Showing papers on "Equations of motion published in 1971"

Dynamics of the Collinear H+H2 Reaction. I. Probability Density and Flux

Abstract: The time evolution of the collinear H+H2 reaction as given by classical mechanics and by time‐dependent quantum mechanics has been studied. The calculations employed the Porter–Karplus potential surface. The relevant equations of motion were solved to high accuracy by direct numerical integration. The evolution of the quantal probability density in the interaction region of the potential surface is shown in a series of perspective plots. Classical mechanics gives an amazingly good description of the probability density and flux patterns during most of the reaction; however, the classical and quantal descriptions begin to diverge near the end of the reaction. Essentially, the classical reaction terminates before the quantal reaction. The dynamic behavior of the reaction is hydrodynamically turbulent, as shown by transient whirlpool formation on the inside of the reaction path. All results reported in this paper are for one average system energy, namely, 0.65 eV (initial average translational energy = 0.38 eV).

Classical S Matrix for Rotational Excitation; Quenching of Quantum Effects in Molecular Collisions

Abstract: A previously developed theory in which exact solutions of the classical equations of motion for a complex collision system (i.e., numerically computed trajectories) can be used to generate the classical limit of the quantum mechanical S matrix (the “classical S matrix”) for the scattering process is applied to rigid rotor–atom collisions (rotational excitation). Comparison with essentially exact quantum results shows that transition probabilities (the square modulus of an S‐matrix element) between individual quantum states are given reasonably accurately by classical dynamics provided the interference terms are properly accounted for; a strictly classical approach (neglect of interference) gives poor agreement with the quantum values. For averaged collision properties, however, it is found that interference and tunneling effects are rapidly quenched. The linear atom–diatom system (vibrational excitation) and the rigid rotor–atom system are both investigated with regard to this question, namely, how much averaging is necessary to quench these quantum effects. Results indicate that even summation over a few quantum states is often sufficient to make a completely classical treatment appropriate.

On the Mathematical Structure of a Large Class of Physical Theories

Abstract: : Many physical theories show formal similarities due to the existence of a common mathematical structure. This structure is independent of the physical contents of the theory and can be found in classical, relativistic and quantum theories; for discrete and continuous systems. In particular many field theories of different tensorial order exhibit such a structure. Traditionally the study of similar structures is the subject of the mathematical field theory. Its starting point is the existence of an action principle from which field equations are deduced. Field theory is particularly suitable for fundamental theories as electromagnestism, gravitation and quantum theories. The approach in the paper is based on a different point of view: the author ascertains the existence of a decomposition of the fundamental equation into three sets of equations. Typically these are balance, definition and constitutive equations.

Elements of Hamiltonian Mechanics

Abstract: (partial) Newtonian mechanics. The hagrangian equations of motion. Small vibrations. Dynamics of rigid bodies. The canonical equations of motion. Hamilton-Jacobi theory. Perturbation theory. Continuous systems.

The equilibrium topography of sputtered amorphous solids II

Abstract: The sputtering of an amorphous solid is considered analytically and the equations of motion of the changing surface topography derived.

Brownian motion of a nonlinear oscillator

Abstract: Starting with the Langevin equation for a nonlinear oscillator (the “Duffing oscillator”) undergoing ordinary Brownian motion, we derive linear transport laws for the motion of the average position and velocity of the oscillator. The resulting linear equations are valid for only small deviations of average values from thermal equilibrium. They contain a renormalized oscillator frequency and a renormalized and non-Markovian friction coefficient, both depending on the nonlinear part of the original equation of motion. Numerical computations of the position correlation function and its spectral density are presented. The spectral density compares favorably with experimental results obtained by Morton using an analog computer method.

Wave Propagation in Nematic Liquid Crystals

Abstract: Basic laws of motion of micropolar continuum are presented, and the adequacy of applying micropolar theory to liquid crystals is indicated. A set of constitutive equations is derived for nematic liquid crystals. Wave propagation problems are solved, and it is shown that the theoretical analysis is in good agreement with the experimental data, which indicates the isotropy of the phase velocity of the longitudinal wave and the anisotropy of the damping coefficient. The coupling, although small, is shown to exist between longitudinal and rotational waves.

Conservation Laws for Gauge-Variant Lagrangians in Classical Mechanics

Abstract: When a physical system has some symmetry properties, it is described by equations of motion invariant under the corresponding transformation group. Its Lagrangian however need not be invariant and may be “gauge-variant,” that is, vary by the addition of a total time derivative. A slightly generalized form of Noether's theorem nevertheless exists in such cases, still leading to conservation laws. The importance of considering such noninvariant Lagrangians and the associated conservation laws is illustrated by several examples: energy conservation, Galilean invariance, dynamical symmetries (harmonic oscillator and Kepler's problem), motion in a uniform electric field.

Theoretical frameworks for testing relativistic gravity. III - Conservation laws, Lorentz invariance, and values of the PPN parameters

Abstract: Lorentz invariant theory for relativistic gravity testing, deriving conservation laws and parameter constraints from parametrized post-Newtonian equations of motion

Particles with Spin in a Gravitational Field

Abstract: Papapetrou's covariant equations of motion for a spinning particle in a gravitational field are discussed. The equations of motion for the spin of a particle at rest outside a rotating mass are derived using the Kerr metric. It is shown that Schiff's formula for the mass‐current effect follows from these equations in the lowest approximation.

Higher moment equations and the distribution function of the solar-wind plasma

Abstract: Study of the higher-moment equations for a collisionless fully ionized plasma. For a collisionless, heat-conducting plasma, the distribution function f is cylindrically symmetric about the direction of the magnetic field. It is shown that under a certain assumption the fourth moments of f can be expressed as simple functions of lower moments. Thus no higher-moment terms appear in the third-moment equations. The two third-moment equations, which are obtained in a simple form, join other lower-moment equations to form a closed set of moment equations. The new equations can be used to study the thermal anisotropy and the heat flux of the solar-wind proton. A special case of the cylindrically symmetric distribution function f is found to resemble the proton distribution function reconstructed from solar-wind data, and this resemblance justifies the assumption needed for decoupling the moment equations.

Numerical design of transonic airfoils

Abstract: Publisher Summary This chapter discusses numerical design of transonic airfoils. It describes an inverse method of computing plane transonic flows past air foils that are not only free of shocks but also have adverse pressure gradients so moderate that no separation of the turbulent boundary layer should take place. Up-to-date existence and uniqueness theorems combine with the experimental evidence to assure that these flows are physically realistic and will occur in practice. The chapter presents the partial differential equations governing steady two-dimensional flow of an inviscid compressible fluid by numerical analysis of characteristic initial value problems for the analytic continuation of the solution into the complex domain. The finite difference scheme presented in the chapter was originally introduced to describe the detached shock wave in front of a blunt body but is actually better suited to the inverse problem of shaping air foils so as to achieve shock-free transonic flow. It is related to Bergman's integral operator method and does exploit simplifications associated with the linearity of the equations of motion in the hodograph plane.

Die umkehrung der stabilitätssätze von Lagrange-Dirichlet und Routh

Abstract: The Lagrange-Dirichlet theorem states that the equilibrium position of a discrete, conservative mechanical system is stable if the potential energy U(q) assumes a minimum in this position. Although everything seems to indicate that the equilibrium is always unstable in case of a maximum of the potential energy, this has yet to be proven. In all existing instability theorems the function U(q) has to satisfy additional requirements which are very restrictive. In this paper instability is proved in the case of a maximum of U(q)eC 2, without further restrictions. The instability follows directly from the existence of certain types of motions which are not found as solutions of differential equations, but as the solutions of a variational problem. Existence theorems are given for the variational problem, based on a result found by Caratheodory. In similar way an “inversion” of Routh's theorem on the stability of steady motions in systems with cyclic coordinates is also given. The result obtained here is not as general as the inversion of the Lagrange-Dirichlet theorem because the equations of motion are of a more complex type.

The choice of conservation equations for plume models

Abstract: Theories on the behavior of turbulent buoyant plumes generally form three classes: (1) strictly self-similar models valid only in a uniform environment; (2) quasi-similar models based on flux conservation of mass, momentum, and heat; and (3) quasi-similar models based on conservation of momentum, heat, and kinetic energy of mean motion. Few direct comparisons of the various numerical solutions for the different models have been reported; and little attempt has been made to compare the formulation of models (2) and (3) in spite of the fact that such a comparison must form the basis for an understanding of their differences. The following contribution has been stimulated by yet another solution published recently by Fox in this journal, but differs from earlier papers in that it concentrates wholly on a discussion of the equations used in the two quasi-similarity approaches. The relationship between the field and gross-flux forms of the equations of motion is considered. In particular, it is shown that the use of the flux equations for vertical momentum and kinetic energy is not equivalent to the use of flux equations for mass and vertical momentum, and that the latter pair is to be preferred.

Dynamic Stability of Tube Conveying Fluid

Abstract: The parametric resonance of a simply supported tube is studied analytically. By employing Galerkin's method, the equation of motion is reduced to a system of coupled Mathieu-Hill-type equations with multiharmonic coefficients. The stability-instability region boundaries are constructed by the methods of Hsu and Bolotin. The results obtained by Hsu's first approximation show that combination resonance is possible and that the coupling terms do not influence the principal and second instability regions. The higher order approximation is obtained by Bolotin's method which shows that the effect of the coupling terms is to lower, in frequency, the instability regions; they have no effect on the size of the instability regions.

Analysis of Motion of Magnetic Levitation Systems: Implications for High‐Speed Vehicles

Abstract: To study the motion of a magnetically suspended high‐speed vehicle, a simple example (the long wire above a thin conducting plate) is considered in detail. The lift and drag forces on the magnet (long wire) are derived for arbitrary motion above the plate. The stability of the system is analyzed for typical parameters (velocity = 300 mph, height = 0.1 m). By using a Laplace‐transform technique, it is shown that two types of modes occur (in the linearized equations of motion). One mode is a vertical oscillation with an amplitude that grows slowly in time. The other mode is an unbounded increase in the horizontal velocity error. This latter instability results from the fact that the drag force decreases with increasing velocity at high speeds. In this connection, an error is pointed out in a recent publication in which it was claimed that the system is stable. Detailed consideration of the effects of horizontal acceleration and vertical velocity on the magnetic forces is given. The effects of aerodynamic dr...

Aperiodic behaviour of a non-linear oscillator

Abstract: The aperiodic behavior of the solution of the equation of motion derived previously (1966) when considering a model thermomechanical oscillator is examined. Periodic solutions of this equation are studied numerically and analytically. Conditions for the instability of the solutions are determined. This instability seems to be the cause of the observed aperiodicity.

Itinerant oscillator models and dielectric absorption in liquids

Abstract: The itinerant oscillator model of a liquid, which was devised previously by Hill (1963) in a study of dielectric absorption, is considered in more detail. Modified equations of motion are more clearly consistent, but the restricted model analysed predicts much narrower resonances than have so far been found in experiment.

Dynamic response of rapidly heated plate elements

Abstract: through the constitutive law. Indirect application of the variational principle yields the classical equations of motion, the force-displacement boundary conditions, and the constitutive relationships between the distortions and bending (and twisting) stresses. A semidirect application of the variational principle that eliminates spatial-coordinate dependence yields generalized time-dependent ordinary differential equations of dynamic equilibrium and constitutive relations between the generalized force and displacement parameters of assumed spatial distributions. Results obtained using the dynamic thermoelastic variational principle demonstrate displacement and bending-moment-convergence characteristics far superior to conventional solutions. A study of the influence of temperature-dependent material properties shows that neglect of temperature dependence is an unconservative assumption; further it is demonstrated that incomplete consideration of temperature dependence can lead to dangerously unconservative results.