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Showing papers on "Equations of motion published in 1990"


Journal ArticleDOI
TL;DR: In this article, a method for carrying out molecular dynamics simulations of processes that involve electronic transitions is proposed, where the time dependent electronic Schrodinger equation is solved self-consistently with the classical mechanical equations of motion of the atoms.
Abstract: A method is proposed for carrying out molecular dynamics simulations of processes that involve electronic transitions. The time dependent electronic Schrodinger equation is solved self‐consistently with the classical mechanical equations of motion of the atoms. At each integration time step a decision is made whether to switch electronic states, according to probabilistic ‘‘fewest switches’’ algorithm. If a switch occurs, the component of velocity in the direction of the nonadiabatic coupling vector is adjusted to conserve energy. The procedure allows electronic transitions to occur anywhere among any number of coupled states, governed by the quantum mechanical probabilities. The method is tested against accurate quantal calculations for three one‐dimensional, two‐state models, two of which have been specifically designed to challenge any such mixed classical–quantal dynamical theory. Although there are some discrepancies, initial indications are encouraging. The model should be applicable to a wide variety of gas‐phase and condensed‐phase phenomena occurring even down to thermal energies.

3,173 citations


Journal ArticleDOI
TL;DR: This work describes a formalism for following the nonlinear propagation of long-wavelength metric and scalar-field fluctuations and performs an expansion in spatial gradients of the Arnowitt-Deser-Misner equations and retains only terms up to first order.
Abstract: Stochastic inflation can be viewed as a sequence of two-step processes In the first step a stochastic impulse from short-distance quantum fluctuations acts on long waves---the interaction In the second step the long waves evolve semiclassically---the propagation Both steps must be developed to address whether fluctuations for cosmic structure formation may be non-Gaussian We describe a formalism for following the nonlinear propagation of long-wavelength metric and scalar-field fluctuations We perform an expansion in spatial gradients of the Arnowitt-Deser-Misner equations and we retain only terms up to first order At each point the fields obey evolution equations like those in a homogeneous universe, but now described by a local scale factor ${e}^{\ensuremath{\alpha}}$ and Hubble expansion rate $H$ However, the different points are joined together through the momentum constraint equation The gradient expansion is appropriate for inflation if the long-wave fields are smoothed over scales below ${e}^{\ensuremath{-}\ensuremath{\alpha}}{H}^{\ensuremath{-}1}$ Our equations are naturally described in the Einstein-Hamilton-Jacobi framework, which governs an ensemble of inhomogeneous universes, and which may be interpreted as a semiclassical approximation to the quantum theory We find that the Hubble parameter, which is a function of the local values of the scalar field, obeys a separated Hamilton-Jacobi equation that also governs the semiclassical phase of the wave functional In our approximation, time hypersurface changes leave the equations invariant However, the stochastic impulses that change the field initial conditions are most simply given on uniform expansion factor hypersurfaces whereas propagation is most easily solved on uniform Hubble hypersurfaces, in terms of $\ensuremath{\alpha}({x}^{j},H)$, the nonlinear analog of $\ensuremath{\zeta}$ of linear perturbation theory; we therefore pay special attention to hypersurface shifting In particular, we describe the transformation process for the fluctuation probability functional Exact general solutions are found for the case of a single scalar field interacting through an exponential potential For example, we show that quantum corrections to long-wavelength evolution of the metric are characteristically small using exact Green's-function solutions of the Wheeler-DeWitt equation for this potential Approximate analytic solutions to our classical system for slowly evolving multiple scalar fields are also easy to obtain in this formalism, contrasting with previous numerical approaches

1,086 citations


Journal ArticleDOI
TL;DR: In this article, a continuous contact force model for the impact analysis of a two-particle collision is presented, where a hysteresis damping function is incorporated in the model which represents the dissipated energy in impact.
Abstract: A continuous contact force model for the impact analysis of a two-particle collision is presented. The model uses the general trend of the Hertz contact law. A hysteresis damping function is incorporated in the model which represents the dissipated energy in impact. The parameters in the model are determined, and the validity of the model is established. The model is then generalized to the impact analysis between two bodies of a multibody system. A continuous analysis is performed using the equations of motion of either the multibody system or an equivalent two-particle model of the colliding bodies. For the latter, the concept of effective mass is presented in order to compensate for the effects of joint forces in the system. For illustration, the impact situation between a slider-crank mechanism and another sliding block is considered.

807 citations


Journal ArticleDOI
TL;DR: In this paper, consistent equations of motion of interacting gauge fields of all spins in 3+1 dimensions are formulated in a closed form, which are explicitly general coordinate invariant, possess all necessary higher spin gauge symmetries and reduce to the usual equations of free massless fields at the linearized level.

754 citations


Journal ArticleDOI
TL;DR: In this article, the equations of motion are cast in a canonical state space form defined by one symmetric and one skew-symmetric differential operator, and the eigenfunctions are orthogonal with respect to each operator.
Abstract: The equations of motion are cast in a canonical state space form defined by one symmetric and one skew-symmetric differential operator When an equation of motion is represented in this form, the eigenfunctions are orthogonal with respect to each operator Following this formulation, a classical vibration theory, comprised of a modal analysis and a Green's function method, is derived for the class of axially moving continua

554 citations



Journal ArticleDOI
M. J. Duff1
TL;DR: In this article, a dual σ-model with coordinates y α ( α = 1, α, n ) for which the roles of field equations and Bianchi identities are interchanged is introduced, and the hidden symmetries are made manifest by considering a target space of double the dimension with coordinates Z M = ( x μ, y α ).

462 citations


Journal ArticleDOI
TL;DR: In this article, the authors derived equations of motions, constitutive equations, and boundary conditions for a class of micromorphic elastic solids whose microelements can undergo expansions and contractions or stretch.

335 citations


Book ChapterDOI
J. G. Charney1
01 Jan 1990
TL;DR: In this paper, the authors pointed out that the problem of integration is greatly complicated by the simultaneous existence of a discrete set of wave motions all of which satisfy the conditions of the problem, namely that the motion be simple-harmonic and of a specified wave length.
Abstract: In a recent publication entitled The Dynamics of Long Waves in a Baroclinic Westerly Current1 (1947) the writer pointed out that, in the study of atmospheric wave motion, the problem of integration is greatly complicated by the simultaneous existance of a discrete set of wave motions all of which satisfy the conditions of the problem, namely that the motion be simple-harmonic and of a specified wave-length Whereas only the long inertially-propagated waves are important for the study of large-scale weather phenomena, one is forced by the generality of the equations of motion to contend with each of the theoretically possible wave types This extreme generality whereby the equations of motion apply to the entire spectrum of possible motions — to sound waves as well as to cyclone waves — constitutes a serious defect of the equations from the meteorological point of view It means that the investigator must take into account modifications to the large-scale motions of the atmosphere which are of little meteorological importance and which only serve to make the integration of the equations a virtual impossibility

319 citations


Journal ArticleDOI
TL;DR: In this article, the structure of pion wave functions of twist 3 and twist 4 is studied for the conformal spin and the first order corrections to asymptotic formulae are calculated by the QCD sum rule method.
Abstract: The restrictions are studied for the general structure of pion wave functions of twist 3 and twist 4 imposed by the conformal symmetry and the equations of motion. A systematic expansion of wave functions in the conformal spin is built and the first order corrections to asymptotic formulae are calculated by the QCD sum rule method. In particular, we have found a multiplicatively renormalizable contribution into the two-particle wave function of twist 4 which cannot be expanded in a finite set of Gegenbauer polynonials.

315 citations


Journal ArticleDOI
TL;DR: In this paper, the authors present the loop equations of motion which define the correlation functions for loop operators in two-dimensional quantum gravity and show that non-perturbative correlation functions constructed from real solutions of the Painleve equation of the first kind violate these equations by nonperturbation terms.
Abstract: We present the loop equations of motion which define the correlation functions for loop operators in two-dimensional quantum gravity. We show that non-perturbative correlation functions constructed from real solutions of the Painleve equation of the first kind violate these equations by non-perturbative terms.

Journal ArticleDOI
TL;DR: In this article, a frequency-domain approach was proposed to model the wave propagation in complex media for multiple source positions, where solutions for multiple sources are required or when only a limited number of frequency components of the solution are required.
Abstract: The migration, imaging, or inversion of wide-aperture cross-hole data depends on the ability to model wave propagation in complex media for multiple source positions. Computational costs can be considerably reduced in frequency-domain imaging by modeling the frequency-domain steady-state equations, rather than the time-domain equations of motion. I develop a frequency-domain approach in this note that is competitive with time-domain modeling when solutions for multiple sources are required or when only a limited number of frequency components of the solution are required.

Journal ArticleDOI
TL;DR: In this article, a new type of molecular dynamics is proposed to solve approximately the many-body problem of interacting identical fermions with spin, where the interacting system is represented by an antisymmetrized manybody wave function consisting of single-particle states which are localized in phase space.

Journal ArticleDOI
TL;DR: In this article, a review of molecular dynamics methods for simulations in the constant-temperature condition is given, where the authors consider a system which is thermally connected with a huge external system to describe a canonical ensemble in statistical mechanics.
Abstract: A review is given of how molecular dynamics methods have been modified to perform simulations in the constant-temperature condition. One usually considers a system which is thermally connected with a huge external system (a heat reservoir) to describe a canonical ensemble in statistical mechanics. The way in which this situation is reflected is a key factor for simulations under isothermal conditions. The total kinetic energy is kept to a constant value in constraint methods. In stochastic methods, interactions with a heat bath are treated as random collisions with hypothetical atoms or random forces acting on particles. In the extended-system method, a degree of freedom which mimics a heat bath is introduced, and the total energy of a physical system is allowed to fluctuate.

Journal ArticleDOI
TL;DR: In this article, a procedure to solve simultaneously the Euler flow equations and modal structural equations of motion is presented for computing aeroelastic responses of wings, which is validated with the experiment, both for a semi-infinite wing and a wall-mounted cantilever rectangular wing.
Abstract: A procedure to solve simultaneously the Euler flow equations and modal structural equations of motion is presented for computing aeroelastic responses of wings. The Euler flow equations are solved by a finite-difference scheme with dynamic grids. The coupled aeroelastic equations of motion are solved using the linear-acceleration method. The aeroelastic configuration adaptive dynamic grids are time-accurately generated using the aeroelastically deformed shape of the wing. The unsteady flow calculations are validated with the experiment, both for a semi-infinite wing and a wall-mounted cantilever rectangular wing. Aeroelastic responses are computed for a rectangular wing using the modal data generated by the finite-element method. The robustness of the present approach in computing unsteady flows and aeroelastic responses that are beyond the capability of earlier approaches using potential equations are demonstrated.

Journal ArticleDOI
TL;DR: In this article, the closed-form solution of the equations of motion of an ideal missile pursuing a nonmaneuvering target according to the pure proportional navigation law is obtained as a function of the polar coordinates for all real navigation constants N>or=2.
Abstract: The closed-form solution of the equations of motion of an ideal missile pursuing a nonmaneuvering target according to the pure proportional navigation law is obtained as a function of the polar coordinates for all real navigation constants N>or=2. The solution is given in the form of a uniformly convergent infinite product which reduces to a product of a finite number of factors if the navigation constant is a rational number. The solution is discussed, and necessary and sufficient conditions are stated for vanishing, bounded, and unbounded missile acceleration in the final phase of pursuit. >

Journal ArticleDOI
TL;DR: In this article, the authors used Boltzmann's superposition principle to express the stress as a time convolution of a fourth rank tensorial relaxation function with the strain tensor.
Abstract: SUMMARY The anisotropic linear viscoelastic rheological relation constitutes a suitable model for describing the variety of phenomena which occur in seismic wavefields. This rheology, known also as Boltzmann’s superposition principle, expresses the stress as a time convolution of a fourth rank tensorial relaxation function with the strain tensor. The first problem is to establish the time dependence of the relaxation tensor in a general and consistent way. Two kernels based on the general standard linear solid are identified with the mean stress and with the deviatoric components of the stress tensor in a. given coordinate system, respectively. Additional conditions are that in the elastic limit the relaxation matrix must give the elasticity matrix, and in the isotropic limit the relaxation matrix must approach the isotropic-viscoelastic matrix. The resulting rheological relation provides the framework for incorporating anelasticity in time-marching methods for computing synthetic seismograms. Through a plane wave analysis of the anisotropic-viscoelastic medium, the phase, group and energy velocities are calculated in function of the complex velocity, showing that those velocities are in general different from each other. For instance, the energy velocity which represents the wave surface, is different from the group velocity unlike in the anisotropic-elastic case. The group velocity loses its physical meaning at the cusps where singularities appear. Each frequency component of the wavefield has a different non-spherical wavefront. Moreover, the quality factors for the different propagating modes are not isotropic. Examples of these physical quantities are shown for transversely isotropic-viscoelastic clayshale and sandstone. As in the isotropic-viscoelastic case, Boltzmann’s superposition principle is implemented in the equation of motion by defining memory variables which circumvent the convolutional relation betweeh stress and strain. The numerical problem is solved by using a new time integration technique specially designed to deal with wave propagation in linear viscoelastic media. As a first application snapshots and synthetic seismograms are computed for 2-D transversely isotropicviscoelastic clayshale and sandstone which show substantial differences in amplitude, waveform and arrival time with the results given by the isotropic and elastic rheologies.

Journal ArticleDOI
TL;DR: In this paper, the wave transmission coefficients of structural joints are calculated for a generic plate/beam junction, which consists of an arbitrary number of plates which are either coupled through a beam or directly coupled along a line, and due allowance is made for offsets between the plate attachment lines and the shear axis of the beam.

Book
01 Jan 1990
TL;DR: In this article, a simple energy balance climate model is proposed to model the dynamics of a rotating system and the effect of transport on the composition of the system, and the generation of eddies by instability is discussed.
Abstract: Preface 1. Introductory remarks 2. Simple energy balance climate models 3. Effect of transport on composition 4. 'Statics' of a rotating system 5. Observed atmospheric structures 6. Equations of motion 7. Symmetric circulation models 8. Internal gravity waves, 1 9. Atmospheric tides 10. Internal gravity waves, 2 11. Rissby waves and the Gulf stream 12. Vorticity and quasi-geostrophy 13. The generation of eddies by instability, 1 14. Instability 2: energetics and climate implications Postscripts Appendix - Gravity wave program References.

Journal ArticleDOI
TL;DR: In this paper, a theory to describe the propagation of elastic waves in a porous medium saturated by a mixture of two immiscible, viscous, compressible fluids is presented.
Abstract: A theory to describe the propagation of elastic waves in a porous medium saturated by a mixture of two immiscible, viscous, compressible fluids is presented. First, using the principle of virtual complementary work, the stress–strain relations are obtained for both anisotropic and isotropic media. Then the forms of the kinetic and dissipative energy density functions are derived under the assumption that the relative flow within the porous medium is of laminar type and obeys Darcy’s law for two‐phase flow in porous media. The equations of motion are derived, and a discussion of the different kinds of body waves that propagate in this type of medium is given. A theorem on the existence, uniqueness, and regularity of the solution of the equations of motion under appropriate initial and boundary conditions is stated.

Journal ArticleDOI
TL;DR: In this article, the impact of shear flow on the isotropic-nematic transition in crystalline liquids was considered and steady-state solutions to the equations of motion for the nematic order parameter and fluid velocity were found in terms of nonequilibrium steady states.
Abstract: We consider the impact of shear flow on the isotropic-nematic transition in crystalline liquids by generalizing Leslie-Ericksen dynamics of nematic systems to include amplitude and biaxial degrees of freedom. Neglecting fluctuations, we find steady-state solutions to the equations of motion for the nematic order parameter and fluid velocity and interpret them in terms of nonequilibrium steady states. We predict a transition temperature increasing with shear rate up to a nonequilibrium critical point, and discuss the singular behavior of the order parameter and external stress near this point.

Journal ArticleDOI
TL;DR: In this paper, the real time dynamics of a moving charged soliton in polyacetylene is studied numerically by using Su-Schrieffer-Heeger's model, where an electric field is introduced to the system through a time dependent vector potential which can be included into the Hamiltonian through the Peierls substitution of the phase factor to the transfer integral.
Abstract: Real time dynamics of a moving charged soliton in polyacetylene is studied numerically by using Su-Schrieffer-Heeger's model. An electric field is introduced to the system through a time dependent vector potential which can be included into the Hamiltonian through the Peierls substitution of the phase factor to the transfer integral. Several interesting properties of the moving soliton are obtained, e.g. , the Brownian-like motion of the soliton due to the soliton-phonon interaction, the saturation of the soliton velocity, the saturation velocity being independent of the applied electric field strength, while the time needed to attain the saturation velocity is, roughly speaking, linearly dependent on the logarithm of the applied field.

Journal ArticleDOI
TL;DR: In this paper, the non-planar responses of a cantilevered beam subject to lateral harmonic baseexcitation are investigated using two non-linear coupled integrodifferential equations of motion.
Abstract: The non-planar responses of a cantilevered beam subject to lateral harmonic baseexcitation are investigated using two non-linear coupled integrodifferential equations of motion. The equations contain cubic non-linearities due to curvature and inertia. Two uniform beams with rectangular cross sections are considered: one has an aspect ratio near unity, and the other has an aspect ratio near 6.27. A combination of the Galerkin procedure and the method of multiple scales is used to construct a first-order uniform expansion for the case of a one-to-one internal resonance and a primary resonance. The results show that the non-linear geometric terms are important for the responses of low-frequency modes because they produce hardening spring effects. On the other hand, the non-linear inertia terms dominate the responses of high-frequency modes. We also obtain quantitative results for non-planar motions and investigate their dynamic behavior. For different range of parameters, the non-planar motions can be steady whirling motions, whirling motions of the beating type, or chaotic motions. Furthermore, we investigate the effects of damping.

Journal ArticleDOI
TL;DR: In this article, the instability of composite laminated plates under uniaxial, harmonically-varying, in-plane loads is investigated, both symmetric cross-ply and antisymmetric angle-ply laminates are analyzed.

Journal ArticleDOI
TL;DR: In this paper, the McLachlan variational principle for the time-dependent Schrodinger equation is utilized in conjunction with localized Guassian wave packet technology to deduce equations of motion for general multidimensional Gaussians.
Abstract: The McLachlan variational principle for the time‐dependent Schrodinger equation is utilized in conjunction with extant localized Guassian wave packet technology to deduce equations of motion for general multidimensional Gaussians. These equations of motion are characterized by the same simplicity as the local quadratic expansion results of Heller [J. Chem. Phys. 62, 1544 (1975)]. However, the resultant variational wave packet evolution is shown to be an improvement over its local quadratic analog as a tool for computing certain photodissociation spectra. Numerical examples drawn from the Beswick–Jortner model of ICN photodissociation [Chem. Phys. 24, 1 (1977)] are presented.

Journal ArticleDOI
TL;DR: The equation of motion for the rotating pair is found analytically and numerical solutions show good agreement with experiments, and the response of bound pairs of magnetic holes subjected to a rotating magnetic field is studied.
Abstract: Experimental studies are made of the response of bound pairs of magnetic holes (nonmagnetic microspheres in ferrofluid) subjected to a rotating magnetic field. For increasing driving frequency, the motion of the system goes through a transition from a state where the pair axis follows the magnetic field with a constant phase delay to a state where the phase delay increases in a series of kinks. The equation of motion for the rotating pair is found analytically and numerical solutions show good agreement with experiments.

Journal ArticleDOI
TL;DR: In this paper, the authors derived various identities and showed that the rigid bond approximation is the high frequency limit of the true dynamic friction coefficient, which is very accurate even for frequencies not much larger than the peak frequency of the solvent spectral density.
Abstract: A wide variety of problems involving molecular motion in liquids can be formulated in terms of the generalized Langevin equation (GLE). The friction coefficient on a molecular bond or on some more complicated reaction coordinate is then required. An often used approximation is to set the dynamic friction constant equal to the autocorrelation function of the fluctuating force exerted on the frozen bond by the remaining unfrozen coordinates. The true friction involves projection operators and should differ from this approximation. In this paper we derive various identities and show that the rigid bond approximation is the high frequency limit of the true dynamic friction coefficient. We compute the ‘‘true’’ dynamic friction and the friction approximated on the basis of the rigid or frozen bond and show that the asymptotic limit is very accurate even for frequencies not much larger than the peak frequency of the solvent spectral density. Two different dynamical systems are studied using MD simulations with our newly devised NAPA integrator for systems with disparate time scales. In one the molecule is not allowed to rotate and in the other it is allowed to rotate. Interestingly, even for very long rotational reorientation times, small but significant differences in the long time decay of the bond dynamic friction are observed for rotational and nonrotational molecules—differences, however, that do not produce large differences in the static friction constants.

Journal ArticleDOI
TL;DR: In this article, a simplification of the canonical Hamiltonian variables for the guiding center motion of a charged particle in a general toroidal field is obtained using the Lagrangian formalism.
Abstract: A simplification of the canonical Hamiltonian variables for the guiding center motion of a charged particle in a general toroidal field is obtained using the Lagrangian formalism.

Journal ArticleDOI
TL;DR: In this paper, a macroscopic model of two-phase flow in packed beds, based on the volume-averaged equations of motion for the gas and liquid phases, was analyzed in an attempt to understand the onset and evolution of fully-developed pulsing flow in trickle beds.
Abstract: A macroscopic model of two-phase flow in packed beds, based on the volume-averaged equations of motion for the gas and liquid phases, was analyzed in an attempt to understand the onset and evolution of fully-developed pulsing flow in trickle beds. By assuming that solutions take the form of waves travelling at constant speed, periodic solutions to these equations are found which can be associated with long-time, asymptotic behavior of pulses in a very long bed. Families of one-dimensional waves which exist at a particular set of mass fluxes can be characterized by infinite period bifurcations in the travelling wave frame. We numerically follow these bifurcations as the fluxes are changed, generating bifurcation diagrams for the original model. The results predict that properties of one-dimensional pulses should correlate with the total superficial velocity through the bed. A hysteresis in the trickling-pulsing transition is also predicted.

Journal ArticleDOI
TL;DR: In this article, a two-dimensional theory of gravity with dynamical metric and torsion is considered, and the equations of motion are reduced to a system of equations for two scalar fields which in particular case yields the Liouville equation.