Newtonian Equations Of Motion Research Articles

Agglomeration behaviors of different types of complex TiN inclusions in high titanium wear resistant steel

Large-sized titanium-containing inclusion clusters seriously affect the continuous casting and performance of high titanium wear-resistant steel. In the current work, thermodynamic calculations were firstly conducted to predict the formation of various complex TiN inclusions, viz, TiOx-TiN, Al2O3–TiN and LaAlO3–TiN. Based on that, laboratory-scale experiments were designed to prepare samples containing a single type of complex TiN inclusions. The agglomeration characterizes of different complex inclusions were compared by combing in-situ observation based on confocal laser scanning microscopy and Newtonian motion equation. Furthermore, the Kralchevsky-Paunov model was used to study the influences of density and size on the agglomeration behavior of complex TiN inclusions. The result shows that the attractive force between TiOx-TiN complex inclusions increases from 1.93 × 10−18 to 4.34 × 10−17 N, while it was 3.41 × 10−17 to 5.12 × 10−16 N for Al2O3–TiN complex inclusions and 6.42 × 10−18 to 1.14 × 10−16 N for LaAlO3–TiN inclusions. Based on the experimental results and theoretical calculation, the agglomeration tendency of Al2O3–TiN is the strongest due to the considered size and compared to the TiOx-TiN inclusions, the stronger agglomeration tendency of LaAlO3–TiN ones can be attributed to their higher density.

Development of equation of motion deciphering locomotion including omega turns of Caenorhabditis elegans.

Locomotion is a fundamental behavior of Caenorhabditis elegans (C. elegans). Previous works on kinetic simulations of animals helped researchers understand the physical mechanisms of locomotion and the muscle-controlling principles of neuronal circuits as an actuator part. It has yet to be understood how C. elegans utilizes the frictional forces caused by the tension of its muscles to perform sequenced locomotive behaviors. Here, we present a two-dimensional rigid body chain model for the locomotion of C. elegans by developing Newtonian equations of motion for each body segment of C. elegans. Having accounted for friction-coefficients of the surrounding environment, elastic constants of C. elegans, and its kymogram from experiments, our kinetic model (ElegansBot) reproduced various locomotion of C. elegans such as, but not limited to, forward-backward-(omega turn)-forward locomotion constituting escaping behavior and delta-turn navigation. Additionally, ElegansBot precisely quantified the forces acting on each body segment of C. elegans to allow investigation of the force distribution. This model will facilitate our understanding of the detailed mechanism of various locomotive behaviors at any given friction-coefficients of the surrounding environment. Furthermore, as the model ensures the performance of realistic behavior, it can be used to research actuator-controller interaction between muscles and neuronal circuits.

Development of equation of motion deciphering locomotion including omega turns of Caenorhabditis elegans

Locomotion is a fundamental behavior of Caenorhabditis elegans (C. elegans). Previous works on kinetic simulations of animals helped researchers understand the physical mechanisms of locomotion and the muscle-controlling principles of neuronal circuits as an actuator part. It has yet to be understood how C. elegans utilizes the frictional forces caused by the tension of its muscles to perform sequenced locomotive behaviors. Here, we present a two-dimensional rigid body chain model for the locomotion of C. elegans by developing Newtonian equations of motion for each body segment of C. elegans. Having accounted for friction-coefficients of the surrounding environment, elastic constants of C. elegans, and its kymogram from experiments, our kinetic model (ElegansBot) reproduced various locomotion of C. elegans such as, but not limited to, forward-backward-(omega turn)-forward locomotion constituting escaping behavior and delta-turn navigation. Additionally, ElegansBot precisely quantified the forces acting on each body segment of C. elegans to allow investigation of the force distribution. This model will facilitate our understanding of the detailed mechanism of various locomotive behaviors at any given friction-coefficients of the surrounding environment. Furthermore, as the model ensures the performance of realistic behavior, it can be used to research actuator-controller interaction between muscles and neuronal circuits.

Modeling the Planar Rocking of a Rigid Body with Irregular Geometry

The nonlinear dynamics of rocking rigid bodies with simple geometries, such as rectangular blocks and cylinders, have been the focus of the rocking community over the last six decades. However, many objects that are prone to rocking or overturning do not conform to such geometries. These objects include museum artifacts and precariously balanced rocks in the natural world. Even in cases where the response of the rocking body is planar, the geometry of the body is much more complicated than the commonly studied geometry of a rocking block or a body with only two rocking corners. This paper introduces a complete model that can examine the planar motion of a body with an irregular in-plane polygonal geometry when subjected to a vibrational excitation, utilizing the geometry of the body as an input—for example, in the form of a stereolithography (STL) file. The model is used for studying the rocking response of an object while taking into account sliding and free flight. The problem is formulated and solved using Newtonian equations of motion, and impacts are treated as hard. A robust framework for integrating the occurring discontinuous equations of motion and for detecting transitions between patterns of motion and impacts, using MATLAB, is presented. Suitably chosen examples demonstrate the importance of accounting for the actual geometry of the studied rocking body, whose dynamic response is substantially richer than an object with simplified geometry.

Modeling Apsidal Motion in Eclipsing Binaries Using ELC

Apsidal motion is the precession of the line of apsides in the orbit of a binary star due to perturbations from General Relativity (GR), tides, or third-body interactions. The rate of precession due to tidal effects depends on the interior structures of the stars, and as a result, binaries in which this precession occurs are of great interest. Apsidal motion is observed through the analysis of eclipse times, which reveal small changes in the average interval between successive primary and secondary eclipses, taking all available observed times of eclipse and yielding an estimate of the apsidal rate. Given that this is a single observed quantity, various degeneracies are unavoidably present. Ideally, one would have a model that predicts eclipse times given the orbital and stellar parameters. These parameters for a given binary could then be computed using least squares, provided a suitably large number of eclipse times. Here we use the eclipsing light curve (ELC) program as such a model. The Newtonian equations of motion with additional force terms accounting for GR contributions and tidal distortions are integrated, yielding precise sky positions as a function of time. Times of mid-eclipse and instantaneous orbital elements are computed as a function of time. In this paper, we outline the method and compare numerically computed apsidal rates with standard formulae using a set of 15 binaries based on real systems. For our simulated systems, the derived apsidal rates agree with the standard formula.

Operando Modeling of Zeolite-Catalyzed Reactions Using First-Principles Molecular Dynamics Simulations.

Within this Perspective, we critically reflect on the role of first-principles molecular dynamics (MD) simulations in unraveling the catalytic function within zeolites under operating conditions. First-principles MD simulations refer to methods where the dynamics of the nuclei is followed in time by integrating the Newtonian equations of motion on a potential energy surface that is determined by solving the quantum-mechanical many-body problem for the electrons. Catalytic solids used in industrial applications show an intriguing high degree of complexity, with phenomena taking place at a broad range of length and time scales. Additionally, the state and function of a catalyst critically depend on the operating conditions, such as temperature, moisture, presence of water, etc. Herein we show by means of a series of exemplary cases how first-principles MD simulations are instrumental to unravel the catalyst complexity at the molecular scale. Examples show how the nature of reactive species at higher catalytic temperatures may drastically change compared to species at lower temperatures and how the nature of active sites may dynamically change upon exposure to water. To simulate rare events, first-principles MD simulations need to be used in combination with enhanced sampling techniques to efficiently sample low-probability regions of phase space. Using these techniques, it is shown how competitive pathways at operating conditions can be discovered and how broad transition state regions can be explored. Interestingly, such simulations can also be used to study hindered diffusion under operating conditions. The cases shown clearly illustrate how first-principles MD simulations reveal insights into the catalytic function at operating conditions, which could not be discovered using static or local approaches where only a few points are considered on the potential energy surface (PES). Despite these advantages, some major hurdles still exist to fully integrate first-principles MD methods in a standard computational catalytic workflow or to use the output of MD simulations as input for multiple length/time scale methods that aim to bridge to the reactor scale. First of all, methods are needed that allow us to evaluate the interatomic forces with quantum-mechanical accuracy, albeit at a much lower computational cost compared to currently used density functional theory (DFT) methods. The use of DFT limits the currently attainable length/time scales to hundreds of picoseconds and a few nanometers, which are much smaller than realistic catalyst particle dimensions and time scales encountered in the catalysis process. One solution could be to construct machine learning potentials (MLPs), where a numerical potential is derived from underlying quantum-mechanical data, which could be used in subsequent MD simulations. As such, much longer length and time scales could be reached; however, quite some research is still necessary to construct MLPs for the complex systems encountered in industrially used catalysts. Second, most currently used enhanced sampling techniques in catalysis make use of collective variables (CVs), which are mostly determined based on chemical intuition. To explore complex reactive networks with MD simulations, methods are needed that allow the automatic discovery of CVs or methods that do not rely on a priori definition of CVs. Recently, various data-driven methods have been proposed, which could be explored for complex catalytic systems. Lastly, first-principles MD methods are currently mostly used to investigate local reactive events. We hope that with the rise of data-driven methods and more efficient methods to describe the PES, first-principles MD methods will in the future also be able to describe longer length/time scale processes in catalysis. This might lead to a consistent dynamic description of all steps-diffusion, adsorption, and reaction-as they take place at the catalyst particle level.

Numerical simulation of non-central collisions of spherical magnets

We present a computational model of non-central collisions of two spherical neodymium-iron-boron magnets, suggested as a demonstration of angular momentum conservation. Our program uses an attractive dipole–dipole force and a repulsive contact force to solve the Newtonian equations of motion for the magnets. We confirm the conservation of angular momentum and study the changes in energy throughout the interaction. Using the exact expression for the dipole–dipole force, including non-central terms, we correctly model the final rotational frequencies, which is not possible with a simple power-law approximation.

Planar central configurations with five different positive masses

We study planar five-body central configurations with different positive masses. From Laura–Andoyer equations, we recover the variational formulation in terms of the potential energy and the total inertia moment, including constraints on the distances. We use coordinates to prove the permanence of the central configuration by Newton equations of motion. The masses appear in our coordinates, forming a rigid orthocentric simplex in four dimensions, with the masses at the vertices. The linear dimensions are determined by a distance. We use different angles, with one angle measuring the rotation around the center of mass. The rest of the coordinates are angles that are constants of motion for central configurations. One angle measures the ratio of the two principal moments of inertia. The rest of the coordinates determine the rotation around the center of mass of the rigid simplex of masses with respect to the plane and to the direction of projection from four to two dimensions. We transform the Laura–Andoyer equations into different expressions in 10 and 5 dimensions, including the properties of the areas formed by five point bodies in the plane. Furthermore, we presented three numerical computed configurations of planar five-body central configurations, one convex and two concave, which give numerical confirmation of our equations with a precision better than 10−12.

Newton-ADE-FDTD Method for Oblique Incident Magnetized Time-varying Plasma

In this paper, the Newtonian equation of motion describing the movement of electrons when electromagnetic waves propagate in a magnetized plasma is combined with the traditional auxiliary differential equation finite difference time domain (ADE-FDTD) method. The FDTD iterative formulas of transverse magnetic (TM) wave and transverse electric (TE) wave of the electromagnetic wave obliquely incident on the magnetized time-varying plasma plate are derived. The biggest difference between this method and the ordinary ADE-FDTD algorithm is the addition of the logarithmic derivative of the time-varying plasma electron density to calculate the current density, which is called the Newton-ADE-FDTD method. Through Example 1, the reflection coefficient of electromagnetic wave incident on the magnetization time-varying plasma plate was calculated, and the correctness of the improved algorithm was verified. At the same time, the Newton-ADE-FDTD algorithm is used to calculate the reflection coefficient of electromagnetic waves incident on the magnetized plasma-dielectric photonic crystal. The results show that different incident angles have a greater impact on the reflection coefficients of left-handed circularly polarized wave (LCP) and right-handed circularly polarizedwave (RCP).

Linear momentum flux from inspiralling compact binaries in quasielliptical orbits at 2.5 post-Newtonian order

Emission of anisotropic gravitational radiation from compact binary system leads to a flux of linear momentum. This results in the recoil of the system. We investigate the rate of loss of Linear momentum flux in the far zone of the source using various mass type and current type multipole moments for inspiralling compact binary mergers in quasi-elliptical orbits at 2.5 Post Newtonian order. We compute the linear momentum flux accurate up to $\mathcal{O}(e_t)$ in harmonic coordinate. A 2.5 Post Newtonian Quasi-Keplarian representation of the parametric solution to the Post Newtonian equation of motion for the compact binary system has been adopted here. We also provide a closed-form expression for the accumulated linear momentum from the remote past through the binary evolution.

Sink versus tilt penetration into shaken dry granular matter: The role of the foundation.

We study the behavior of cylindrical objects as they sink into a dry granular bed fluidized due to lateral oscillations. Somewhat unexpectedly, we have found that, within a large range of lateral shaking powers, cylinders with flat bottoms sink vertically, while those with a "foundation" consisting of a shallow ring attached to their bottom, tilt besides sinking. The latter scenario seems to dominate independently from the nature of the foundation when strong enough lateral vibrations are applied. We are able to explain the observed behavior by quasi-2D numerical simulations, which also demonstrate the influence of the intruder's aspect ratio. The vertical sink dynamics is explained with the help of a Newtonian equation of motion for the intruder. Our findings may shed light on the behavior of buildings and other manmade structures during earthquakes.

Dynamical Invariants for Generalized Coherent States via Complex Quantum Hydrodynamics

For time dependent Hamiltonians like the parametric oscillator with time-dependent frequency, the energy is no longer a constant of motion. Nevertheless, in 1880, Ermakov found a dynamical invariant for this system using the corresponding Newtonian equation of motion and an auxiliary equation. In this paper it is shown that the same invariant can be obtained from Bohmian mechanics using complex Hamiltonian equations of motion in position and momentum space and corresponding complex Riccati equations. It is pointed out that this invariant is equivalent to the conservation of angular momentum for the motion in the complex plane. Furthermore, the effect of a linear potential on the Ermakov invariant is analysed.

Statistics of Sliding on Periodic and Atomically Flat Surfaces

Among the so-called analytical models of friction, the most popular and widely used one, the Prandtl-Tomlinson model in one and two dimensions is considered here to numerically describe the sliding of the tip within an atomic force microscope over a periodic and atomically flat surface. Because in these PT-models, the Newtonian equations of motion for the AFM-tip are Langevin-type coupled stochastic differential equations the resulting friction and reaction forces must be statistically correctly determined and interpreted. For this, it is firstly shown that the friction and reaction forces as averages of the time-resolved ones over the sliding part, are normally (Gaussian) distributed. Then based on this, an efficient numerical scheme is developed and implemented to accurately estimate the means and standard deviations of friction and reaction forces without performing too many repetitions for the same sliding experiments. The used corrugation potential is the simplest one obtained from the Fourier series expansion of the two-dimensional (2D) periodic potential, e.g., for an fcc(111) surface, which permits sliding on both commensurate and incommensurate paths. In this manner, it is proven that the PT-models predict both frictional regimes, namely the structural superlubricity and stick-slip along (in)commensurate sliding paths, if the ratio of mean corrugation and elastic energies is properly set.

Understanding simple liquids through statistical and deep learning approaches.

Statistical and deep learning-based methods are employed to obtain insights into the quasi-universal properties of simple liquids. In the first part, a statistical model is employed to provide a probabilistic explanation for the similarity in the structure of simple liquids interacting with different pair potential forms, collectively known as simple liquids. The methodology works by sampling the radial distribution function and the number of interacting particles within the cutoff distance, and it produces the probability density function of the net force. We show that matching the probability distribution of the net force can be a direct route to parameterize simple liquid pair potentials with a similar structure, as the net force is the main component of the Newtonian equations of motion. The statistical model is assessed and validated against various cases. In the second part, we exploit DeepILST [A. Moradzadeh and N. R. Aluru, J. Phys. Chem. Lett. 10, 1242-1250 (2019)], a data-driven and deep-learning assisted framework to parameterize the standard 12-6 Lennard-Jones (LJ) pair potential, to find structurally equivalent/isomorphic LJ liquids that identify constant order parameter [τ=∫0 ξcf gξ-1ξ2dξ, where gξ and ξ(=rρ13) are the reduced radial distribution function and radial distance, respectively] systems in the space of non-dimensional temperature and density of the LJ liquids. We also investigate the consistency of DeepILST in reproducibility of radial distribution functions of various quasi-universal potentials, e.g., exponential, inverse-power-law, and Yukawa pair potentials, quantified based on the radial distribution functions and Kullback-Leibler errors. Our results provide insights into the quasi-universality of simple liquids using the statistical and deep learning methods.

Solution of Lagrange's equation of motion form the first principle for volumetric modulated arc therapy delivery

To calculate the Lagrange’s equation of motion from first principle for volumetric modulated arc therapy delivery. The delivery of volumetric modulated arc therapy (VMAT) is based on a rotational co-ordinate system. This delivery technique differs from any other delivery technique with static beams. VMAT delivery is a complex many-body problem, to represent the VMAT uniquely, it is necessary to find the equation of motion in a non-inertial (rotating) fame. In the non-inertial frame of reference, Newton's equations of motion do not hold, hence Lagrange's equation of motion (Lagrange's equation of motion of second kind) needs to be solved. If A is the aperture created by Multileaf collimator (MLC), MU is delivered Monitor Unit of radiation dose and θ is the gantry angle, then equation of motion for volumetric arc therapy delivery is represented by following three equations 1. $$\frac{d\theta }{dt}=\dot{\theta }=$$ constant between two control points, 2. $$m\ddot{x}-m x \dot{\theta }2= Fx$$ ; where Fx is the external force along $$\overrightarrow{x}$$ direction, causing the MLC movement along the x-direction only and m is the mass of the MLC, and $${MU}_{delivered}\left(t\right)={\int }_{t=0}^{t=t}MU{(t)}_{planned}dt$$ . This piece of derivation presents the calculation of the equation of motion for VMAT delivery in terms of MLC movement, gantry angle, and delivered dose by deriving Lagrangian formulation from the first principle.

Investigating Total Collisions of the Newtonian N-Body Problem on Shape Space

We analyze the points of total collision of the Newtonian gravitational system on shape space (the relational configuration space of the system). While the Newtonian equations of motion, formulated with respect to absolute space and time, are singular at the point of total collision due to the singularity of the Newton potential at that point, this need not be the case on shape space where absolute scale doesn’t exist. We investigate whether, adopting a relational description of the system, the shape degrees of freedom, which are merely angles and their conjugate momenta, can be evolved through the points of total collision. Unfortunately, this is not the case. Even without scale, the equations of motion are singular at the points of total collision (and only there). This follows from the special behavior of the shape momenta. While this behavior induces the singularity, it at the same time provides a purely shape-dynamical description of total collisions. By help of this, we are able to discern total-collision solutions from non-collision solutions on shape space, that is, without reference to (external) scale. We can further use the shape-dynamical description to show that total-collision solutions form a set of measure zero among all solutions.

Effects of intrachain disorder on photoexcitation in conjugated polymer chains

The luminescence efficiency of conjugated polymers has been a central topic in the study of light emitting. The effect of disorder plays an important role in generating excitons after the conjugated polymers have been excited by photons. In this paper, by using the Su-Schriffer-Heeger model, which has been modified to include intrachain disorder and electron correlation, we investigate the effects of disorder on the photoexcitation, especially on the yield of excitons in a conjugated polymer chain. We adopt the multi-configurational time-dependent Hartree–Fock method to solve the multi-electron time-dependent Schrödinger equation and the Newtonian equation of motion for the lattice vibration. The results show that after the photoexcitation relaxation process, the products of the disordered polymer chain are qualitatively distinct from those of the pure polymer chain. While the pairs of polarons can be generated directly after the photoexcitation in a pure polymer chain, the disorder favors excitons as the products of the photoexcitation, and the yield of excitons depends crucially on the kind and strength of the disorder. Furthermore, the influences of the electron correlation and the conjugation length on the yield of excitons are also discussed. Specifically, we find that in the case of diagonal disorder, when the conjugation length is short and the diagonal disorder is weak, the excitons are mainly generated by the recombination of two lattice defects with a high yield of excitons which will be reduced as the conjugation length increases. The excitons tend to be generated directly with a low yield of excitons which is enlarged as the disorder gets stronger when the conjugation length is long or the diagonal disorder is strong. The on-site Coulomb repulsion favors the generation of excitons as well. The case of off-diagonal disorder is similar to that of diagonal disorder except that the on-site Coulomb potential favors the generation of excitons in the weak disorder regime but depresses the generation of excitons in the strong disorder regime. When both diagonal and off-diagonal disorders are considered, the yield of excitons is dominated by the off-diagonal disorder. We hope that our investigations can provide useful guidance and help for designing organic photoelectric materials and devices.

Effectiveness of KS elements in satellite orbit prediction using earth's gravity, drag and solar radiation pressure

Satellite moving under the gravitational field of Earth deviates from its two-body elliptic orbit, due to the combined effects of the gravitational field of Earth, atmospheric drag, solar radiation pressure, third-body gravitational effects, etc. This paper utilizes the KS regular element equations to solve Newtonian equations of motion to obtain numerical solution with respect to perturbing forces, like, Earth's gravity (includes zonal, sectorial and tesseral harmonics terms), atmospheric drag and solar radiation pressure. Effectiveness of the theory is illustrated by comparing the results with some of the existing theories in literature.Â

The Connection between Bohmian Mechanics and Many-Particle Quantum Hydrodynamics

Bohm developed the Bohmian mechanics (BM), in which the Schrödinger equation is transformed into two differential equations: a continuity equation and an equation of motion similar to the Newtonian equation of motion. This transformation can be executed both for single-particle systems and for many-particle systems. Later, Kuzmenkov and Maksimov used basic quantum mechanics for the derivation of many-particle quantum hydrodynamics (MPQHD) including one differential equation for the mass balance and two differential equations for the momentum balance, and we extended their analysis in a prework (K. Renziehausen, I. Barth in Prog. Theor. Exp. Phys. 2018:013A05, 2018) for the case that the particle ensemble consists of different particle sorts. The purpose of this paper is to show how the differential equations of MPQHD can be derived for such a particle ensemble with the differential equations of BM as a starting point. Moreover, our discussion clarifies that the differential equations of MPQHD are more suitable for an analysis of many-particle systems than the differential equations of BM because the differential equations of MPQHD depend on a single position vector only while the differential equations of BM depend on the complete set of all particle coordinates.

Fuel Conservation for Launch Vehicles: Falcon Heavy Case Study

Space exploration has recently been growing at an increasing pace and has caused a significant burden to the environment, in particular, during the launch of rockets, when a large amount of fuel is burned and the exhaust gases are released in the air. For this case study, we selected the SpaceX Falcon Heavy reusable heavy-lift launch vehicle, which is one of the most promising rockets for the low-cost lifting of heavy payloads into orbit and beyond. We evaluated several strategies for optimisation of fuel consumption and for minimisation of environmental impact during launch through the atmosphere for the case of its first launch on February 6, 2018, when the rocket carried a red Tesla Roadster with a “Starman” in the direction toward Mars. In addition to the flight plan and Newtonian equations of motion, we have taken into account the thermodynamic properties of the rocket engines. Results are similar but slightly different if one minimises the total fuel consumption for the desired flight plan or if one minimises the environmental pollution during the initial stage of the launch through the atmosphere. The same methodology can be extended for launches in other directions including the Earth orbit and the Moon.