First-order Equations Of Motion Research Articles

Spacetime-bridge solutions in vacuum gravity

Spacetimes, which are representations of a bridge-like geometry in gravity theory, are constructed as vacuum solutions to the first order equations of motion. Each such configuration consists of two copies of an asymptotically flat sheet, connected by a bridge of finite extension where tetrad is noninvertible. These solutions can be classified into static and non-static spacetimes. The associated SO(3,1) invariant fields, namely the metric, affine connection and field-strength tensor, are all continuous across the hypersurfaces connecting the invertible and noninvertible phases of tetrad and are finite everywhere. These regular spacetime-bridge solutions do not have any analogue in Einsteinian gravity in vacuum.

Tensor Fields on Self-Dual Warped AdS3 Background

Considering rank s fields obey first order equation of motion, we study the dynamics of such fields in a 3-dimensional self-dual space-like warped AdS3 black hole background. We obtain a Klein–Gordon-like equation for tensor fields. By using gauge constraint and traceless condition, we will find the exact solutions of the equations of motion. Then, we will compute the quasi-normal modes by imposing appropriate boundary conditions at horizon and infinity.

First-order form, Lagrange’s form, and Gibbs–Appell’s form of Kane’s equations

This paper proposes that Kane’s equations for a simple nonholonomic system are the first-order form of generalized speeds. When the first-order form of Kane’s equations is put in matrix form, the element of the mass matrix is identical to the inertia coefficient. Because the orthogonal set of partial velocities will decouple the first-order equations, one can use the orthogonal criterion to generate efficient equations of motion. With the presented first-order form, Kane’s equations are different from Maggi’s or Gibbs–Appell’s equations. Moreover, in order to clarify the relationship between Kane’s approach and classical approaches, we start from Kane’s equations and introduce kinetic energy or acceleration energy functions to derive Lagrange’s or Gibbs–Appell’s forms of Kane’s equations for the system. We found that the Lagrange’s forms of Kane’s equations can be used to solve nonholonomic systems without introducing Lagrangian multipliers. At last, the first-order form, Lagrange’s forms, and Gibbs–Appell’s forms of Kane’s equations are, respectively, used to depict the derivation of first-order equations of motion for a rolling coin.

Construction of Lagrangian and Hamiltonian structures starting from one constant of motion

The problem of the construction of Lagrangian and Hamiltonian structures starting from two first-order equations of motion is presented. This approach requires the knowledge of one (time independent) constant of motion for the dynamical system only. The Hamiltonian and Lagrangian structures are constructed, the Hamilton–Jacobi equation is then written and solved, and the second (time dependent) constant of the motion for the problem is explicitly exhibited.

Correlated electron dynamics with time-dependent quantum Monte Carlo: Three-dimensional helium

Here the recently proposed time-dependent quantum Monte Carlo method is applied to three dimensional para- and ortho-helium atoms subjected to an external electromagnetic field with amplitude sufficient to cause significant ionization. By solving concurrently sets of up to 20,000 coupled 3D time-dependent Schrödinger equations for the guide waves and corresponding sets of first order equations of motion for the Monte Carlo walkers we obtain ground state energies in close agreement with the exact values. The combined use of spherical coordinates and B-splines along the radial coordinate proves to be especially accurate and efficient for such calculations. Our results for the dipole response and the ionization of an atom with un-correlated electrons are in good agreement with the predictions of the conventional time-dependent Hartree-Fock method while the calculations with correlated electrons show enhanced ionization that is due to the electron-electron repulsion.

Higher order variational origin of the Dixonʼs system and its relation to the quasi-classical ‘Zitterbewegung’ in General Relativity

We show how the Dixonʼs system of first order equations of motion for the particle with inner dipole structure together with the side Mathisson constraint follows from rather general construction of the ‘Hamilton system’ developed by Weyssenhoff, Rund and Grässer to describe the phase space counterpart of the evolution under the ordinary Euler–Poisson differential equation of the parameter-invariant variational problem with second derivatives. One concrete expression of the ‘Hamilton function’ leads to the General Relativistic form of the fourth order equation of motion known to describe the quasi-classical ‘quiver’ particle in Special Relativity. The corresponding Lagrange function including velocity and acceleration coincides in the flat space of Special Relativity with the one considered by Bopp in an attempt to give an approximate variational formulation of the motion of self-radiating electron, when expressed in terms of geometric quantities.

Exact null tachyons from renormalization group flows

We construct exact two-dimensional conformal field theories, corresponding to closed string tachyon and metric profiles invariant under shifts in a null coordinate, which can be constructed from any two-dimensional renormalization group flow. These solutions satisfy first order equations of motion in the conjugate null coordinate. The direction along which the tachyon varies is identified precisely with the world sheet scale, and the tachyon equations of motion are the renormalization group flow equations.

Evaluation of the reduction of dynamical coupling for robot manipulators

The problem of evaluation of the reduction of dynamical coupling in rigid serial manipulators is presented in the current paper. The proposed strategy is based on first-order equations of motion expressed in terms of normalized generalized velocity components, which result from decomposition of the manipulator mass matrix. The principal activity of the study relies on an algorithm in which some evaluation criteria as well as some indexes are used. The presented approach is important, mainly for manipulators with a direct drive arm at their design stage. Simulation results on a three-degree-of-freedom (3-DOF) spatial manipulator are given in order to illustrate the offered method.

Investigation of manipulator dynamics with a viscous damping model

In this article, some remarks concerning dynamics investigation of a manipulator described using first-order equations of motion with a viscous damping model is conducted. The viscous damping model arises from the Rayleigh dissipation potential and decomposition of the manipulator mass matrix. As a result, it takes into account both kinematic and mechanical parameters of the system. Moreover, the use of first-order equations of motion leads to obtaining some interesting insight into the manipulator dynamics. The proposed approach was tested on a three-degrees-of-freedom, three-dimensional Yasukawa-like robot.

Quantization of theories with non-Lagrangian equations of motion

We present an approach to the canonical quantization of systems with non-Lagrangian equations of motion. We first construct an action principle for equivalent first-order equations of motion. Hamiltonization and canonical quantization of the constructed Lagrangian theory is a nontrivial problem, since this theory involves time-dependent constraints. We adapt the general approach to hamiltonization and canonical quantization for such theories (Gitman, Tyutin, 1990) to the case under consideration. There exists an ambiguity (not reduced to a total time derivative) in associating a Lagrange function with a given set of equations. We give a complete description of this ambiguity. It is remarkable that the quantization scheme developed in the case under consideration provides arguments in favor of fixing this ambiguity. Finally, as an example, we consider the canonical quantization of a general quadratic theory. Bibliography: 23 titles.

Conserved quantities in the noncommutative principal chiral model with Wess–Zumino term

We construct noncommutative extension of U(N) principal chiral model with Wess-Zumino term and obtain an infinite set of local and non-local conserved quantities for the model using iterative procedure of Brezin {\it et.al} \cite{BIZZ}. We also present the equivalent description as Lax formalism of the model. We expand the fields perturbatively and derive zeroth- and first-order equations of motion, zero-curvature condition, iteration method, Lax formalism, local and non-local conserved quantities.

Sliding mode control of manipulators using first-order equations of motion with diagonal mass matrix

This paper presents a proposition of a sliding mode controller for a rigid manipulator expressed in terms of the generalized velocity components (GVC) vector. Introduction of GVC (Trans. ASME J. Appl. Mech. 62 (1995) 216) together with generalized positions leads to two first-order decoupled equations of motion instead of a single second-order equation. It is shown that the new controller, stable in the sense of Lyapunov, has different properties and can, according to Slotine and Li (Int. J. Robotics Res. 6 (1987) 49), give better performance than the classical sliding mode controller. Both control algorithms were tested on a 3 d.o.f., 3- D Yasukawa-like robot.

자유-자유보의 동적해석에 대한 섭동법의 적용

This paper is concerned with the application of perturbation method to the dynamic analysis of free-free beam. In general, the rigid-body motions and elastic vibrations are analyzed separately. However, the rigid-body motions cause vibrations and elastic vibrations also affect rigid-body motions in turn, which indicates that the rigid-body motions and elastic vibrations are coupled in nature. The resulting equations of motion are hybrid and nonlinear. We can discretize the equations of motion by means of admissible functions but still we have to cope with nonlinear equations. In this paper, we propose the use of perturbation method to the coupled equations of motion. The resulting equations consist of zero-order equations of motion which depict the rigid-body motions and first-order equations of motion which depict the perturbed rigid-body motions and elastic vibrations. Numerical results show the efficacy of the proposed method.

섭동법을 이용한 부유 한성체의 동역학 해석

This paper is concerned with the application of perturbation method to the dynamic analysis of floating flexible body. In dealing with the dynamics of free-floating body, the rigid-body motions and elastic vibrations are analyzed separately. However, the rigid-body motions cause vibrations and elastic vibrations also affect rigid-body motions in turn, which indicates that the rigid-body motions and elastic vibrations are coupled in nature. The resulting equations of motion are hybrid and nonlinear. We can discretize the equations of motion by means of admissible functions but still we have to cope with nonlinear equations. In the previous paper, we proposed the use of perturbation method to the coupled equations of motion and derived zero-order and first-order equations of motion. The derivation process was lengthy and tedious. Hence, in this paper, we propose a new approach to the same problem by applying the perturbation method to the Lagrange's equations, thus providing a systematic approach to the addressed problem. Theoretical derivations show the efficacy of the proposed method.

First-Order Equations of Motion in the Supersymmetric Yang–Mills Theory with a Scalar Multiplet

We propose a system of first-order equations of motion all solutions of which are solutions of a system of second-order equations of motion for the supersymmetric Yang–Mills theory with a scalar multiplet. We find N = 1 transformations under which the systems of first- and second-order equations of motion are invariant.

Effects of bath relaxation on dissipative two-state dynamics

A formal solution to the two-state Liouville equations is used to derive quantum equations of motion for dissipative two-state systems without making the assumption of a harmonic bath. The first-order equation of motion thus obtained is equivalent to the noninteracting blip approximation and can be systematically improved by introducing high-order cumulants. The second-order equation of motion incorporates effects of bath relaxation on two-state dynamics and leads to an effective nonadiabatic rate expression, which in the classical limit reduces to the well-known electron transfer rate formula. Numerical results with an Ohmic bath show saturation at large coupling constants due to the rate-limiting effect of relatively slow bath relaxation, and a comparison with classical calculations demonstrates larger rate constants at low temperature when quantum coherence is taken into account.

Two derivations of the Feynman formulae for the fields of a moving charge

The Feynman expressions for the electric and magnetic fields of a charge moving on an arbitrary trajectory are found by two simple methods. The first involves a direct integration of Maxwell's time-dependent equations. The second is based on the formalism of quantum optics which uses the first order equations of motion for creation and annihilation operators. As an illustration of the simplicity of these methods, the fields of an oscillating electric dipole are obtained.

Active control of waves in a cochlear model with subpartitions.

Multiscale asymptotic methods developed previously to study macromechanical wave propagation in cochlear models are generalized here to include active control of a cochlear partition having three subpartitions, the basilar membrane, the reticular lamina, and the tectorial membrane. Activation of outer hair cells by stereocilia displacement and/or by lateral wall stretching result in a frequency-dependent force acting between the reticular lamina and basilar membrane. Wavelength-dependent fluid loads are estimated by using the unsteady Stokes' equations, except in the narrow gap between the tectorial membrane and reticular lamina, where lubrication theory is appropriate. The local wavenumber and subpartition amplitude ratios are determined from the zeroth order equations of motion. A solvability relation for the first order equations of motion determines the subpartition amplitudes. The main findings are as follows: The reticular lamina and tectorial membrane move in unison with essentially no squeezing of the gap; an active force level consistent with measurements on isolated outer hair cells can provide a 35-dB amplification and sharpening of subpartition waveforms by delaying dissipation and allowing a greater structural resonance to occur before the wave is cut off; however, previously postulated activity mechanisms for single partition models cannot achieve sharp enough tuning in subpartitioned models.

Further remarks on the Hamiltonian of quantum optics

The equations of motion and the class of Hamiltonians for electrodynamics, both classical and quantal, are considered for the interaction of radiation with bound charges within the electric-dipole approximation. It is emphasized that, despite the uniqueness of the equations of motion for physical variables, the form of the Hamiltonian is not unique and that a physical system is associated with an equivalence class of Hamiltonians. Two of the members of this class are the minimal-coupling and multipolar forms extensively used in quantum optics. It is shown that although different choices of Hamiltonians lead to different first-order equations of motion that connect canonical coordinates and momenta, they lead to the same second-order equations of motion for the physical variables and physical electromagnetic fields. In the quantum theory of electrodynamics the role played by the choice of representation for the operators and for the states is analyzed with care. The choice of representation leads to a freedom additional to that provided by canonical transformations in classical electrodyamics, although the two lead to the same operator equations of motion.

Bogomolny equations and dimensional reduction

Abstract A variant of the bosonic sector of N = 4 Yang-Mills theory is constructed which admits Bogomolny equations (i.e. first order equations of motion which minimize the action), but which does not, unlike similar cases hitherto known, posses a straightforward interpretation as a pure Yang-Mills field in extra dimensions.