifshitz theory for arbitrary z using the path integral representation. The entanglement measures are expressed in terms of certain cross ratio functions we specify, and satisfy the c-function monotonicity theorems. We also consider the holographic description of the Lifshitz theory. In order to match with the field theory result for the entanglement entropy, we propose a z-dependent radius scale for the Lifshitz background. This relation is consistent with the z-dependent scaling symmetry respected by the Lifshitz vacuum. Furthermore, the time-like entanglement entropy is determined using holography. Our result suggests that there should exist a fundamental definition of time-like entanglement other than employing analytic continuation as performed in relativistic field theory.

Newton vs. Euler–Lagrange approach, or how and when beam equations are variational

AbstractThere is a clear and compelling need to correctly write the equations of motion of structures in order to adequately describe their dynamics. Two routes, indeed very different from a philosophical standpoint, can be used in classical mechanics to derive such equations, namely the Newton vectorial approach (i.e., roughly, sum of forces equal to mass times acceleration) or the Euler–Lagrange variational formulation (i.e., roughly, stationarity of a certain functional). However, it is desirable that whichever derivation strategy is chosen, the equations are the same. Since many structures of interest often consist of slender and highly flexible beams operating in regimes of large displacement and large rotation, we restrict our attention to the Euler-Bernoulli assumptions with a generic initial configuration. In this setting, the question that arises is: What conditions must the constitutive assumptions satisfy in order for the equations of motion obtained by Newton’s approach to be identical to the Euler–Lagrange equations derived from an appropriate Lagrangian, natural or virtual, for any arbitrary initial configuration? The aim of this paper is to try to answer this basic question, which indeed does not have an immediate and simple answer, in particular as a consequence of the fact that bending moment could be related to two different notions of flexural curvature.

Dynamics around small irregularly shaped objects modeled as a mass dipole

In this work, we investigate the dynamics of a spacecraft near two primary bodies. The massive body is considered to have a spherical shape, while the less massive one is elongated and modeled as a dipole. The dipole consists of two connected masses, one is spherical and the other is an oblate spheroid. The gravitational potential of the elongated body is determined by four independent parameters. To study the dynamics, we construct the equations of motion of a spacecraft with negligible mass under the effect of the current force model. The existence and locations of the equilibrium points are analyzed for various values of the system parameters. We found that the existence and locations of the points are affected by the system parameters. Also, we studied the linear stability of the equilibrium points. We found some stable collinear points when the oblateness parameter is negative, otherwise the points are not stable. We used the curves of zero velocity to identify the regions of allowed motion. Furthermore, we discussed the 2001 SN263 asteroid system and found some stable collinear points when the oblateness parameter is negative. In addition, the triangular points of the system are stable in a linear sense.

Electron-photon exchange-correlation approximation for quantum-electrodynamical density-functional theory

Quantum-electrodynamical density-functional theory (QEDFT) provides a promising avenue for exploring complex light-matter interactions in optical cavities for real materials. Similar to conventional density-functional theory, the Kohn-Sham formulation of QEDFT needs approximations for the generally unknown exchange-correlation functional. In addition to the usual electron-electron exchange-correlation potential, an approximation for the electron-photon exchange-correlation potential is needed. A recent electron-photon exchange functional [C. Schäfer , ], derived from the equation of motion of the nonrelativistic Pauli-Fierz Hamiltonian, shows robust performance in one-dimensional systems across weak- and strong-coupling regimes. Yet, its performance in reproducing electron densities in higher dimensions remains unexplored. Here we consider this QEDFT functional approximation from one- to three-dimensional finite systems and across weak to strong light-matter couplings. The electron-photon exchange approximation provides excellent results in the ultrastrong-coupling regime. However, to ensure accuracy also in the weak-coupling regime across higher dimensions, we introduce a computationally efficient renormalization factor for the electron-photon exchange functional, which accounts for part of the electron-photon correlation contribution. These findings extend the applicability of photon-exchange-based functionals to realistic cavity-matter systems, fostering the field of cavity QED (quantum-electrodynamics) materials engineering. Published by the American Physical Society 2024

MinActionPath2: path generation between different conformations of large macromolecular assemblies by action minimization.

Recent progress in solving macromolecular structures and assemblies by cryogenic electron microscopy techniques enables sampling of their conformations in different states that are relevant to their biological function. Knowing the transition path between these conformations would provide new avenues for drug discovery. While the experimental study of transition paths is intrinsically difficult, in-silico methods can be used to generate an initial guess for those paths. The Elastic Network Model (ENM), along with a coarse-grained representation (CG) of the structures are among the most popular models to explore such possible paths. Here we propose an update to our software platform MinActionPath that generates non-linear transition paths based on ENM and CG models, using action minimization to solve the equations of motion. The new website enables the study of large structures such as ribosomes or entire virus envelopes. It provides direct visualization of the trajectories along with quantitative analyses of their behaviors at http://dynstr.pasteur.fr/servers/minactionpath/ minactionpath2_submission.

A worldsheet description of flux compactifications

We demonstrate how recent developments in string field theory provide a framework to systematically study type II flux compactifications with non-trivial Ramond-Ramond profiles. We present an explicit example where physical observables can be computed order by order in a small parameter which can be effectively viewed as string coupling constant. We obtain the corresponding background solution of the string field equations of motions up to the second order in the expansion. Along the way, we show how the tadpole cancellations of the string field equations lead to the minimization of the F-term potential of the low energy supergravity description. String field action expanded around the obtained background solution furnishes a “worldsheet” description of the flux compactifications.

Imaginary past of a quantum particle moving on imaginary time

The analytical continuation of classical equations of motion to complex times suggests that a tunneling particle spends in the barrier an imaginary duration i|T|. Does this mean that it takes a finite time to tunnel, or should tunneling be seen as an instantaneous process? It is well known that examination of the adiabatic limit in a small additional AC field points towards |T| being the time it takes to traverse the barrier. However, this is only half the story. We probe the transmitted particle's history, and find that it very little of the field's past behavior, as if the transit time were close to zero. The ensuing contradiction suggests that the question is ill posed, and we explain why. Published by the American Physical Society 2024

Vibration analysis of graphene-reinforced porous aluminum-based variable-walled thickness sandwich joined conical-conical panel with elastic boundary conditions using differential quadrature method

In this paper, a unified solution is proposed to investigate the vibration characteristics of the variable-walled thickness graphene-reinforced porous aluminum-based (GRPA) sandwich joined conical-conical panel (JCCP) with the arbitrary elastic support boundary conditions by using the differential quadrature method (DQM). The two surfaces of the sandwich JCCP are made of metallic aluminum and the central core layer is the GRPA. The core thickness of each conical plate varies linearly with its generatrix. Three types of the graphene distributions and two types of porosity distributions are considered along the core thickness direction. Based on the first-order shear deformation theory (FSDT), von-Karman strain displacement relationship, constitutive relationship and Hamiltonian principle, the partial differential governing equations of motion are obtained for the variable wall thickness GRPA sandwich JCCP. Using the DQM, the dynamic equation is discretized into the ordinary differential equation. The matrix of the characteristic equation is analyzed to solve the frequencies and mode shapes of the variable wall thickness GRPA sandwich JCCP. The effects of the spring stiffness, boundary conditions, graphene distributions, porosity distributions and geometric parameters on the vibration properties are studied for the variable wall thickness GRPA sandwich JCCP with two interesting elastic supported boundary conditions. At the same time, this article provides a useful approach for studying the arbitrary boundary coupled plate and shell structures with the variable wall thickness.

Effects of Earth’s Oblateness on Black Hole Imaging through Earth–Space and Space–Space Very Long Baseline Interferometry

Earth-based very long baseline interferometry (VLBI) has made rapid advances in imaging black holes. However, due to the limitations imposed on terrestrial VLBI by the Earth’s finite size and turbulent atmosphere, it is imperative to have a space-based component in future VLBI missions. This paper investigates the effect of the Earth’s oblateness, also known as the J 2 effect, on orbiters in Earth–space and space–space VLBI. The paper provides an extensive discussion on how the J 2 effect can directly impact orbit selection for black hole observations and how, through informed choices of orbital parameters, the effect can be used to a mission’s advantage, a fact that has not been addressed in previous space VLBI investigations. We provide a comprehensive study of how the orbital parameters of several current space VLBI proposals will vary specifically due to the J 2 effect. For black hole accretion flow targets of interest, we demonstrate how the J 2 effect leads to a modest increase in shorter-baseline coverage, filling gaps in the (u, v) plane. Subsequently, we construct a simple analytical formalism that allows isolation of the impact of the J 2 effect on the (u, v) plane without requiring computationally intensive orbit propagation simulations. By directly constructing (u, v) coverage using J 2-affected and J 2-invariant equations of motion, we obtain distinct coverage patterns for M87* and Sgr A* that show extremely dense coverage on short baselines as well as long-term orbital stability on longer baselines.

Analysis of bending vibrations of a three-layered pre-twisted sandwich beam with an exact dynamic stiffness matrix

This paper presents the novel development of the dynamic stiffness matrix for a three-layered symmetric pre-twisted sandwich beam (PTSB), aiming to investigate its free vibration characteristics. The outer layers of the beam are modeled using the Euler–Bernoulli theory, while the core is assumed to deform solely in shear. The boundary conditions and governing partial differential equations of motion are derived based on Hamilton's principle. By applying harmonic variations of displacements, the governing equations of motion are expressed as a tenth-order equation, which is solved to obtain the desired dynamic stiffness matrix. To compute the natural frequencies of in-plane and out-of-plane free vibration for both uniform and PTSBs, the Wittrick–Williams algorithm is employed. The computed frequencies are compared with the results obtained by other authors as well as those obtained from ABAQUS simulations. Various vibration modes of uniform and twisted sandwich beams are plotted and thoroughly discussed. Interestingly, contrary to straight symmetric sandwich beams, the results indicate that flexural displacements in pre-twisted symmetric sandwich beams exhibit coupling in two planes. Additionally, although there are minor changes in vibration frequencies, the mode shapes undergo significant transformations as the pre-twist angle is altered.

Noether’s Theorem of Herglotz Type for Fractional Lagrange System with Nonholonomic Constraints

This research aims to investigate the Noether symmetry and conserved quantity for the fractional Lagrange system with nonholonomic constraints, which are based on the Herglotz principle. Firstly, the fractional-order Herglotz principle is given, and the Herglotz-type fractional-order differential equations of motion for the fractional Lagrange system with nonholonomic constraints are derived. Secondly, by introducing infinitesimal generating functions of space and time, the Noether symmetry of the Herglotz type is defined, along with its criteria, and the conserved quantity of the Herglotz type is given. Finally, to demonstrate how to use this method, two examples are provided.