Generalisations of the Laplace–Runge–Lenz Vector

Abstract

The characteristic feature of the Kepler Problem is the existence of the so-called Laplace–Runge–Lenz vector which enables a very simple discussion of the properties of the orbit for the problem. It is found that there are many classes of problems, some closely related to the Kepler Problem and others somewhat remote, which share the possession of a conserved vector which plays a significant rôle in the analysis of these problems.     

On the Initial Boundary Value Problem for the Vlasov–Poisson–Boltzmann System

We prove existence of DiPerna-Lions renormalized solutions to the Boltzmann equation and to the Vlasov-Poisson-Boltzmann system for the initial boundary value problem.     

The Vlasov–Maxwell–Boltzmann System in the Whole Space

The Vlasov–Maxwell–Boltzmann system is one of the most fundamental models to describe the dynamics of dilute charged particles, where particles interact via collisions and through their self-consistent electromagnetic field. We prove existence of global in time classical solutions to the Cauchy problem near Maxwellians.     

Solution of Duffin–Kemmer–Petiau Equation for the Step Potential

The Duffin–Kemmer–Petiau (DKP) equation for spin 0 and 1 in the pressence of the step potential is solved. The problem is reduced to the solution of an equation of Feshbach–Villars type.The reflection and transmission coefficients are correctly deduced. The Klein paradox persists and is discussed using the charge interpretation.     

Global Existence for the Einstein–Boltzmann Equation in the Flat Robertson–Walker Spacetime

The initial value problem for the Einstein–Boltzmann equation in the spatially homogeneous and isotropic case is considered. The global in time mild solution is obtained. Received: 18 May 1998 / Accepted: 23 November 1998     

Selective aluminum dissolution as a means to observe the microstructure of nanocrystalline intermetallic phases from Al–Fe–Cr–Ti–Ce rapidly solidified alloy

Rapidly solidified aluminum alloys are promising materials with very fine microstructure. The microscopy observation of these materials is complicated due to overlay of fcc-Al matrix and different intermetallic phases. A possible way to solve this problem is to dissolve the Al matrix. By this process powder formed by single intermetallic phase particles is obtained. In this paper a new aqueous based dissolving agent for Al-based alloy is presented. The influence of oxidation agent (FeCl₃) concentration on quality of extraction process was studied.     

Construction of the Dirichlet–Voronoi translation-compatible polyhedrons based on the Zelling parameters

A method of constructing the Dirichlet–Voronoi translation-compatible polyhedrons for sublattices of complex crystal compounds is presented. With the use of the Zelling parameters, this problem is reduced to a linear programming problem which can be solved on a personal computer for a finite ratio of translation-compatible polyhedron volumes. Particular solutions are derived for cubic and tetragonal systems with different orientations of the Bravais frame and polyhedron types.     

Generalized Lane–Emden Equation and the Structure of Galactic Dark Matter

Starting from the collisionless Boltzmann equation and the Plummer's distribution function, the Lane–Emden (LE) equation is derived and applied to the problem of galactic dark halos. Some generalizations of the LE equation are considered and their implications for the structure of dark halos are investigated. The relevance of a relativistic scalar field to the problem under investigation is also discussed.     

A least-squares/finite element method for the numerical solution of the Navier–Stokes-Cahn–Hilliard system modeling the motion of the contact line

In this article we discuss the numerical solution of the Navier–Stokes-Cahn–Hilliard system modeling the motion of the contact line separating two immiscible incompressible viscous fluids near a solid wall. The method we employ combines a finite element space approximation with a time discretization by operator-splitting. To solve the Cahn–Hilliard part of the problem, we use a least-squares/conjugate gradient method. We also show that the scheme has the total energy decaying in time property under certain conditions. Our numerical experiments indicate that the method discussed here is accurate, stable and efficient.     

Processes Proceeding in Fe–0.12C–1Cr–1Mo–1V Steel after Pressure Cycling

The evolution of the phase content and morphology and crystalline defects during temperature and pressure cycling is studied by the methods of transmission electron microscopy and x-ray diffraction analysis. Thermostable Fe–0.12C–1Cr–1Mo–1V steel in the initial ferrite-pearlite state is examined. The evolution of the grain structure and dislocation density is quantitatively described. A low stability of large pearlite grains at temperatures above 600°C is established. The problem of oxidizing of steel of this class is also discussed.