New Mathematical Models for Particle Flow Dynamics

Abstract

A new class of integro-partial differential equation models is derived for the prediction of granular flow dynamics. These models are obtained using a novel limiting averaging method (inspired by techniques employed in the derivation of infinite-dimensional dynamical systems models) on the Newtonian equations of motion of a many-particle system incorporating widely used inelastic particle-particle force formulas. By using Taylor series expansions, these models can be approximated by a system of partial differential equations of the Navier-Stokes type. The exact or approximate governing equations obtained are far from simple, but they are less complicated than most of the continuum models now being used to predict particle flow behavior. Solutions of the new models for granular flows down inclined planes and in vibrating beds are compared with known experimental and analytical results and good agreement is obtained.     

Triangular Newton Equations with Maximal Number of Integrals of Motion

Abstract

We study two-dimensional triangular systems of Newton equations (acceleration = velocity-independent force) admitting three functionally independent quadratic integrals of motion. The main idea is to exploit the fact that the first component M ₁(q ₁) of a triangular force depends on one variable only. By using the existence of extra integrals of motion we reduce the problem to solving a simultaneous system of three linear ordinary differential equations with nonconstant coefficients for M ₁(q ₁). With the help of computer algebra we have found and solved these ordinary differential equations in all cases. A complete list of superintegrable triangular equations in two dimensions is been given. Most of these equations were not known before.     

Changes in the Effective Parameters of Averaged Motion in Nonlinear Systems Subject to Noise

We discuss how the effective parameters characterising averaged motion in nonlinear systems are affected by noise (random fluctuations). In this approach to stochastic dynamics, the stochastic system is replaced by its deterministic equivalent but with noise-dependent parameters. We show that it can help to resolve certain paradoxes and that it has a utility extending far beyond its usual application in passing from the microscopic equations of motion to the macroscopic ones. As illustrative examples, we consider the diode-capacitor circuit, a Brownian ratchet, and a generic stochastic resonance system. In the latter two cases we calculate for the first time their effective parameters of averaged motion as functions of noise intensity. We speculate that many other stochastic problems can be treated in a similar way. PACS: 05.10.Gg, 05.40.-a, 05.40.Jc     

Critical dynamics of slightly disordered spin systems

A new approach to the analysis of magnetic dipole motion in external magnetic field and fields generated by neighboring magnetic dipoles is suggested, and original general kinetic equations for the dipole density are derived. Special cases of these general equations are the Bloch, Redfield, and Provotorov equations, which are widely used in NMR theory. A comparison between NMR spectra calculated with the new theory and published experimental data also shows good agreement in regions to which the equations listed above do not apply. Zh. éksp. Teor. Fiz. 113, 967–980 (March 1998)     

Dynamical coupling in the differential equations approach to atom-diatom exchange reactions

The atomic exchange reaction A + BC → AB + C is investigated quantum mechanically employing a coupled differential equations approach. The relative motion in reactant and product channels is described in the common coordinate R ₃ (the AC nuclear separation) and is developed in three-dimensional space. The total wave functions of the system are expressed as a superposition of valence bond electronic states of the initial (A, BC) and final (AB, C) configurations, with the coefficients describing the relative and internal (vibrational, rotational) nuclear motions. Choosing convenient trial functions with the appropriate boundary conditions and using the Kohn variational principle, a set of differential (rather than the usual integro-differential) equations is obtained for the relative motion wave functions in R ₃. The potential matrix elements turn out to be dynamical in that they depend on the initial k ₁ and final k ₂ wave vectors. Two-state coupled channel calculations of the differential and integral cross sections for the isotopic species D + H₂, H + H₂ and D + D₂ are presented for collision energies up to 0·8 eV.     

Dynamics of systems of extended bodies in monad representation

A universal form proposed earlier by the author for the equations of motion is used to find a monad representation of the equations of motion of a system of N extended bodies in general relativity theory. An explicit form of the equations of motion in a fixed chronometric reference system is presented. It is shown that it differs from the known coordinate motion equations obtained by the Fock method.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 72–78, January, 1995.     

A Non-Commutative Approach to Ordinary Differential Equations

We adapt ideas coming from Quantum Mechanics to develop a non-commutative strategy for the analysis of some systems of ordinary differential equations. We show that the solution of such a system can be described by an unbounded, self-adjoint and densely defined operator H which we call, in analogy with Quantum Mechanics, the Hamiltonian of the system. We discuss the role of H in the analysis of the integrals of motion of the system. Finally, we apply this approach to several examples. Pacs Numbers: 02.30.Hq, 03.65.-w, 03.65.Db     

Dark Equations

Abstract

Observing the Universe, astronomers have concluded that the motion of stars can not be accounted for unless one assumes that most of the mass in the Universe is carried on by a “dark matter”, so far impervious to all attempts at being detected. There is now a similar concept of “dark energy”. I shall discuss a different subject, “dark equations”. These have never indicated that they influence anything or even exist, but if one supposes that they do exist, one can systematically discover them and study their properties, some of which turn out to be strange and others mysterious. These equations are similar in spirit to what one gets when linearizing a given system, or studies how an external linear wave interacts with a particular solution of a given system. We define and study linear extensions of dynamical systems in general, and integrable and Hamiltonian systems in particular. Systems discussed include the KdV and mKdV equations and the associated Miura maps, the Burgers hierarchy and the associated Hopf–Cole transformations, long wave equations, the Benney hierarchy, and the KP hierarchy.     

An ODE Model of the Motion of Pelagic Fish

A system of ordinary differential equations (ODEs) is derived from a discrete system of Vicsek, Czirók et al. [Phys. Rev. Lett. 75(6):1226–1229, 1995], describing the motion of a school of fish. Classes of linear and stationary solutions of the ODEs are found and their stability explored using equivariant bifurcation theory. The existence of periodic and toroidal solutions is also proven under deterministic perturbations and structurally stable heteroclinic connections are found. Applications of the model to the migration of the capelin, a pelagic fish that undertakes an extensive migration in the North Atlantic, are discussed and simulation of the ODEs presented.     

Symmetry constraint of MKdV equations by binary nonlinearization

Abstract

A symmetry constraint for the MKdV integrable hierarchy is presented by binary nonlinearization. The spatial and temporal parts of the Lax pairs and adjoint Lax pairs of MKdV equations are all constrained as finite-dimensional Liouville integrable Hamiltonian systems, whose integrals of motion are explicitly given. In terms of the proposed symmetry constraint, MKdV equations are decomposed into two finite-dimensional Liouville integrable constrained systems and thus a kind of separation of variables for MKdV equations is established.     

Theory of nonlinear directional couplers

A method is proposed for obtaining exact analytical solutions for the system of nonlinear equations for the propagation of electromagnetic waves in directional couplers with an arbitrary nonlinearity in the propagation constant. The resulting solutions can be used to determine the operating characteristics of nonlinear directional couplers. Zh. Tekh. Fiz. 69, 69–71 (August 1999)     

Constant of Motion,Lagrangian and Hamiltonian of the Gravitational Attraction of Two Bodies with Variable Mass

The Lagrangian, the Hamiltonian and the constant of motion of the gravitational attraction of two bodies when one of them has variable mass is considered. The relative and center of mass coordinates are not separated, and choosing the reference system in the body with much higher mass, it is possible to reduce the system of equations to 1-D problem. Then, a constant of motion, the Lagrangian, and the Hamiltonian are obtained. The trajectories found in the space position-velocity,(x,v), are qualitatively different from those on the space position-momentum,(x,p). PACS numbers: 03.20.+i     

Study of nonlinear system stability using eigenvalue analysis: Gyroscopic motion

General computational multibody system (MBS) algorithms allow for the linearization of the highly nonlinear equations of motion at different points in time in order to obtain the eigenvalue solution. This eigenvalue solution of the linearized equations is often used to shed light on the system stability at different configurations that correspond to different time points. Different MBS algorithms, however, employ different sets of orientation coordinates, such as Euler angles and Euler parameters, which lead to different forms of the dynamic equations of motion. As a consequence, the forms of the linearized equations and the eigenvalue solution obtained strongly depend on the set of orientation coordinates used. This paper addresses this fundamental issue by examining the effect of the use of different orientation parameters on the linearized equations of a gyroscope. The nonlinear equations of motion of the gyroscope are formulated using two different sets of orientation parameters: Euler angles and Euler parameters. In order to obtain a set of linearized equations that can be used to define the eigenvalue solution, the algebraic equations that describe the MBS constraints are systematically eliminated leading to a nonlinear form of the equations of motion expressed in terms of the system degrees of freedom. Because in MBS applications the generalized forces can be highly nonlinear and can depend on the velocities, a state space formulation is used to solve the eigenvalue problem. It is shown in this paper that the independent state equations formulated using Euler angles and Euler parameters lead to different eigenvalue solutions. This solution is also different from the solution obtained using a form of the Newton-Euler matrix equation expressed in terms of the angular accelerations and angular velocities. A time-domain solution of the linearized equations is also presented in order to compare between the solutions obtained using two different sets of orientation parameters and also to shed light on the important issue of using the eigenvalue analysis in the study of MBS stability. The validity of using the eigenvalue analysis based on the linearization of the nonlinear equations of motion in the study of the stability of railroad vehicle systems, which have known critical speeds, is examined. It is shown that such an eigenvalue analysis can lead to wrong conclusions regarding the stability of nonlinear systems.     

A gas of Brownian particles in statistical dynamics

We consider a large number of particles on a one-dimensional latticel Z in interaction with a heat particle; the latter is located on the bond linking the position of the particle to the point to which it jumps. The energy of a single particle is given by a potentialV(x), x∈Z. In the continuum limit, the classical version leads to Brownian motion with drift. A quantum version leads to a local drift velocity which is independent of the applied force. Both these models obey Einstein's relation between drift, diffusion, and applied force. The system obeys the first and second laws of thermodynamics, with the time evolution given by a pair of coupled non linear heat equations, one for the density of the Brownian particles and one for the heat occupation number; the equation for a tagged Brownian particle can be written as a stochastic differential equation.     

On the definition of the natural frequency of oscillations in nonlinear large rotation problems

Computational multibody system algorithms allow for performing eigenvalue analysis at different time points during the simulation to study the system stability. The nonlinear equations of motion are linearized at these time points, and the resulting linear equations are used to determine the eigenvalues and eigenvectors of the system. In the case of linear systems, the system eigenvalues remain the same under a constant coordinate transformation; and zero eigenvalues are always associated with rigid body modes, while nonzero eigenvalues are associated with non-rigid body motion. These results, however, cannot in general be applied to nonlinear multibody systems as demonstrated in this paper. Different sets of large rotation parameters lead to different forms of the nonlinear and linearized equations of motion, making it necessary to have a correct interpretation of the obtained eigenvalue solution. As shown in this investigation, the frequencies associated with different sets of orientation parameters can differ significantly, and rigid body motion can be associated with non-zero oscillation frequencies, depending on the coordinates used. In order to demonstrate this fact, the multibody system motion equations associated with the system degrees of freedom are presented and linearized. The resulting linear equations are used to define an eigevalue problem using the state space representation in order to account for general damping that characterizes multibody system applications. In order to demonstrate the significant differences between the eigenvalue solutions associated with two different sets of orientation parameters, a simple rotating disk example is considered in this study. The equations of motion of this simple example are formulated using Euler angles, Euler parameters and Rodriguez parameters. The results presented in this study demonstrate that the frequencies obtained using computational multibody system algorithms should not in general be interpreted as the system natural frequencies, but as the frequencies of the oscillations of the coordinates used to describe the motion of the system.     

Relativistic particles in a quadrupole waveguide

A solution is obtained for the equations of motion of a particle in the field of a traveling electromagnetic wave propagating between two pairs of metal surfaces of hyperbolic shape. The motion of the particle on excitation of the waveguide by monochromatic and pulsed voltage sources is considered. The possibility of applying a quadrupole waveguide to beam focusing and the separation of particles according to their specific charge and longitudinal velocity is investigated. Zh. Tekh. Fiz. 67, 98–102 (July 1997)     

A generally covariant formulation of classical electrodynamics for charges of finite extension

A generally covariant formulation of classical electrodynamics for charges of finite extension has been developed. The charges are required to maintain a prescribed “rigid” shape throughout the course of their motion. An action principle is formulated for the coupled system consisting of the charges plus electromagnetic and gravitational fields. The action principle yields a complicated set of coupled integro-differential equations for the motion and fields. A perturbation expansion in powers of the size of the charge distribution is obtained. In the limit that the size of the charge tends to zero, only a few kinematical features survive in the equations of motion. The resulting equations of motion have the DeWitt-Brehme [Ann. Phys.9 (1969), 220] form, but with additional curvature-coupling terms which were omitted by them owing to an algebraic error.     

Control of quasiparticle self-fluctuations in a CuCl crystal

A theory is developed for regular and chaotic self fluctuations in crystal CuCl for a ring resonator geometry. A system of nonlinear differential equations is derived for the dynamic evolution of coherent excitons, photons, and biexcitons. It is shown that, in the unstable portions of the optical bistability curves, nonlinear periodic and chaotic self fluctuations can develop with the creation of limit cycles and strange attractors in the phase space of the system. A computer simulation is used to determine the parameters for which reliable switching takes place in the system and the parameter ranges are found within which the system undergoes a transition from strange attractor to limit cycle. The possibility of experimentally observing the phenomena studied here is discussed. Fiz. Tverd. Tela (St. Petersburg) 41, 1939–1943 (November 1999)     

Electron-lattice kinetics of metals heated by ultrashort laser pulses

We propose a kinetic model of transient nonequilibrium phenomena in metals exposed to ultrashort laser pulses when heated electrons affect the lattice through direct electron-phonon interaction. This model describes the destruction of a metal under intense laser pumping. We derive the system of equations for the metal, which consists of hot electrons and a cold lattice. Hot electrons are described with the help of the Boltzmann equation and equation of thermoconductivity. We use the equations of motion for lattice displacements with the electron force included. The lattice deformation is estimated immediately after the laser pulse up to the time of electron temperature relaxation. An estimate shows that the ablation regime can be achieved. Zh. éksp. Teor. Fiz. 115, 149–157 (January 1999) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

A unified quantum theory of mechanics and thermodynamics. Part IIb. Stable equilibrium states

Part IIb presents some of the most important theorems for stable equilibrium states that can be deduced from the four postulates of the unified theory presented in Part I. It is shown for the first time that the canonical and grand canonical distributions are the only distributions that are stable. Moreover, it is shown that reversible adiabatic processes exist which cannot be described by the dynamical equation of quantum mechanics. A number of conditions are discussed that must be satisfied by the general equation of motion which is yet to be discovered.Part I of this paper appeared inFound. Phys. 6(1), 15 (1976). Part IIa appeared inFound. Phys. 6(2), 127 (1976). The numbering of the sections, equations, and references in this part of the paper continues from those in Part IIa.