Lie and Noether symmetries of geodesic equations and collineations

The Lie symmetries of the geodesic equations in a Riemannian space are computed in terms of the special projective group and its degenerates (affine vectors, homothetic vector and Killing vectors) of the metric. The Noether symmetries of the same equations are given in terms of the homothetic and the Killing vectors of the metric. It is shown that the geodesic equations in a Riemannian space admit three linear first integrals and two quadratic first integrals. We apply the results in the case of Einstein spaces, the Schwarzschild spacetime and the Friedman Robertson Walker spacetime. In each case the Lie and the Noether symmetries are computed explicitly together with the corresponding linear and quadratic first integrals.     

PROLONGATION STRUCTURE,BACKLUND TRAESFORKATION AND PRINCIPAL HOMOGENEOUS HILBERT PROBLEM IN GENERAL RELATIVITY+

We will briefly review the prolongation structure theory for nonlinear systems, such as nonlinear evolution equations and soli-ton equations, and its applications to the solutlon generating techniques for the gravitational fields with two commuting Killing vectors.     

Fundamental Equations for Submanifolds

We notice that in the embedding of submanifolds, the fundamental equations are only the Gauss equations for the tangent vectors while the Weingarten equations for the normal equations can essentially be determined by them, and furthermore that the integrability condition of the Weingarten equations, the Ricci equations, are consitently satisfied under that of the Gauss equations. Therefore, the Weingarten and the Ricci equations do not describe essentially independent conditions for embedding. We demonstrate these facts by explicitly constructing the normal vectors from the tangent vectors.     

Shear-free,twisting Einstein-Maxwell metrics in the Newman-Penrose formalism

The problem of finding algebraically special solutions of the vacuum Einstein-Maxwell equations is investigated using the spin coefficient formalism of Newman and Penrose. The general case, in which the degenerate null vectors are not hypersurface orthogonal, is reduced to a problem of solving five coupled differential equations that are no longer dependent on the affine parameter along the degenerate null directions. It is shown that the most general regular, shearfree, nonradiating solution of these equations is the Kerr-Newman metric.Based in part on a doctoral thesis submitted to the University of Pittsburgh (1970) while the author was NASA Predoctoral Trainee. Research also supported in part by the National Science Foundation under Grant GP-19378.     

Elasticity and dislocations in quasicrystals with 18-fold symmetry

Elasticity problems of quasicrystals with 18-fold rotational symmetry are studied. Constitutive equations and governing equations are obtained. For static elasticity problems, the displacement vectors in two phason fields are expressed in terms of two pairs of associated harmonic functions or two analytic functions. For dynamic problems, the displacement vectors can be represented in terms of an auxiliary function satisfying a fourth-order partial differential equation. A general solution of phasons is given by the solution of two diffusion equations. Phason elastic fields induced by a dislocation in a quasicrystal with 18-fold symmetry are determined and exhibit an inverse singularity.     

Electromagnetic field representation in the source region

Representation of electromagnetic fields in the source region in inhomogeneous anisotropic media is developed in terms of Hertz vectors. Differential equations satisfied by these vectors as functions of stream potentials of the sources are obtained. Conditions for representation in terms of uncoupled transverse electric and magnetic modes satisfying second-order differential equations are derived.     

Palatini-type variational principle for the SL(2, C) gauge theory of gravitation

An explicitly SL(2, C) gauge-invariant Lagrangian density, equivalent to Hilbert's Lagrangian density, is written, and a Palatini-type variational principle is applied to it. The resulting field equations are Einstein's equations written in dyad notation and a set of equations defining the spin coefficients in terms of the components of the null tetrad vectors and their directional derivatives. The techniques employed throughout this article are those which were developed in the SL(2, C) gauge theory of gravitation.     

Gaudin model,Bethe Ansatz and critical level

We propose a new method of diagonalization oif hamiltonians of the Gaudin model associated to an arbitrary simple Lie algebra, which is based on the Wakimoto modules over affine algebras at the critical level. We construct eigenvectors of these hamiltonians by restricting certain invariant functionals on tensoproducts of Wakimoto modules. This gives explicit formulas for the eigenvectors via bosonic correlation functions. Analogues of the Bethe Ansatz equations naturally appear as equations on the existence of singular vectors in Wakimoto modules. We use this construction to explain the connection between Gaudin's model and correlation functios of WZNW models.     

数值求解Navier-Stokes方程的一个新的系数矩阵分裂法

在一个单步格式的基础上构造了一个新的隐式系数矩阵分裂法来数值求解可压Navier-St-okes方程。对方程中的无粘项部分利用守恒型方程中流通量向量为一齐次函数这一特性,根据Jacobian矩阵特征值的符号而将流通向量分裂成两部分。在此基础上据风向而构造逼近于无粘项的差分格式。对方程中的粘性项部分利用算子附加修正的方法来改进计算的收敛过程。所建立的差分格式被用来数值求解Couette流以考查这一方法。     

啁啾超短激光脉冲对二能级体系特性的调控

 研究了啁啾超短激光脉冲与二能级体系近共振作用下的特性.采用光与物质相互作用的半经典理论,建立了含啁啾相位项的修正光学Bloch方程,并用高准确度、快速和可靠的四阶龙格-库塔法数值求解了该方程.通过数值计算,得到啁啾符号和啁啾量与Bloch矢量的关系,以及共振失谐量符号和啁啾符号对Bloch矢量性质影响的规律.结果表明,啁啾的符号和大小对Bloch矢量的瞬态相干过程和稳态特性都会产生明显的调制作用,由此可以通过调节激光脉冲的啁啾特性来实现对二能级体系性质的调控.     

On quadratic first integrals of the geodesic equations for type {22} spacetimes

It is shown that every type {22} vacuum solution of Einstein's equations admits a quadratic first integral of the null geodesic equations (conformal Killing tensor of valence 2), which is independent of the metric and of any Killing vectors arising from symmetries. In particular, the charged Kerr solution (with or without cosmological constant) is shown to admit a Killing tensor of valence 2. The Killing tensor, together with the metric and the two Killing vectors, provides a method of explicitly integrating the geodesics of the (charged) Kerr solution, thus shedding some light on a result due to Carter.     

稠密共振介质中近偶极-偶极相互作用的局域场效应

 为研究存在由近偶极-偶极相互作用诱导的局域场效应时超短强激光脉冲与稠密共振介质相互作用的特性,采用光与物质相互作用的半经典理论,建立了稠密二能级体系中考虑原子间近偶极-偶极相互作用的修正光学Bloch方程,并用四阶Runge-Kutta法数值求解了该方程。研究结果表明:局域场效应对稠密二能级体系中Bloch矢量的瞬态相干过程和稳态性质都具有强烈的调制作用。并由此提出了调控稳态粒子数布居的两种方案。     

Equations of the electromagnetic field in the chronometric-radial invariant formulation of the general theory of relativity

Chronometric-radial invariant (CRI) vectors of the electromagnetic field are introduced in the CRI formulation of the general theory of relativity proposed by Vladimirov. The general theory of relativity Maxwell equations for the electromagnetic field are transformed into equations for CRI quantities. It is postulated that the CRI components of vectors and tensors are the physically measurable quantities of the general theory of relativity.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 110–115, February, 1992.The author is grateful to Yu. S. Vladimirov for useful discussions and interest in this work.     

双折射晶体入射、折射光电场矢量的琼斯描述及界面处菲涅耳方程的修正

为了用琼斯矢量更明确地表示双折射晶体入射光和折射光的偏振状态,利用波法线椭球和物质方程,将入射光和折射光的电场矢量均视为由o振动和e振动两个方向的分量叠加而成,并将之投影到垂直于和平行于入射面的两个方向上。其中,入射光和折射光电场分量之间的关系用菲涅耳方程中的透射系数表示。考虑到e光电场矢量与电位移矢量的差别,对菲涅耳方程进行了一定的修正以满足e光的边界条件。最终给出了入射光和折射光电场矢量的琼斯矢量形式。并通过具体的数值计算说明,菲涅耳方程修正前后e光的透射系数有一定的差别,且e光的电场矢量和电位移矢量之间的分裂不可忽略。     

声诱导电磁场的赫兹矢量表示与多极声电测井模拟

在假设声场不受电磁场影响的前提下，将Pride声电耦合方程组化为具有电流源的麦克斯韦方程组.与空间位置固定的电流源产生的电磁场不同，孔隙地层中声波诱导的电磁场是由空间波动的电流源产生的.通过引入赫兹矢量，将求解麦克斯韦方程组问题转化为求解关于赫兹矢量的非齐次矢量赫姆霍兹方程组.通过求解该方程组，得出电磁场表达式.利用此方法，针对声电效应测井，分别计算了由单极声源、偶极声源、四极声源激发的井内声场及其诱导电磁场的全波波形. 关键词： 孔隙介质 诱导电磁场 测井 多极声源     

Viscoelastic micropolar continuum and viscoelastic waves

Generalized Kelvin model is applied to isotropic viscoelastic micropolar continuum. Constitutive equations in integral and differential form, generalized Lamé equations, and wave equations for displacement and microrotation are derived for this model. For damped viscoelastic waves, which are realized in the continuum under examination, the wave vectors and vectors of decay are explicitly given as functions of elastic moduli, viscosity coefficients, angular frequency, density, and microinertia coefficient. Analogous relations are derived for further eleven simpler mass models. Results are compared for three models in current use. For two of them (classical elastic and micropolar elastic medium) are the derived results in agreement with the ones usually used, for the third model (viscoelastic continuum) are the usual formulas a limiting case of the relations derived in this paper.     

Quaternion physical quantities

Quaternions consist of a scalar plus a vector and result from multiplication or division of vectors by vectors. Division of vectors is equivalent to multiplication divided by a scalar. Quaternions as used here consist of the scalar product with positive sign plus the vector product with sign determined by the right-hand rule. Units are specified by the multiplication process. Trigonometric functions are quaternions with units that can satisfy Hamilton's requirements. The square of a trigonometric quaternion is a real number provided that the product of the scalar number and the vector is not commutative. Maxwell's electromagnetic equations for empty space can be represented by a single quaternion equation.     

Scalar Product of Bethe Vectors from Functional Equations

In this work the scalar product of Bethe vectors for the six-vertex model is studied by means of functional equations. The scalar products are shown to obey a system of functional equations originated from the Yang-Baxter algebra and its solution is given as a multiple contour integral.     

Complete Classification of Conformal Killing Symmetries of Plane-Symmetric Static Spacetimes in Teleparallel Theory

We have studied the conformal, homothetic and Killing vectors in the context of teleparallel theory of gravitation for plane-symmetric static spacetimes. We have solved completely the non-linear coupled teleparallel conformal Killing equations. This yields the general form of teleparallel conformal vectors along with the conformal factor for all possible cases of metric functions. We have found four solutions which are divided into one Killing symmetries and three conformal Killing symmetries. One of these teleparalel conformal vectors depends on x only and other is a function of all spacetime coordinates. The three conformal Killing symmetries contain three proper homothetic symmetries where the conformal factor is an arbitrary non-zero constant.     

(Conformal) Killing Vectors in the Newman-Penrose Formalism

Finding (conformal) Killing vectors of a given metric can be a difficult task. This paper presents an efficient technique for finding Killing, homothetic, or even proper conformal Killing vectors in the Newman-Penrose (NP) formalism. Leaning on, and extending, results previously derived in the GHP formalism we show that the (conformal) Killing equations can be replaced by a set of equations involving the commutators of the Lie derivative with the four NP differential operators, applied to the four coordinates.It is crucial that these operators refer to a preferred tetrad relative to the (conformal) Killing vectors, a notion to be defined. The equations can then be readily solved for the Lie derivative of the coordinates, i.e. for the components of the (conformal) Killing vectors. Some of these equations become trivial if some coordinates are chosen intrinsically (where possible), i.e. if they are somehow tied to the Riemann tensor and its covariant derivatives.If part of the tetrad, i.e. part of null directions and gauge, can be defined intrinsically then that part is generally preferred relative to any Killing vector. This is also true relative to a homothetic vector or a proper conformal Killing vector provided we make a further restriction on that intrinsic part of the tetrad. If because of null isotropy or gauge isotropy, where part of the tetrad cannot even in principle be defined intrinsically, the tetrad is defined only up to (usually) one null rotation parameter and/or a gauge factor, then the NP-Lie equations become slightly more involved and must be solved for the Lie derivative of the null rotation parameter and/or of the gauge factor as well. However, the general method remains the same and is still much more efficient than conventional methods.Several explicit examples are given to illustrate the method.