Classical mechanics of nonspherical bodies. I. Binary collisions in two dimensions

1982 ◽  
Vol 23 (4) ◽  
pp. 539-546 ◽  
Author(s):  
Yves Elskens ◽  
David Speiser
Keyword(s):  
Classical Mechanics ◽  
Two Dimensions ◽  
Binary Collisions

Velocity distribution function for a dilute granular material in shear flow

1997 ◽  
Vol 340 ◽  
pp. 319-341 ◽  
Author(s):  
V. KUMARAN
Keyword(s):  
Distribution Function ◽  
Shear Flow ◽  
Velocity Distribution ◽  
Scaling Laws ◽  
Three Dimensional ◽  
Particle Diameter ◽  
Velocity Distribution Function ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Binary Collisions

The velocity distribution function for the steady shear flow of disks (in two dimensions) and spheres (in three dimensions) in a channel is determined in the limit where the frequency of particle–wall collisions is large compared to particle–particle collisions. An asymptotic analysis is used in the small parameter ε, which is naL in two dimensions and n2L in three dimensions, where n is the number density of particles (per unit area in two dimensions and per unit volume in three dimensions), L is the separation of the walls of the channel and a is the particle diameter. The particle–wall collisions are inelastic, and are described by simple relations which involve coefficients of restitution et and en in the tangential and normal directions, and both elastic and inelastic binary collisions between particles are considered. In the absence of binary collisions between particles, it is found that the particle velocities converge to two constant values (ux, uy) =(±V, 0) after repeated collisions with the wall, where ux and uy are the velocities tangential and normal to the wall, V=(1−et) Vw/(1+et), and Vw and −Vw are the tangential velocities of the walls of the channel. The effect of binary collisions is included using a self-consistent calculation, and the distribution function is determined using the condition that the net collisional flux of particles at any point in velocity space is zero at steady state. Certain approximations are made regarding the velocities of particles undergoing binary collisions in order to obtain analytical results for the distribution function, and these approximations are justified analytically by showing that the error incurred decreases proportional to ε1/2 in the limit ε→0. A numerical calculation of the mean square of the difference between the exact flux and the approximate flux confirms that the error decreases proportional to ε1/2 in the limit ε→0. The moments of the velocity distribution function are evaluated, and it is found that 〈u2x〉→V2, 〈u2y〉 ∼V2ε and − 〈uxuy〉 ∼ V2εlog(ε−1) in the limit ε→0. It is found that the distribution function and the scaling laws for the velocity moments are similar for both two- and three-dimensional systems.

Three-body ion-neutral association

1974 ◽  
Vol 339 (1616) ◽  
pp. 13-28 ◽  
Keyword(s):  
Angular Momentum ◽  
Internal Energy ◽  
Orbital Angular Momentum ◽  
Rate Coefficient ◽  
Classical Mechanics ◽  
Impulse Approximation ◽  
Rate Coefficients ◽  
Quasi Equilibrium ◽  
Binary Collisions ◽  
Three Body

Consideration is given to the calculation of the rate coefficient of processes of the type A + + B + C → AB + + C. Classical mechanics is used. The impulse approximation is adopted and hard-sphere interactions are taken to describe C – A + and C – B collisions. Formulae are derived for the rate coefficients of binary collisions giving specified changes in the internal energy of the associating pair (without reference to their orbital angular momentum) and also of binary collisions giving specified changes in both the internal energy and in the square of the orbital angular momentum. By using quasi-equilibrium statistical theory, the rate co­efficient for three-body ion-neutral association is expressed in terms of either set of binary rate coefficients. Computations are carried out only Hg + + Hg + He → Hg + 2 + He. As expected, the predicted rate co­efficient is too high if specific account is not taken of the orbital angular momentum. If such account is taken excellent agreement is obtained with a measurement at 370 K made by Biondi (1953, 1972, private com­munication). The rate coefficient falls off slowly as the temperature is increased.

On the rigorous foundation of short-wave asymptotics

1966 ◽  
Vol 62 (2) ◽  
pp. 227-244 ◽  
Author(s):  
F. Ursell
Keyword(s):  
Wave Equations ◽  
Classical Mechanics ◽  
Short Wave ◽  
Two Dimensions ◽  
Linear Wave ◽  
Wave Optics ◽  
Wave Mechanics ◽  
Linear Wave Equations ◽  
Wave Limit ◽  
Time Periodic

AbstractCertain physical theories are short-wave limits of more general theories. Thus ray optics is the short-wave limit of wave optics, and classical mechanics is the short-wave limit of wave mechanics. In principle it must be possible to deduce the former from the latter theories by a rigorous mathematical limiting process; in fact the arguments found in the literature are formal, plausible and non-rigorous. (We are here concerned with linear wave equations and time-periodic phenomena.) For some wave equations there are, however, a few explicit rigorous canonical solutions relating to simple geometrical configurations, e.g. to conics in two dimensions for the equations of acoustics, and for these the asymptotics can be found rigorously. For more general configurations the solution of a typical boundary-value problem can be reduced to the solution of a Fredholm integral equation of the second kind.

scholarly journals N=2 Supersymmetry with Central Charge: A Twofold Implementation

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
L. P. R. Ospedal ◽  
R. C. Terin
Keyword(s):  
Sigma Model ◽  
Central Charge ◽  
Classical Mechanics ◽  
Theoretical Framework ◽  
Two Dimensions ◽  
Nonlinear Sigma Model ◽  
Two Dimensional ◽  
One Dimensional ◽  
Matter Model

In this work, we analyze an extended N=2 supersymmetry with central charge and develop its superspace formulation under two distinct viewpoints. Initially, in the context of classical mechanics, we discuss the introduction of deformed supersymmetric derivatives and their consequence on the deformation of one-dimensional nonlinear sigma model. After that, considering a field-theoretical framework, we present an implementation of this superalgebra in two dimensions, such that one of the coordinates is related to the central charge. As an application, in this two-dimensional scenario, we consider topological (bosonic) configurations of a special self-coupled matter model and present a nontrivial fermionic solution.

scholarly journals SEMICLASSICAL ANALYSIS FOR SPECTRAL SHIFT FUNCTIONS IN MAGNETIC SCATTERING BY TWO SOLENOIDAL FIELDS

2008 ◽  
Vol 20 (10) ◽  
pp. 1249-1282
Author(s):  
HIDEO TAMURA
Keyword(s):  
Classical Mechanics ◽  
Classical System ◽  
Spectral Shift ◽  
Two Dimensions ◽  
Magnetic Scattering ◽  
Shift Function ◽  
Semiclassical Analysis ◽  
Trapping Effect ◽  
Bohm Effect ◽  
Aharonov Bohm

We study the Aharonov–Bohm effect through the semiclassical analysis for the spectral shift function and its derivative in magnetic scattering by two solenoidal fields in two dimensions, assuming that the total magnetic flux vanishes. The corresponding classical system has a trajectory oscillating between the centers of two solenoidal fields. The emphasis is placed on analyzing how the trapping effect is reflected in the semiclassical asymptotic formula. We also make a comment on the case of scattering by a finite number of solenoidal fields and discuss the relation between the Aharonov–Bohm effect from quantum mechanics and the trapping effect from classical mechanics.

The tangled web of agency

2018 ◽  
Vol 41 ◽  
Author(s):  
Alain Pe-Curto ◽  
Julien A. Deonna ◽  
David Sander
Keyword(s):  
Two Dimensions ◽  
Bad Company

AbstractWe characterize Doris's anti-reflectivist, collaborativist, valuational theory along two dimensions. The first dimension is socialentanglement, according to which cognition, agency, and selves are socially embedded. The second dimension isdisentanglement, the valuational element of the theory that licenses the anchoring of agency and responsibility in distinct actors. We then present an issue for the account: theproblem of bad company.

A transmission electron microscopic study of structural transitions in fast ionic conductors

1987 ◽  
Vol 45 ◽  
pp. 156-157
Author(s):  
R. B. Queenan ◽  
P. K. Davies
Keyword(s):  
Optical Properties ◽  
Ionic Conduction ◽  
Recent Observation ◽  
Ionic Conductivities ◽  
Sodium Aluminate ◽  
Two Dimensions ◽  
Ionic Conductors ◽  
Electron Microscopic ◽  
Transmission Electron ◽  
Trivalent Cations

Na ß“-alumina (Na1.67Mg67Al10.33O17) is a non-stoichiometric sodium aluminate which exhibits fast ionic conduction of the Na+ ions in two dimensions. The Na+ ions can be exchanged with a variety of mono-, di-, and trivalent cations. The resulting exchanged materials also show high ionic conductivities.Considerable interest in the Na+-Nd3+-ß“-aluminas has been generated as a result of the recent observation of lasing in the pulsed and cw modes. A recent TEM investigation on a 100% exchanged Nd ß“-alumina sample found evidence for the intergrowth of two different structure types. Microdiffraction revealed an ordered phase coexisting with an apparently disordered phase, in which the cations are completely randomized in two dimensions. If an order-disorder transition is present then the cooling rates would be expected to affect the microstructures of these materials which may in turn affect the optical properties. The purpose of this work was to investigate the affect of thermal treatments upon the micro-structural and optical properties of these materials.

Laser Scanning Confocal Microscopy and Three-Dimesnsional Reconstruction of the Mitotic Spindles of Unequally Dividing Embryonic Cells

1990 ◽  
Vol 48 (3) ◽  
pp. 166-167
Author(s):  
J. Holy ◽  
G. Schatten
Keyword(s):  
Confocal Microscopy ◽  
Mitotic Spindle ◽  
Laser Scanning ◽  
Three Dimensional ◽  
Laser Scanning Confocal Microscopy ◽  
Two Dimensions ◽  
Embryonic Cells ◽  
Mitotic Spindles ◽  
Cleavage Division ◽  
Scanning Confocal Microscopy

One of the classic limitations of light microscopy has been the fact that three dimensional biological events could only be visualized in two dimensions. Recently, this shortcoming has been overcome by combining the technologies of laser scanning confocal microscopy (LSCM) and computer processing of microscopical data by volume rendering methods. We have employed these techniques to examine morphogenetic events characterizing early development of sea urchin embryos. Specifically, the fourth cleavage division was examined because it is at this point that the first morphological signs of cell differentiation appear, manifested in the production of macromeres and micromeres by unequally dividing vegetal blastomeres.The mitotic spindle within vegetal blastomeres undergoing unequal cleavage are highly polarized and develop specialized, flattened asters toward the micromere pole. In order to reconstruct the three-dimensional features of these spindles, both isolated spindles and intact, extracted embryos were fluorescently labeled with antibodies directed against either centrosomes or tubulin.

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