Terms In Equations Of Motion Research Articles

Theoretical investigation of atomic dynamics of molten metals at melting points

In molten metals, the atomic dynamics have been studied in the present study with the help of equations of motion in terms of the velocity autocorrelation function (VACF), power spectrum G ( ω ) , and mean square displacement r 2 ( t ) . The pair correlation function g ( r ) is used to produce VACF for molten metals in the current study with the help of three different types of structure factor methods i.e. hard sphere Yukawa (HSY) method, optimised random phase approximation (ORPA) method and charged hard sphere (CHS) method. The presently obtained results for molten metals of different groups like Na (Z = 1), Mg (Z = 2), Al (Z = 3), Sn (Z = 4), and Sb (Z = 5) are compared with the molecular dynamic (MD) study and also with the other theoretical data available in the literature. For the present study, our own developed pseudopotential has been used for the calculations of the above-mentioned properties. The screening effect is also been observed over and beyond the bare ion potential with Hartree (H) and the effects of exchanges and correlation were computed using procedures given by Taylor (T) and Nagy (N).

MEMS Electrostatically Driven Coupled Beam Filter Banks.

MEMS bandpass filters based on electrostatically driven, mechanically coupled beams with in-plane motion have been demonstrated up to the VHF band. Filters higher than second order with parallel plate drives have inherent tuning difficulties, which may be resolved by adding mass-loaded beams to the ends of the array. These beams deflect for DC voltages, and thus allow synchronized electrostatic tuning, but do not respond to in-band AC voltages and hence do not interfere with dynamic synchronization. Additional out-of-band responses may be damped, leaving the desired response. The principle is extended here to close-packed banks of filters, with adjacent arrays sharing mass-loaded beams that localize modes to sub-arrays. The operating principles are explained using a lumped element model (LEM) of the equations of motion in terms of resonant modes and the reflection of acoustic waves at discontinuities. Performance is simulated using the LEM and verified using the more realistic stiffness matrix method (SMM) for banks of up to eight filters. Similar or dissimilar filters may be combined in a compact arrangement, and the method may be extended to higher order resonances and alternative coupled resonator systems.

Generalized multiple scale approach to the problem of a taut string traveled by a single force

The strongly nonlinear dynamics of taut strings, traveled by a force moving with uniform velocity, is analyzed. A change of variable is performed, which recasts the equations of motion in terms of a linearized dynamic displacement, measured from the nonlinear quasi-static response. Under the hypothesis the load velocity is far enough from the celerity of the string, the system appears in the form of linear PDEs whose coefficients are slowly variable in time. Since the classic perturbation methods are not applicable to such kind of equations, the Generalized Method of Multiple Scales is developed, by directly attacking the PDEs, to derive asymptotic solutions. The validity of the analytical predictions is assessed by comparisons with numerical simulations, aimed to prove the accuracy of (1) linearization, and (2) the asymptotic approach.

Mathematical Analysis of the Vibratory Pile Driving Rate

Vibratory piling technology does not require analytical tools to predict displacement rates and arising forces. The authors consider the problem of vibratory driving of a pile into a homogeneous unsaturated sandy massif under the action of static and dynamic loads. The purpose of this study is to develop a new analytical solution to the problem of the vibratory pile driving rate in a homogeneous sand base taking vibration creep into account. The solution is provided for the quasi-dynamic problem statement (inertial terms in equations of motion are neglected): the sand medium develops viscous properties due to vibration under the action of the dynamic component of the load, and a pile is driven into the viscous sand base due to the static component of the vertical load. The obtained mathematical model converges with the results of laboratory flume and field experiments performed by other researchers earlier, where the pile vibratory embedding rate increased along with an increase in static loading, the amplitude of dynamic load, and vibration frequency. It can be used to predict the pile or sheet pile driving rate into the unsaturated sand base under the action of vibration, and also to evaluate the necessary parameters of pile driving to obtain the required value of the pile embedding rate.

Equivalence between finite state stochastic machine, non-dissipative and dissipative tight-binding and Schrödinger model

The mathematical equivalence between finite state stochastic machine and non-dissipative and dissipative quantum tight-binding and Schrödinger model is derived. Stochastic Finite state machine is also expressed by classical epidemic model and can reproduce the quantum entanglement emerging in the case of electrostatically coupled qubits described by von-Neumann entropy both in non-dissipative and dissipative case. The obtained results show that quantum mechanical phenomena might be simulated by classical statistical model as represented by finite state stochastic machine. It includes the quantum like entanglement and superposition of states. Therefore coupled epidemic models expressed by classical systems in terms of classical physics can be the base for possible incorporation of quantum technologies and in particular for quantum like computation and quantum like communication. The classical density matrix is derived and described by the equation of motion in terms of anticommutator. Existence of Rabi like oscillations is pointed in classical epidemic model. Furthermore, the existence of Aharonov–Bohm effect in quantum systems (Becker et al., 2019) can also be reproduced by the classical epidemic model or in broader sense by finite state stochastic machine. Every quantum system made from quantum dots and described by simplistic tight-binding model by use of position-based qubits can be effectively described by classical statistical model encoded in finite stochastic state machine with very specific structure of S matrix that has twice bigger size as it is the case of quantum matrix Hamiltonian. Furthermore, the description of linear and non-linear stochastic finite state machine is mapped to tight-binding and Schrödinger model. The concept of N dimensional complex time is incorporated into tight-binding model, so the description of dissipation in most general case is possible. Obtained results help in approximation of non-dissipative or dissipative quantum mechanical phenomena by classical statistical physics expressed by finite state stochastic machine. Furthermore, the equivalence of Wannier tight-binding dissipative or non-dissipative formalism to dissipative or non-dissipative Schrödinger formalism and to finite state stochastic machine was shown.

A structure preserving stochastic perturbation of classical water wave theory

The inclusion of stochastic terms in equations of motion for fluid problems enables a statistical representation of processes which are left unresolved by numerical computation. Here, we derive stochastic equations for the behaviour of surface gravity waves using an approach which is designed to preserve the geometric structure of the equations of fluid motion beneath the surface. In doing so, we find a stochastic equation for the evolution of a velocity potential and, more significantly, demonstrate that the stochastic equations for water wave dynamics have a Hamiltonian structure which mirrors that found by Zakharov for the deterministic theory. This involves a perturbation of the velocity field which, unlike the deterministic velocity, need not be irrotational for the problem to close.

Spherical photon orbits around a rotating black hole with quintessence and cloud of strings

In this paper, we calculate the analytical solutions for the radii of planar and polar spherical photon orbits around a rotating black hole that is associated with quintessential field and cloud of strings. This includes a full analytical treatment of a quintic that describes orbits on the equatorial plane. Furthermore, The radial profile of the impact parameters is studied and the radii corresponding to the extreme cases are derived. For the more general cases, we also discuss the photon regions that form around this black hole. To simulate the orbits that appear in different inclinations, we analytically solve the latitudinal and azimuth equations of motion in terms of the Weierstrassian elliptic functions, by considering the radii of spherical orbits, in their general form, as the initial conditions. The period and the stability conditions of the orbits are also obtained analytically.

Free Vibration Characteristics of Bidirectional Graded Porous Plates with Elastic Foundations Using 2D-DQM

This manuscript develops for the first time a mathematical formulation of the dynamical behavior of bi-directional functionally graded porous plates (BDFGPP) resting on a Winkler–Pasternak foundation using unified higher-order plate theories (UHOPT). The kinematic displacement fields are exploited to fulfill the null shear strain/stress at the bottom and top surfaces of the plate without needing a shear factor correction. The bi-directional gradation of materials is proposed in the axial (x-axis) and transverse (z-axis) directions according to the power-law distribution function. The cosine function is employed to define the distribution of porosity through the transverse z-direction. Equations of motion in terms of displacements and associated boundary conditions are derived in detail using Hamilton’s principle. The two-dimensional differential integral quadrature method (2D-DIQM) is employed to transform partial differential equations of motion into a system of algebraic equations. Parametric analysis is performed to illustrate the effect of kinematic shear relations, gradation indices, porosity type, elastic foundations, geometrical dimensions, and boundary conditions (BCs) on natural frequencies and mode shapes of BDFGPP. The effect of the porosity coefficient on the natural frequency is dependent on the porosity type. The natural frequency is dependent on the coupling of gradation indices, boundary conditions, and shear distribution functions. The proposed model can be used in designing BDFGPP used in nuclear, marine, aerospace, and civil structures based on their topology and natural frequency constraints.

A Simplified Panel Method (sPM) for Hydrodynamics of Air Cushion Assisted Platforms

Air-cushion-assisted platforms (ACAPs) are floating platforms supported by both buoyancy pontoon and air cushion, which have merits of wave bending moment reduction, better stability, and hydrodynamic performance. However, there is barely a concise method that can quickly predict the motion response of ACAPs. In this paper, a simplified panel method (sPM) was presented for evaluating the hydrodynamics of ACAPs. The sPM extends the conventional boundary integral equation (BIE) to include the radiation solutions of pulsating air pressure but ignores some unimportant air-water cross terms in motion equations whose coefficients cannot be directly derived from conventional Green’s function methods. The effectiveness of the sPM was validated by experimental data from an ACAP model with one air chamber and analytical results from an oscillating water column (OWC). The numerical results demonstrate that the sPM can give desirable predictions for motion responses of the ACAP and inner pressure of the OWC as compared with results from the literature, which suggests the sPM could be approximately applied to evaluation of hydrodynamic performance of ACAPs and OWCs.

Dynamics of active polar ring polymers.

The conformational and dynamical properties of isolated semiflexible active polar ring polymers are investigated analytically. A ring is modeled as a continuous Gaussian polymer exposed to tangential active forces. The analytical solution of the linear non-Hermitian equationof motion in terms of an eigenfunction expansion shows that ring conformations are independent of activity. In contrast, activity strongly affects the internal ring dynamics and yields characteristic time regimes, which are absent in passive rings. On intermediate timescales, flexible rings show an activity-enhanced diffusive regime, while semiflexible rings exhibit ballistic motion. Moreover, a second active time regime emerges on longer timescales, where rings display a snake-like motion, which is reminiscent to a tank-treading rotational dynamics in shear flow, dominated by the mode with the longest relaxation time.

Axion-gauge field dynamics with backreaction

Phenomenological success of inflation models with axion and SU(2) gauge fields relies crucially on control of backreaction from particle production. Most of the previous study only demanded the backreaction terms in equations of motion for axion and gauge fields be small on the basis of order-of-magnitude estimation. In this paper, we solve the equations of motion with backreaction for a wide range of parameters of the spectator axion-SU(2) model. First, we find a new slow-roll solution of the axion-SU(2) system in the absence of backreaction. Next, we obtain accurate conditions for stable slow-roll solutions in the presence of backreaction. Finally, we show that the amplitude of primordial gravitational waves sourced by the gauge fields can exceed that of quantum vacuum fluctuations in spacetime by a large factor, without backreaction spoiling slow-roll dynamics. Imposing additional constraints on the power spectra of scalar and tensor modes measured at CMB scales, we find that the sourced contribution can be more than ten times the vacuum one. Imposing further a constraint of scalar modes non-linearly sourced by tensor modes, the two contributions can still be comparable.

Holographic multicondensate with nonlinear terms

We study the influence of nonlinear terms quartic of the condensates on the phase structure of a holographic model with a multicondensate in the probe limit. We include one s-wave order and one p-wave order charged under the same U(1) gauge field in the holographic model and study the influence of the three quartic nonlinear terms with coefficients ${\ensuremath{\lambda}}_{s}$, ${\ensuremath{\lambda}}_{p}$ and ${\ensuremath{\lambda}}_{sp}$ on the phase structure. The quartic terms in action result in cubic nonlinear terms in equations of motion, which do not affect the critical points of a single condensate because the infinitesimal condensate value vanishes the nonlinear contributions. However, the quartic terms have a clear influence on the phase structure of systems containing multicondensates. We show the influence of each of the three parameters on the phase diagram with the other two set to zero, respectively. With these nonlinear terms, we get new power on tuning the phase structure of the holographic systems showing multicondensates, and realize a reentrant phase transition as an example.

Collisional strong-field QED kinetic equations from first principles

Starting from nonequilibrium quantum field theory on a closed time path, we derive kinetic equations for the strong-field regime of quantum electrodynamics (QED) using a systematic expansion in the gauge coupling $e$. The strong field regime is characterized by a large photon field of order $\mathcal{O}(1/e)$, which is relevant for the description of, e.g., intense laser fields, the initial stages of off-central heavy ion collisions, and condensed matter systems with net fermion number. The strong field enters the dynamical equations via both quantum Vlasov and collision terms, which we derive to order $\mathcal{O}(e^2)$. The kinetic equations feature generalized scattering amplitudes that have their own equation of motion in terms of the fermion spectral function. The description includes single photon emission, electron-positron pair photoproduction, vacuum (Schwinger) pair production, their inverse processes, medium effects and contributions from the field, which are not restricted to the so-called locally-constant crossed field approximation. This extends known kinetic equations commonly used in strong-field QED of intense laser fields. In particular, we derive an expression for the asymptotic fermion pair number that includes leading-order collisions and remains valid for strongly inhomogeneous fields. For the purpose of analytically highlighting limiting cases, we also consider plane-wave fields for which it is shown how to recover Furry-picture scattering amplitudes by further assuming negligible occupations. Known on-shell descriptions are recovered in the case of simply peaked ultrarelativistic fermion occupations. Collisional strong-field equations are necessary to describe the dynamics to thermal equilibrium starting from strong-field initial conditions.

Note on gauge invariance of second order cosmological perturbations * *Z.C. and Q.-H.Z. are supported by the National Natural Science Foundation of China (12075249, 11675182, 11690022), and S.W. is supported by the grants from the Institute of High Energy Physics (Y954040101, H95120P0U7), the Key Research Program of the Chinese Academy of Sciences (XDPB15), and the science research grants from the

We study the gauge invariant cosmological perturbations up to the second order. We demonstrate that there are infinite families of gauge invariant variables at both the first and second orders. The conversion formulae among different families are verified to be described by a finite number of bases that are gauge invariant. For the second order cosmological perturbations induced by the first order scalar perturbations, we explicitly represent their equations of motion in terms of the gauge invariant Newtonian, synchronous and hybrid variables, respectively.

Actively deforming porous media in an incompressible fluid: A variational approach

Many parts of biological organisms are comprised of deformable porous media. The biological media is both pliable enough to deform in response to an outside force and can deform by itself using the work of an embedded muscle. For example, the recent work (Ludeman et al., 2014) has demonstrated interesting ‘sneezing’ dynamics of a freshwater sponge, when the sponge contracts and expands to clear itself from surrounding polluted water. We derive the equations of motion for the dynamics of such an active porous media (i.e., a deformable porous media that is capable of applying a force to itself with internal muscles), filled with an incompressible fluid. These equations of motion extend the earlier derived equation for a passive porous media filled with an incompressible fluid. We use a variational approach with a Lagrangian written as the sum of terms representing the kinetic and potential energy of the elastic matrix, and the kinetic energy of the fluid, coupled through the constraint of incompressibility. We then proceed to extend this theory by computing the case when both the active porous media and the fluid are incompressible, with the porous media still being deformable, which is often the case for biological applications. For the particular case of a uniform initial state, we rewrite the equations of motion in terms of two coupled telegraph-like equations for the material (Lagrangian) particles expressed in the Eulerian frame of reference, particularly suitable for numerical simulations, formulated for both the compressible media/incompressible fluid case and the doubly incompressible case. We derive interesting conservation laws for the motion, perform numerical simulations in both cases and show the possibility of self-propulsion of a biological organism due to particular running wave-like application of the muscle stress.

A new pendulum motion with a suspended point near infinity

In this paper, a pendulum model is represented by a mechanical system that consists of a simple pendulum suspended on a spring, which is permitted oscillations in a plane. The point of suspension moves in a circular path of the radius (a) which is sufficiently large. There are two degrees of freedom for describing the motion named; the angular displacement of the pendulum and the extension of the spring. The equations of motion in terms of the generalized coordinates varphi and xi are obtained using Lagrange’s equation. The approximated solutions of these equations are achieved up to the third order of approximation in terms of a large parameter varepsilon will be defined instead of a small one in previous studies. The influences of parameters of the system on the motion are obtained using a computerized program. The computerized studies obtained show the accuracy of the used methods through graphical representations.

Thermal, trapped and chromo-natural inflation in light of the swampland criteria and the trans-Planckian censorship conjecture

We consider thermal, trapped and chromo-natural inflation in light of the swampland criteria and the Trans-Planckian Censorship Conjecture (TCC). Since thermal inflation occurs at energies low compared to those of Grand Unification, it is consistent with the TCC, and it is also consistent with the refined swampland conditions. Trapped and chromo-natural inflation are candidates for primordial (high energy scale) inflation. Since in both of these scenarios there are effective damping terms in the scalar field equation of motion, the models can easily be consistent with the swampland criteria. The TCC, on the other hand, constrains these scenarios to only take place at low energies.

Exact solution for dynamic analysis of a shape memory polymer plate subjected to the initial conditions

A theoretical solution was presented for the dynamic behavior of a simply supported shape memory polymer (SMP) plate subjected to the initial conditions. According to the time-dependent behavior of SMP, Hamilton’s principle, and the first-order shear deformation theory (FSDT), the equations of motion in terms of time and displacement were derived. It was seen that there are the fourth- and sixth-order differential equations for the in-plane and out-plane deformation components (u and w), respectively. The differential equations were solved theoretically and introduced in terms of time and displacement. In order to validate the theoretical solution, the theoretical results are compared to those obtained by the numerical code written based on the Runge-Kutta fourth-order method for the set of differential equations. The results indicate a good agreement between them. Moreover, the effects of the elastic modulus, viscosity, and the geometric dimensions of the plate on its deformation were investigated. The results showed that the mechanical and geometric parameters, as well as the initial conditions, have more effect on the deformation of the plate.

Analyzing the influence of a particle's linear and angular velocity on the equations of liquid motion

This paper has analyzed the equation of motion in terms of stresses (Navier), as well as its two special cases for an incompressible viscous current. One is the Stokes (Navier-Stokes) equation, and the other was derived with fewer restrictions. It has been shown that the Laplace equation of linear velocity can be represented as a function of two variables ‒ the linear and angular speed of particle rotation. To describe the particle acceleration, all motion equations employed a complete derivative from speed in the Gromeka-Lamb form, which depends on the same variables. Taking into consideration the joint influence of linear and angular velocity allows solving a task of the analytical description of a turbulent current within the average model. A given method of analysis applies the provision of general physics that examines the translational and rotational motion. The third type of mechanical movement, oscillatory (pulsation), was not considered in the current work. A property related to the Stokes equation decomposition has been found; a block diagram composed of equations and conditions has been built. It is shown that all equations for viscous liquid have their own analog in a simpler model of non-viscous fluid. That makes it easier to find solutions to the equations for the viscous flow. The Stokes and Navier equations were used to solve two one-dimensional problems, which found the distribution of speed along the normal to the surface at the flow on a horizontal plate and in a circular pipe. Both solution methods produce the same result. No solution for the distribution of speed along the normal to the surface in a laminar sublayer could be found. A relevant task related to the mathematical part is to solve the problem of closing the equations considered. A comparison of the theoretical and empirical equations has been performed, which has made it possible to justify the assumption that a rarefied gas is the Stokes liquid

Nanoscale pattern formation on solid surfaces bombarded by two broad ion beams in the regime in which sputtering is negligible

We study nanoscale pattern formation on the surface of a solid that is bombarded with two diametrically opposed, broad ion beams for ion energies low enough that sputtering can be neglected. We focus on the case in which the angle of ion incidence is just above the threshold angle for pattern formation. The equation of motion at sufficiently long times is derived using a generalized crater function formalism. This formalism also yields expressions for the coefficients in the equation of motion in terms of crater function moments. We find that virtually defect-free ripples with a sawtooth profile can emerge at sufficiently long times. The ripples also coarsen as time passes, in contrast to the near-threshold behavior of ripples in the higher energy regime in which sputtering is significant.