Induced rotation in the Papapetrou equations

We obtain an integral form of the Papapetrou equations, which describes the motion of an extended body in an external gravitational field. Using the Fock method, we calculate an explicit form for the components of the spin tensor and derive relativistic equations of rotational motion in the Schwarzschild space V₄. We show that the spin of the body becomes proportional not only to the angular velocity but also to the angular momentum. Thus, induced rotation also follows from the Papapetrou equations.Astrophysics Institute, Kazan Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 117–124, October, 1992.     

Force multipole moments for a spherically symmetric particle in solution

We present a general theorem for the force multipole moments of arbitrary order induced in a spherically symmetric particle immersed in a fluid whose motion satisfies the linear Navier-Stokes equation for steady incompressible viscous flow. The multipole moments are expressed in terms of the unperturbed fluid velocity field. It is shown that for a particle with a finite extension only a few terms give rise to fluid perturbations which are not confined to the interior of the particle. We give explicit results for a polymer satisfying the Debye-Bueche-Brinkman equations and for a hard sphere with mixed slip-stick boundary conditions.     

Teleparallel versions of Friedmann and Lewis–Papapetrou spacetimes

This paper is devoted to investigating the teleparallel versions of the Friedmann models as well as the Lewis–Papapetrou solution. We obtain the tetrad and the torsion fields for both spacetimes. It is shown that the axial-vector vanishes for the Friedmann models. We discuss the different possibilities for the axial-vector, depending on the arbitrary functions ω and ψ in the Lewis–Papapetrou metric. The vector related to spin has also been evaluated.     

The Influence of Supplementary Conditions for Papapetrou Equations on the Lyapunov Stability of Motion

The Lyapunov stability of circular motions of spinning test bodies in the Schwarzschild field has been considered. For Papapetrou equations the generalized supplementary conditions S⁰ⁱu₀ = kS^iju_j = k const, i, j = 1, 2, 3, have been chosen. Four of the first integrals of motion have been indicated. With the aid of Lyapunov functions the domains of stable motions have been found.     

Gravitational waves and Papapetrou metric in the six-dimensional treatment of gravitation

Six-dimensional treatment of gravitation based on the principle of simplicity to which there corresponds motion of particles with the speed of light in the Compton neighborhood of the three-dimensional space along the geodesics complying with the Fermat principle is given to the Papapetrou metric and gravitational waves. The envelope of the geodesics has the form of a tubular surface with the Compton transverse sizes in the additional subspace where the radius and speed of light vary along the tube. Gravitational waves, which are perturbations of these radii and speed of light, turn out to attenuate exponentially here. Their amplitudes are considered in the near-field zone of the rotator with n Maltese cross lobes and calculated at n = 4.     

Fast multipole accelerated boundary element method for the Helmholtz equation in acoustic scattering problems

We apply the fast multipole method (FMM) accelerated boundary element method (BEM) for the three-dimensional (3D) Helmholtz equation, and as a result, large-scale acoustic scattering problems involving 400000 elements are solved efficiently. This is an extension of the fast multipole BEM for two-dimensional (2D) acoustic problems developed by authors recently. Some new improvements are obtained. In this new technique, the improved Burton-Miller formulation is employed to over-come non-uniqueness difficultie...     

A stochastic particle system modeling the Carleman equation

Two species of Brownian particles on the unit circle are considered; both have diffusion coefficient >0 but different velocities (drift), 1 for one species and –1 for the other. During the evolution the particles randomly change their velocity: if two particles have the same velocity and are at distance ( being a positive parameter), they both may simultaneously flip their velocity according to a Poisson process of a given intensity. The analogue of the Boltzmann-Grad limit is studied when goes to zero and the total number of particles increases like ^–1. In such a limit propagation of chaos and convergence to a limiting kinetic equation are proven globally in time, under suitable assumptions on the initial state. If, furthermore, depends on and suitably vanishes when goes to zero, then the limiting kinetic equation (for the density of the two species of particles) is the Carleman equation.Dedicated to the memory of Paola Calderoni.     

Phase particle integrations of the nuclear vlasov equation

The non-linear Vlasov equation, describing the time development of the one-body Wigner distribution function, is integrated as an initial value problem by following the trajectories of notional phase particles which evolve along its hamiltonian characteristic curves. Initial conditions are generated by galilean transformations of the self-consistent solutions of the static equation obtained assuming a Thomas-Fermi form for the Wigner distribution.Fusion, deep inelastic collisions and fragmentation are all exhibited depending on the bombarding energy per nucleon, in qualitative agreement with the results of comparable TDHF calculations. A simple criterion for determining the boundaries between these phenomena, based on classical penetration of the collective mean field by the phase particles, is found not to be accurate, presumably due to an inadequate allowance for the effects of self-consistency.     

Twistfree and Papapetrou vacuum spacetimes with a spacelike killing vector

We show that, for the case of vacuum solutions of the Einstein equations with a spacelike hypersurface orthogonal Killing vector /³ and associated metricds ² =e ^2U (dx ³)² +e ^–2U _ab dx ^a dx ^b whereU is not a constant, there exists at every point of the quotient 3-space a plane of vectorsK ^a such that £_KR_ab=0 andK ^a R_ab=0 whereR{inab} is the Ricci tensor formed from _ab . Then in the case whereU{in,a} is a timelike or spacelike vector in the quotient 3-space, Petrov type I solutions of the vacuum field equations are obtained. In the simpler case whereU{in,a} is a null vector in the quotient 3-space, the complete solution of the vacuum field equations is obtained. It is shown that this solution is Petrov type III of Kundt's class. For the case of Papapetrou solutions where there is a twist potential which is a function ofU, solutions corresponding to the twistfree solutions are given.     

The black-box fast multipole method

A new O(N)$O(N)$ fast multipole formulation is proposed for non-oscillatory kernels. This algorithm is applicable to kernels K(x,y)$K(x\text{,}y)$ which are only known numerically, that is their numerical value can be obtained for any (x,y)$(x\text{,}y)$. This is quite different from many fast multipole methods which depend on analytical expansions of the far-field behavior of K  , for |x-y|$|x-y|$ large. Other “black-box” or “kernel-independent” fast multipole methods have been devised. Our approach has the advantage of requiring a small pre-computation time even for very large systems, and uses the minimal number of coefficients to represent the far-field, for a given L²$L^2$ tolerance error in the approximation. This technique can be very useful for problems where the kernel is known analytically but is quite complicated, or for kernels which are defined purely numerically.     

新的对角形式快速多极边界元法求解声学Helmholtz方程

传统外部声学Helmholtz边界积分方程无法在个人计算机上求解大规模工程问题. 为了有效解决这个问题, 将快速多极方法引入到边界积分方程中, 加速系统矩阵方程组的迭代求解. 由于在边界积分方程中引入基本解的对角形式多极扩展, 新的快速多极边界元法的计算效率与传统边界元相比显著提高, 计算量和存储量减少到O(N)量级(N为问题的自由度数). 包括含有420000个自由度的大型潜艇模型数值算例验证了快速多极边界元法的准确性和高效性, 清楚表明新算法在求解大规模声学问题中的优势,     

新的对角形式快速多极边界元法求解声学Helmholtz方程

传统外部声学Helmholtz边界积分方程无法在个人计算机上求解大规模工程问题. 为了有效解决这个问题, 将快速多极方法引入到边界积分方程中, 加速系统矩阵方程组的迭代求解. 由于在边界积分方程中引入基本解的对角形式多极扩展, 新的快速多极边界元法的计算效率与传统边界元相比显著提高, 计算量和存储量减少到O(N)量级(N为问题的自由度数). 包括含有420000个自由度的大型潜艇模型数值算例验证了快速多极边界元法的准确性和高效性, 清楚表明新算法在求解大规模声学问题中的优势, 具有良好的工程应用前景.     

静磁外部球谐多极矩展开

本文讨论静磁球谐多极矩展开,包括静磁标势和静磁矢势的外部球谐多极矩展开.在磁矢势外部球谐多极矩分析中介绍了两种不同的方法,方法一基于磁标势外部球谐多极矩展开求惯用磁矢势外部球谐多极矩展开,方法二从电流对远源场点磁矢势场贡献出发借助德拜势直接求得静磁矢势外部球谐多极矩展开的惯用表示.     

The two-body multipole problem of electrodynamics

A 2-body system composed of two objects having arbitrary distributions of charge and current is discussed. An expression for the velocity dependent potential between these two objects has been obtained in the non-relativistic approximation. This potential consists of two parts viz. a velocity independent scalar potential Φ_eff and another part which is linearly dependent on the relative velocity between the objects. The second part naturally suggests a vector potential A_eff. The potentials have been expanded into multipole terms. It has been found that Φ_eff is a sum of two components viz. Φ_EE and Φ_MM such that each multipole term in Φ_EE represents an interaction between the electric multipoles of the two systems, each term in Φ_MM represents an interaction between their magnetic multipoles whereas each term in A_eff represents an interaction between an electric multipole of one and a magnetic multipole of the other. The results have been applied to the interaction between an electric dipole and a magnetic dipole. The symmetry among the multipole terms in A_eff suggests vanishing vector potential between two identical objects. A corollary of this appears to be absence of spin orbit interaction between two identical particles in the same spin state.     

Convergence to equilibrium distribution. The Klein-Gordon equation coupled to a particle

The Hamiltonian system formed by a Klein-Gordon vector field and a particle in ℝ³ is considered. The initial data of the system are given by a random function, with finite mean energy density, which also satisfies a Rosenblatt- or Ibragimov-type mixing condition. Moreover, initial correlation functions are assumed to be translation invariant. The distribution μ _t of the solution at time t ∈ ℝ is studied. The main result is the convergence of μ _t to a Gaussian measure as t → ∞, where μ_∞ is translation invariant.     

哈密顿正则方程在生物膜与胶体粒子相互作用研究中的应用

介绍经典分析力学中的哈密顿正则方程在生物膜与胶体粒子相互作用研究中的一个具体应用.由Helfrich理论模型得到体系的哈密顿,用正则方程给出一组常微分方程,并用打靶法对其进行求解得到体系的稳定构型随膜参数变化的规律.     

The theory of particle diffusion in a strong random magnetic field with an allowance for higher multipole moments of the distribution function

The kinetic equation including a small-scale collisional integral for the particles propagating in a strong random and regular magnetic field [29] is solved by expanding the distribution function into series in spherical harmonics of the particle momentum angles. Using methods of the quantum theory of the angular moment [41], the equations for higher multipole moments of the distribution function in the space of momentum angles are derived and solved in the stationary case for the galactic cosmic rays in interplanetary space. The observed amplitudes and phases of the diurnal variation harmonics can be explained using the results of measurements of the interplanetary magnetic field performed on board the Ulysses spacecraft [12–14] and other satellites [45, 46] with an allowance for redistribution of the interplanetary and interstellar magnetic field lines. The spatial structure of the convection and diffusion fluxes of the galactic cosmic rays is refined. Formulas taking into account a change in the Earth’s axis tilt relative to the direction toward the Sun are derived, which allow the annual changes in contributions to the diurnal variation harmonics to be determined. The equation of diffusion taking into account the 2nd harmonic is obtained, and the contribution of this effect to the relative particle density in the cosmic rays in a spherically symmetric case is analyzed.