磁单极和电磁对偶性

根据狄拉克-麦克斯韦方程组和推广的洛伦兹力公式,讨论了磁单极和电磁对偶性的基本概念和物理意义.麦克斯韦方程组和洛伦兹力公式可以通过对偶变化转化为电荷与磁荷并存的形式;但是狄拉克磁单极假设改变了麦克斯韦方程组的结构,任何对偶变换都不能将狄拉克-麦克斯韦方程组简化为只有电荷而没有磁荷的原始形式.采用推广的洛伦兹力公式,还证明了狄拉克-麦克斯韦电磁场的能量密度和玻印亭矢量,以及动量密度和动量流张量形式不变.最后,我们还将狄拉克-麦克斯韦方程组分解为仅仅分别包含电荷与磁荷的两组麦克斯韦方程,传统的电动力学理论可以直接推广应用于磁单极问题.     

建立麦克斯韦方程组的其他途径

麦克斯韦方程组是电磁场遵从的基本规律,是经典电磁现象的理论基础.十九世纪六十年代,麦克斯韦根据库仑、安培和法拉第等人分别在特定条件下获得的实验定律,经过分析推广到普遍情形,发现存在着不协调的因素,在此基础上作了两个重要的假设:变化的磁场产生涡旋电场,变化的电场产生涡旋磁场,从而建立了电磁场的统一理论,其基础就是电磁场普遍遵从的麦克斯韦方程组. 在麦克斯韦以后,许多物理学家研究从别的方面建立麦克斯韦方程组的可能性,取得一些成果.本文拟介绍三种建立麦克斯韦方程组的其他途径:(1)根据能量原理和近距作用原理建立麦克斯韦…     

对麦克斯韦方程组的理解

对麦克斯韦方程组涵义做了全面的分析与阐述.     

电磁波方程及折射、反射定律的一些思考

本文由电磁波的麦克斯韦方程组出发,介绍导出折射定律和反射定律的一种证明方法.其证明方法,使用了空间微元近似,然后推广至全空间传播的方法,从而简化了麦克斯韦方程组求解的烦琐过程,提出了一种可教学推广的实用性方法.通过使用微元法,求解得到麦克斯韦方程的行波解形式,即得出电磁场是一种行波.由电磁场的向量形式推导空间中电磁波的折射、反射定律,得到折射、反射定律的证明并不需要电磁波的解析形式,在连续函数的情形下是普遍成立的.求解过程中加深对麦克斯韦方程组的理解,体现了电磁过程的深刻物理图像,也为由几何光学向波动光学过渡提供一种思想上的指导.     

从麦克斯韦方程组到电磁波方程

<正>麦克斯韦方程组的威名可谓如雷贯耳,是经典电磁学的最高成就。先来看看麦克斯韦方程组的样子:方程组写成以上形式,需要一个前提:空间中的媒质是各向同性的(即,媒质中的每一点的物理性质不随方向改变)。方程组的一式描述了电荷产生电场的高斯定律,二式描述了变化的磁场产生电场的法拉第电磁感应定律,三式描述了磁单极子不存在的高斯磁定律,四式描述了电流和变化的电场产生磁场的麦克斯韦-安培环路定律。有物理意义上的场必定存在物理意义上的源,因此,我们可以发现,麦克斯韦方程组等式的左边是场,     

轴对称电荷分布非静态轴外场的一种计算方法

利用麦克斯韦方程组和轴对称性,给出了轴对称电荷分布非静态轴外场的一种计算方法.     

关于伸缩变换与麦克斯韦方程组对称性的讨论

指出麦克斯韦方程组在坐标伸缩变换下不具有不变性.     

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     

关于麦克斯韦方程组的讨论

本文讨论了麦克斯韦方程组的来龙去脉、形式和地位，对麦克斯韦主程组的教学具有一定的参考价值。     

轴对称非静态电流分布轴外场的一种计算方法

在轴对称非静态电流分布的情况下,利用麦克斯韦方程组和轴对称性,给出了轴外电磁场的一种计算方法.     

From the Becker–Döring to the Lifshitz–Slyozov–Wagner Equations

Connections between two classical models of phase transitions, the Becker–Döring (BD) equations and the Lifshitz–Slyozov–Wagner (LSW) equations, are investigated. Homogeneous coefficients are considered and a scaling of the BD equations is introduced in the spirit of the previous works by Penrose and Collet, Goudon, Poupaud and Vasseur. Convergence of the solutions to these rescaled BD equations towards a solution to the LSW equations is shown. For general coefficients an approach in the spirit of numerical analysis allows to approximate the LSW equations by a sequence of BD equations. A new uniqueness result for the BD equations is also provided.     

On the Geometry of Darboux Transformations for the KP Hierarchy and its Connection with the Discrete KP Hierarchy

We tackle the problem of interpreting the Darboux transformation for the KP hierarchy and its relations with the modified KP hierarchy from a geometric point of view. This is achieved by introducing the concept of a Darboux covering. We construct a Darboux covering of the KP equations and obtain a new hierarchy of equations, which we call the Darboux-KP hierarchy (DKP). We employ the DKP equations to discuss the relationships among the KP equations, the modified KP equations, and the discrete KP equations. Our approach also handles the various reductions of the KP hierarchy. We show that the KP hierarchy is a projection of the DKP, the mKP hierarchy is a DKP restriction to a suitable invariant submanifold, and that the discrete KP equations are obtained as iterations of the DKP ones. Received: 23 July 1996 / Accepted: 6 January 1997     

Mode competition in the orotron

A nonlinear, self-consistent and multimode analysis of the orotron is presented. The field in the cavity is expanded into the Hermite-Gaussian modes with time-dependent amplitudes, for which a set of ordinary differential equations is obtained from Maxwell's equations. The equations for the amplitudes are coupled to the equations of motion for the electrons. To yield a self-consistent solution, this set of coupled equations is solved simultaneously. The calculations yield transient and steady state behaviour, saturated efficiency, mode competition and multi-frequency behaviour.     

Conservation equations and equations of motion in the null formalism

The object of this paper is to relate three equations in the Newman-Penrose system of equations to the conservation laws and, hence, to the equations of motion. To do so, the corresponding result is first obtained using the Einstein equations in a null coordinate system. The Newman-Penrose equations are then analyzed. They are separated into hypersurface, propagation, supplementary, and conservation equations. When all field equations except the three conservation equations have been appropriately satisfied, the desired result follows.Research supported by the National Science Foundation under Grant GP-34641X.     

对称分类在非线性偏微分方程组边值问题中的应用

研究了微分方程对称分类在非线性偏微分方程组边值问题中的应用.首先,利用偏微分方程(组)完全对称分类微分特征列集算法确定了给定非线性偏微分方程组边值问题的完全对称分类;其次,利用一个扩充对称将非线性偏微分方程组边值问题约化为常微分方程组初值问题;最后,利用龙格-库塔法求解了常微分方程组初值问题的数值解.     

On the Calculation of Finite-Gap Solutions of the KdV Equation

Abstract

A simple and general approach for calculating the elliptic finite-gap solutions of the Korteweg-de Vries (KdV) equation is proposed. Our approach is based on the use of the finite-gap equations and the general representation of these solutions in the form of rational functions of the elliptic Weierstrass function. The calculation of initial elliptic finite-gap solutions is reduced to the solution of the finite-band equations with respect to the parameters of the representation. The time evolution of these solutions is described via the dynamic equations of their poles, integrated with the help of the finite-gap equations. The proposed approach is applied by calculating the elliptic 1-, 2- and 3-gap solutions of the KdV equations.     

Can the macroscopic Maxwell equations be obtained from the microscopic Maxwell-Lorentz equations by performing averages?

It is shown that the usual procedures of obtaining the macroscopic Maxwell equations from the microscopic Maxwell-Lorentz equations by performing averages contain an arbitrary choice of gauge. By a suitable different choice of the gauge the so-obtained Maxwell equations can be cast back to the form of the starting Maxwell-Lorentz equations. Therefore one cannot consider the Maxwell equations to be obtainable from the Maxwell-Lorentz equations by simply performing averages. The implication of this result is that besides the electromagnetic fields produced by the moving electric charges, as given by the Maxwell-Lorentz equations, there may be some other agents that cannot be identified as some kind of motion of the electric charges and that participate in the production of the electromagnetic fields.     

On the singular sector of the Hermitian random matrix model in the large N limit

The one-cut case of the Hermitian random matrix model in the large N limit is considered. Its singular sector in the space of coupling constants is analyzed from the point of view of the hodograph equations of the underlying dispersionless Toda hierarchy. A deep connection with the singular sector of the hodograph equations of the 1-layer Benney (classical long wave equation) hierarchy is stablished. This property is a consequence of the fact that the hodograph equations for both hierarchies describe the critical points of solutions of Euler-Poisson-Darboux equations.     

Time-fractional extensions of the Liouville and Zwanzig equations

This paper presents extensions to the classical stochastic Liouville equation of motion that contain the Riemann-Liouville and Caputo time-fractional derivatives. At first, the dynamic equations with the time-fractional derivatives are formally obtained from the classical Liouville equation. A feature of these new equations is that they have the same common formal solution as the classical Liouville equation and therefore may be used for study of the Hamiltonian system dynamics. Two cases of the time-dependent and time-independent Hamiltonian are considered separately. Then, the time-fractional Liouville equations are deduced from the short- and long-time asymptotic expansions of the obtained dynamic equations. The physical meaning of the resulting equations is discussed. The statements of the Cauchy-type problems for the derived time-fractional Liouville equations are given, and the formal solutions of these problems are presented. At last, the projection operator formalism is employed to derive the time-fractional extensions of the Zwanzig kinetic equations and the corresponding formal statistical operators from the time-fractional Liouville equations.