Showing papers on "Equations of motion published in 2004"

The Motion of Point Particles in Curved Spacetime

Abstract: This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise to a self-force that prevents the particle from moving on a geodesic of the background spacetime. The field's action on the particle is difficult to calculate because of its singular nature: the field diverges at the position of the particle. But it is possible to isolate the field's singular part and show that it exerts no force on the particle. What remains after subtraction is a smooth field that is fully responsible for the self-force. The mathematical tools required to derive the equations of motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime are developed here from scratch. The review begins with a discussion of the basic theory of bitensors. It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle's word line. It continues with a thorough discussion of Green's functions in curved spacetime. The review presents a detailed derivation of each of the three equations of motion. Because the notion of a point mass is problematic in general relativity, the review concludes with an alternative derivation of the equations of motion that applies to a small body of arbitrary internal structure.

Hall effect of light.

Abstract: We derive the semiclassical equation of motion for the wave packet of light taking into account the Berry curvature in momentum-space. This equation naturally describes the interplay between orbital and spin angular momenta, i.e., the conservation of the total angular momentum of light. This leads to the shift of wave-packet motion perpendicular to the gradient of the dielectric constant, i.e., the polarization-dependent Hall effect of light. An enhancement of this effect in photonic crystals is also proposed.

Statistical Orbit Determination

Abstract: 1 Orbit Determination Concepts 2 The Orbit Problem 3 Observations 4 Fundamentals of Orbit Determination 5 Square-root Solution Methods 6 Consider Covariance Analysis A Probability and Statistics B Review of Matrix Concepts C Equations of Motion D Constants E Analytical Theory for Near-Circular Orbits F Example of State Noise and Dynamic Model Compensation G Solution of the Linearized Equations of Motion H ECI and ECF Transformation

Point based animation of elastic, plastic and melting objects

Abstract: We present a method for modeling and animating a wide spectrum of volumetric objects, with material properties anywhere in the range from stiff elastic to highly plastic. Both the volume and the surface representation are point based, which allows arbitrarily large deviations form the original shape. In contrast to previous point based elasticity in computer graphics, our physical model is derived from continuum mechanics, which allows the specification of common material properties such as Young's Modulus and Poisson's Ratio.In each step, we compute the spatial derivatives of the discrete displacement field using a Moving Least Squares (MLS) procedure. From these derivatives we obtain strains, stresses and elastic forces at each simulated point. We demonstrate how to solve the equations of motion based on these forces, with both explicit and implicit integration schemes. In addition, we propose techniques for modeling and animating a point-sampled surface that dynamically adapts to deformations of the underlying volumetric model.

Geometrically Exact, Intrinsic Theory for Dynamics of Curved and Twisted Anisotropic Beams

Abstract: A formulation is presented for the nonlinear dynamics of initially curved and twisted anisotropic beams. When the applied loads at the ends of, and distributed along, the beam are independent of the deformation, neither displacement nor rotation variables appear: an intrinsic formulation. Like well-known special cases of these equations governing nonlinear dynamics of rigid bodies and nonlinear statics of beams, the complete set of intrinsic equations has a maximum degree of nonlinearity equal to two. Advantages of such a formulation are demonstrated with a simple example. When the initial curvature and twist are constant along the beam, two space-time conservation laws are shown to exist, one being a work-energy relation and the other a generalized impulse-momentum relation. These laws can be used, for example, as benchmarks to check the accuracy of any proposed solution, including time-marching and finite element schemes. The structure of the intrinsic equations suggests parallel approaches to spatial and temporal discretization. A particularly simple spatial discretization scheme is presented for the special case of the nonlinear static behavior of end-loaded beams that, by virtue of the Kirchhoff analogy, leads to a time-marching scheme for the dynamics of a pivoted rigid body in a gravity field. This time-marching scheme conserves both the angular momentum about a vertical line passing through the pivot and total mechanical energy, whereas the analogous spatial discretization scheme for the nonlinear static behavior of end-loaded beams satisfies analogous integrals of deformation along the beam span. Remarkably, a straightforward generalization of these discretization schemes is shown to satisfy both space-time conservation laws for the nonlinear dynamics of beams when the applied loads are constant within a space-time element.

Neurons compute internal models of the physical laws of motion

Abstract: A critical step in self-motion perception and spatial awareness is the integration of motion cues from multiple sensory organs that individually do not provide an accurate representation of the physical world. One of the best-studied sensory ambiguities is found in visual processing, and arises because of the inherent uncertainty in detecting the motion direction of an untextured contour moving within a small aperture1,2,3,4. A similar sensory ambiguity arises in identifying the actual motion associated with linear accelerations sensed by the otolith organs in the inner ear5,6. These internal linear accelerometers respond identically during translational motion (for example, running forward) and gravitational accelerations experienced as we reorient the head relative to gravity (that is, head tilt). Using new stimulus combinations, we identify here cerebellar and brainstem motion-sensitive neurons that compute a solution to the inertial motion detection problem. We show that the firing rates of these populations of neurons reflect the computations necessary to construct an internal model representation of the physical equations of motion.

The role of viscosity in the impulse diffraction field of elastic waves induced by the acoustic radiation force

Abstract: Several ultrasound-based techniques for the estimation of soft tissue elasticity are currently being investigated. Most of them study the medium response to dynamic excitations. Such responses are usually modeled in a purely elastic medium using a Green's function solution of the motion equation. However, elasticity by itself is not necessarily a discriminant parameter for malignancy diagnosis. Modeling viscous properties of tissues could also be of great interest for tumor characterization. We report in this paper an explicit derivation of the Green's function in a viscous and elastic medium taking into account shear, bulk, and coupling waves. From this theoretical calculation, 3D simulations of mechanical waves in viscoelastic soft tissues are presented. The relevance of the viscoelastic Green's function is validated by comparing simulations with experimental data. The experiments were conducted using the supersonic shear imaging (SSI) technique which dynamically and remotely excites tissues using acoustic radiation force. We show that transient shear waves generated with SSI are modeled very precisely by the Green's function formalism. The combined influences of out-of-plane diffraction, beam shape, and shear viscosity on the shape of transient waves are carefully studied as they represent a major issue in ultrasound-based viscoelasticity imaging techniques.

Finite element formulation of viscoelastic sandwich beams using fractional derivative operators

Abstract: This paper presents a finite element formulation for transient dynamic analysis of sandwich beams with embedded viscoelastic material using fractional derivative constitutive equations. The sandwich configuration is composed of a viscoelastic core (based on Timoshenko theory) sandwiched between elastic faces (based on Euler– Bernoulli assumptions). The viscoelastic model used to describe the behavior of the core is a four-parameter fractional derivative model. Concerning the parameter identification, a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Curve-fitting aspects are focused, showing a good agreement with experimental data. In order to implement the viscoelastic model into the finite element formulation, the Grunwald definition of the fractional operator is employed. To solve the equation of motion, a direct time integration method based on the implicit Newmark scheme is used. One of the particularities of the proposed algorithm lies in the storage of displacement history only, reducing considerably the numerical efforts related to the non-locality of fractional operators. After validations, numerical applications are presented in order to analyze truncation effects (fading memory phenomena) and solution convergence aspects.

Solitary Wave Dynamics in an External Potential

Abstract: We study the behavior of solitary-wave solutions of some generalized nonlinear Schrodinger equations with an external potential. The equations have the feature that in the absence of the external potential, they have solutions describing inertial motions of stable solitary waves. We consider solutions of the equations with a non-vanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that, over a large interval of time, they describe a solitary wave whose center of mass motion is a solution of Newton's equations of motion for a point particle in the given external potential, up to small corrections corresponding to radiation damping.

Large spin limits of AdS/CFT and generalized Landau-Lifshitz equations

Abstract: We consider AdS_5 x S^5 string states with several large angular momenta along AdS_5 and S^5 directions which are dual to single-trace Super-Yang-Mills (SYM) operators built out of chiral combinations of scalars and covariant derivatives. In particular, we focus on the SU(3) sector (with three spins in S^5) and the SL(2) sector (with one spin in AdS_5 and one in S^5), generalizing recent work hep-th/0311203 and hep-th/0403120 on the SU(2) sector with two spins in S^5. We show that, in the large spin limit and at the leading order in the effective coupling expansion, the string sigma model equations of motion reduce to matrix Landau-Lifshitz equations. We then demonstrate that the coherent-state expectation value of the one-loop SYM dilatation operator restricted to the corresponding sector of single trace operators is also effectively described by the same equations. This implies a universal leading order equivalence between string energies and SYM anomalous dimensions, as well as a matching of integrable structures. We also discuss the more general 5-spin sector and comment on SO(6) states dual to non-chiral scalar operators.

Aeroelastic stability analysis of wind turbines using an eigenvalue approach

Abstract: A design tool for performing aeroelastic stability analysis of wind turbines is presented in this paper. The method behind this tool is described in a general form, as independent of the particular aeroelastic modelling as possible. Here, the structure is modelled by a Finite beam Element Method, and the aerodynamic loads are modelled by the Blade Element Momentum method coupled with a Beddoes-Leishman type dynamic stall model in a state-space formulation. The linearization of the equations of motion is performed about a steady-state equilibrium, where the deterministic forcing of the turbine is neglected. To eliminate the periodic coefficients and avoid using the Floquet Theory, the multi-blade transformation is utilized. From the corresponding eigenvalue problem, the eigenvalues and eigenvectors can be computed at any operation condition to give the aeroelastic modal properties: Natural frequencies, damping and mode shapes. An example shows a good agreement between predicted and measured aeroelastic damping of a stall-regulated 600 kW turbine. Copyright © 2004 John Wiley & Sons, Ltd.

Tracking solutions in tachyon cosmology

Abstract: We perform a thorough phase-plane analysis of the flow defined by the equations of motion of a FRW Universe filled with a tachyonic fluid plus a barotropic one. The tachyon potential is assumed to be of inverse square form, thus allowing for a two-dimensional autonomous system of equations. The Friedmann constraint, combined with a convenient choice of coordinates, renders the physical state compact. We find the fixed-point solutions, and discuss whether or not they represent attractors. The way the two fluids contribute at late times to the fractional energy density depends on how fast the barotropic fluid redshifts. If it does it fast enough, the tachyonic fluid takes over at late times, but if the opposite happens, the situation will not be completely dominated by the barotropic fluid; instead there will be a residual non-negligible contribution from the tachyon subject to restrictions coming from nucleosynthesis.

A direct simulation method for particle‐fluid systems

Abstract: A coupled numerical method for the direct simulation of particle‐fluid systems is formulated and implemented. The Navier‐Stokes equations governing fluid flow are solved using the lattice Boltzmann method, while the equations of motion governing particles are solved with the discrete element method. Particle‐fluid coupling is realized through an immersed moving boundary condition. Particle forcing mechanisms represented in the model to at least the first‐order include static and dynamic fluid‐induced forces, and intergranular forces including particle collisions, static contacts, and cementation. The coupling scheme is validated through a comparison of simulation results with the analytical solution of cylindrical Couette flow. Simulation results for the fluid‐induced erosive failure of a cemented particulate constriction are presented to demonstrate the capability of the method.

Theory of spin-charge-coupled transport in a two-dimensional electron gas with Rashba spin-orbit interactions

Abstract: We use microscopic linear response theory to derive a set of equations that provide a complete description of coupled spin and charge diffusive transport in a two-dimensional electron gas (2DEG) with the Rashba spin-orbit (SO) interaction. These equations capture a number of interrelated effects including spin accumulation and diffusion, Dyakonov-Perel spin relaxation, magnetoelectric, and spin-galvanic effects. They can be used under very general circumstances to model transport experiments in 2DEG systems that involve either electrical or optical spin injection. We comment on the relationship between these equations and the exact spin and charge density operator equations of motion. As an example of the application of our equations, we consider a simple electrical spin injection experiment and show that a voltage will develop between two ferromagnetic contacts if a spin-polarized current is injected into a 2DEG, that depends on the relative magnetization orientation of the contacts. This voltage is present even when the separation between the contacts is larger than the spin diffusion length.

Multipole moments of isolated horizons

Abstract: To every axi-symmetric isolated horizon we associate two sets of numbers, Mn and Jn with n = 0, 1, 2, ..., representing its mass and angular momentum multipoles. They provide a diffeomorphism invariant characterization of the horizon geometry. Physically, they can be thought of as the 'source multipoles' of black holes in equilibrium. These structures have a variety of potential applications ranging from equations of motion of black holes and numerical relativity to quantum gravity.

Third post-Newtonian accurate generalized quasi-Keplerian parametrization for compact binaries in eccentric orbits

Abstract: We present Keplerian-type parametrization for the solution of third post-Newtonian (3PN) accurate equations of motion for two nonspinning compact objects moving in an eccentric orbit. The orbital elements of the parametrization are explicitly given in terms of the 3PN accurate conserved orbital energy and angular momentum in both Arnowitt-Deser-Misner--type and harmonic coordinates. Our representation will be required to construct post-Newtonian accurate ``ready to use'' search templates for the detection of gravitational waves from compact binaries in inspiralling eccentric orbits. Because of the presence of certain 3PN accurate gauge invariant orbital elements, the parametrization should be useful to analyze the compatibility of general relativistic numerical simulations involving compact binaries with the corresponding post-Newtonian descriptions. If required, the present parametrization will also be needed to compute post-Newtonian corrections to the currently employed ``timing formula'' for the radio observations of relativistic binary pulsars.

Random Vibrations with Impacts: A Review

Abstract: Analytical methods and specific results are presented for random vibrations of systems with lumped parameters and “classical” impacts whereby finite relations between impact/rebound velocities are imposed at the impact instants that are not known in advance but rather governed by the equations of motion. Emphasis is placed on the procedures using special piecewise-linear transformation of state variables that exclude the velocity jumps at impacts or makes them small if impact losses are present. In the former case, exact analyses for stationary probability densities of the response to white-noise excitation are possible, whereas the stochastic averaging method is applied in the latter case. Furthermore, the special case of an isochronous system permits a more profound response analysis, such as predicting the spectral density of the response or subharmonic response to narrow-band excitation. The method of direct energy balance is also illustrated, based on direct application of the stochastic differential equation calculus between impacts. Certain two-degree-of-freedom impacting systems are considered, with application to moored systems, as used in ocean engineering.

Fundamentals of a Vector Form Intrinsic Finite Element: Part I. Basic Procedure and A Plane Frame Element

Abstract: In a series of three articles, fundamentals of a vector form intrinsic finite element procedure (VFIFE) are summarized. The procedure is designed to calculate motions of a system of rigid and deformable bodies. The motion may include large rigid body motions and large geometrical changes. Newton's law, or a work principle, for particle is assumed to derive the governing equations of motion. They are obtained by using a set of deformation coordinates for the description of kinematics. A convected material frame approach is proposed to handle very large deformations. Numerical results are calculated by using an explicit algorithm. In the first article, using the plane frame element as an example, basic procedures are described. In the accompanied articles, plane solid elements, convected material frame procedures and numerical results of patch tests are given.

Casimir-Polder forces: A nonperturbative approach

Abstract: Within the frame of macroscopic QED in linear, causal media, we study the radiation force of Casimir-Polder type acting on an atom which is positioned near dispersing and absorbing magnetodielectric bodies and initially prepared in an arbitrary electronic state. It is shown that minimal and multipolar coupling lead to essentially the same lowest-order perturbative result for the force acting on an atom in an energy eigenstate. To go beyond perturbation theory, the calculations are based on the exact center-of-mass equation of motion. For a nondriven atom in the weak-coupling regime, the force as a function of time is a superposition of force components that are related to the electronic density matrix elements at a chosen time. Even the force component associated with the ground state is not derivable from a potential in the ususal way, because of the position dependence of the atomic polarizability. Further, when the atom is initially prepared in a coherent superposition of energy eigenstates, then temporally oscillating force components are observed, which are due to the interaction of the atom with both electric and magnetic fields.

Equations of motion in a non-integer-dimensional space

Abstract: Equations of motion are derived for a fractional dimensional system of n-spatial coordinates to be used as an effective description of anisotropic and confined systems. An existing measure theoretic approach is extended to multiple variables and different degrees of confinement in orthogonal directions and comparisons are made with the analytic continuation of Gaussian integrals. This is applied to the variational principle, and equations of motion for a field described by a Lagrange density are found. A specific example is looked at in Schrodinger wave mechanics, particularly in three-coordinate systems.

A Reduced-order Model for Electrically Actuated Microplates

Abstract: We present a reduced-order model for electrically actuated microplate-based MEMS. The model accounts for the electric force nonlinearity and the mid-plane stretching of the plate. The linear undamped vibration modes are found numerically using the hierarchical finite-element method. These mode shapes are used in a Galerkin approximation to reduce the partial-differential equations of motion and associated boundary conditions into a finite-dimensional system of nonlinearly coupled second-order ordinary-differential equations. The model is validated by comparing its results with those obtained experimentally and those obtained by solving the distributed-parameter system. The model is used to calculate the deflection of the microplate under dc voltages and study the pull-in phenomenon. The natural frequencies and mode shapes around these deflected positions of the microplate are calculated by solving the linear eigenvalue problem. The effects of various design parameters on both the static and dynamic characteristics of microplates are studied. The reduced-order model provides an effective and accurate design tool, useful in design optimization and determination of the stable operation range of MEMS devices.

Field quantization in inhomogeneous absorptive dielectrics

Abstract: The quantization of the electromagnetic field in a three-dimensional inhomogeneous dielectric medium with losses is carried out in the framework of a damped-polariton model with an arbitrary spatial dependence of its parameters. The equations of motion for the canonical variables are solved explicitly by means of Laplace transformations for both positive and negative time. The dielectric susceptibility and the quantum noise-current density are identified in terms of the dynamical variables and parameters of the model. The operators that diagonalize the Hamiltonian are found as linear combinations of the canonical variables, with coefficients depending on the electric susceptibility and the dielectric Green function. The complete time dependence of the electromagnetic field and of the dielectric polarization is determined. Our results provide a microscopic justification of the phenomenological quantization scheme for the electromagnetic field in inhomogeneous dielectrics.