平方反比有心力系统的统一的Runge-Lenz矢量

本文给出平方反比有心力系统的Runge-Lenz矢量的统一的表达形式,给出了其守恒性的统一的数学证明;尤其是推导出该矢量的一种新的张量表达形式;罗列并证明了该矢量的一些性质,并基于该守恒矢量讨论了质点作各类开普勒运动的判据.     

龙格-楞次矢量的张量积形式及其应用

导出了平方反比中心力场的龙格-楞次矢量的一种张量积形式,即由运动质点的能动张量与位置矢量的张量积构成;给出了其守恒性的统一的数学证明;在此张量积表达形式的基础上罗列并证明了该守恒矢量的若干性质;推导并讨论了质点作各类开普勒运动的判据,尤其是束缚态椭圆轨道的能量公式.     

两质点孤立系统相对动力学方程及其应用

两质点孤立系统可以简化为单个质点的运动来研究,质点相对质心的运动与质点的相对运动是等效的,相对动力学方程在解决两体问题中,具有明显的优越性.     

平方反比有心力作用下的二体系统的一套初等教案

以平方反比有心力作用下的椭圆轨道运动为例,文章基于开普勒第一、第二定律和牛顿的万有引力公式,配合有心力系统普遍适用的机械能守恒律,推演了平方反比有心力作用下的二体系统运动规律的一套初等的教学方案,而无需求解非线性的比耐(Binet)微分方程。用初等方法推导了椭圆轨道的能量与偏心率公式、圆轨道条件以及特征点的若干运动参数(速度及曲率半径)并给出了开普勒第三定律的一种新颖而简单的初等证明方法。     

开普勒问题的矢量解法

直接利用矢量解法求解开普勒问题,不必借助于比奈方程．该方法简明扼要,物理图像清晰．     

开普勒轨道的进动

我们在本文中研究:一质量为m的质点,在引力作用下进入和离开S系的运动,S系是由均匀分布在半径为R的球内的大量质点组成的.我们发现:如果穿越S系时运动质点没有碰撞,则运动质点的轨道不再是一个稳定的开普勒轨道.轨道的长轴将绕通过系统的中心并垂直于物体运动平面的轴进动.在一定的条件下。物体作有限次的周期运转后,将返回到它的初位置.某一颗星运动进入一星系,尔后又离开星系,可作为我们实际分析的例子. 我们采用极坐标系来分析这一质量为m的星的运动.设系统的质量为M,而极坐标的极点在系统的质心,且 m<相似文献   

全面感知水声信息的新传感器技术——矢量水听器及其应用

文章介绍了一种新型水声接收换能器——矢量水听器(矢量传感器),它可以同时共点地测量声场的声压和矢量(水媒质质点振速,振动加速度或声压梯度等),使用矢量水听器比常规声压水听器能获得更全面的声场信息,因此,在水声技术中获得了广泛的应用.文中描述了矢量水听器的结构设计和工作原理及特性,给出了一些重要的应用例子.     

浅谈拉普拉斯-隆格-楞次矢量

介绍了拉普拉斯-隆格-楞次矢量在经典力学、广义相对论以及量子力学中的应用.利用该矢量在经典和量子开普勒问题中的守恒性,可以不求解运动方程而直接得到系统的重要性质,如轨道方程、能量本征值等;广义相对论对牛顿引力的修正会破坏该矢量的守恒性,而它随时间的演化方程,也可以完全决定轨道进动角和光线在引力场中的偏移.这些都说明,拉普拉斯-隆格-楞次矢量是反应系统动力学对称性的一个基本物理量,将其引入大学物理教学是十分有意义的.     

矢量场穿过运动高斯面的通量随时间变化率的严密理论

证明了当计算矢量场穿过运动高斯面的通量随时间的变化率时,积分符号与求导符号的顺序是可以交换的.导出了运动曲面上的面元矢量随时间变化率的公式.给出了矢量场穿过运动高斯面的通量随时间变化率的几种算法.     

基于等效源法和双面质点振速测量的声场分离方法

目前基于等效源法的声场分离方法有两种输入方式, 一种以双测量面上的声压为输入, 另一种以单测量面上的声压和质点振速为输入. 本文以双测量面上的质点振速为输入, 提出一种新的基于等效源法的声场分离方法. 首先给出了该方法的理论推导, 然后通过数值仿真和实验验证了该方法的有效性. 通过与基于双面声压测量的声场分离方法的比较, 证明了该方法在分离质点振速方面的优越性. 此外, 在仿真中还研究了干扰声源强度和测量面间距对分离精度的影响. 关键词： 声场分离 质点振速测量 等效源法 近场声全息     

On the global symmetry of the classical Kepler problem

A model for the classical Kepler problem is presented in which both the temporal evolution and the symmetry group act globally in a simple and canonical way. These actions are generated by the Hamiltonian function, the angular momentum and the Runge-Lenz vector. The symmetry group is SO(4) for negative and SO(1,3)₀ for positive energy.     

The geodesic motion on generalized Taub-NUT gravitational instantons

A class of generalized Taub-NUT gravitational instantons is reported in five-dimensional Einstein gravity coupled to a non-linear sigma model. The geodesic dynamics of a spinless particle of unit mass on these static gravitational instantons is studied. This is accomplished by finding a generalized Runge-Lenz vector. Unlike the Kepler problem, or, the dynamics of a spinless particle on the familiar Taub-NUT gravitational instantons, the orbits are not conic sections.     

Dynamical groups and spherical potentials in Classical Mechanics

The one particle problem in a spherical potential is examined in Classical Mechanics from a group theorical point of view. The constants of motion are classified according to their behaviour under the rotation groupSO(3), i.e. according to the irreducible representationsD _j ofSO(3) (section 1).The Lie algebras ofSO(4) andSO(3) are explicitly built in terms of Poisson brackets for an arbitrary potential, from global considerations. The Kepler and the 3 dimensional oscillator problems are shown to play particular roles with respect to these groups (sections 2 and 3).In the last section, the Kepler problem is analyzed with the aid of theSO(4) group instead of the Lie algebra. It is proved that the transformations generated by the angular momentum and the Runge-Lenz vector form indeed a group of canonical transformations isomorphic toSO(4). Consequences with respect to the quantization problem are examined.     

电荷运动轨迹方程解法的讨论

文章讨论了在库仑有心力作用下点电荷二维运动轨迹方程的不同解法：比耐方程法、Runge-Lenz矢量法以及速度积分法.比耐方程法是利用比耐公式求解微分方程,得出点电荷的运动轨迹;Runge-Lenz矢量法和速度积分法都是从点电荷的动力学方程出发,利用矢量积分得出一个常矢量,并应用该矢量分析得出点电荷的轨迹方程.3种方法得到的轨迹方程是一致的.计算表明,库仑有心力作用下的点电荷的二维运动轨迹为圆锥曲线,并分析了不同初始条件下圆锥曲线的类型.     

Variations on the Kepler problem

The elliptical orbits resulting from Newtonian gravitation are generated with a multifaceted symmetry, mainly resulting from their conservation of both angular momentum and a vector fixing their orientation in space—the Laplace or Runge-Lenz vector. From the ancient formalisms of celestial mechanics, I show a rather counterintuitive behavior of the classical hydrogen atom, whose orbits respond in a direction perpendicular to a weak externally-applied electric field. I then show how the same results can be obtained more easily and directly from the intrinsic symmetry of the Kepler problem. If the atom is subjected to an oscillating electric field, it enjoys symmetry in the time domain as well, which is manifest by quasi-energy states defined only modulo ħω. Using the Runge-Lenz vector in place of the radius vector leads to an exactly-solvable model Hamiltonian for an atom in an oscillating electric field—embodying one of the few meaningful exact solutions in quantum mechanics, and a member of an even more exclusive set of exact solutions having a time-dependent Hamiltonian. I further show that, as long as the atom suffers no change in principal quantum number, incident radiation will produce harmonic radiation with polarization perpendicular to the incident radiation. This unusual polarization results from the perpendicular response of the wavefunction, and is distinguished from most usual harmonic radiation resulting from a scalar nonlinear susceptibility. Finally, I speculate on how this radiation might be observed.     

Runge-Lenz矢量与升降算子

分析表明，对于二维氢原子和各向同性谐振子，从径向Schr?dinger方程的因式分解导出的升降算子与Runge-Lenz矢量等价．这表明径向方程能够因式分解，与经典粒子轨道的闭合性及体系的动力学对称性密切相关 关键词：     

A conservative discretization of the Kepler problem based on the L-transformations

Conservative numerical schemes for the two- and three-dimensional Kepler motion were recently proposed in [Y. Minesaki, Y. Nakamura, Phys. Lett. A 306 (2–3) (2002) 127; Y. Minesaki, Y. Nakamura, Phys. Lett. A 324 (4) (2004) 282]. They are based on Levi-Civita and Kustaanheimo–Stiefel transformations respectively. The schemes preserve all first integrals of Kepler motion: the Hamiltonian function, the angular momentum and Runge–Lenz vector. In the present Letter we extend this approach to the L-transformations (a generalization of Levi-Civita and Kustaanheimo–Stiefel transformations). Thus, we can consider more general family of the conservative numerical schemes for the two- and three-dimensional Kepler motion as well as construct conservative schemes for higher-dimensional Kepler problems. Conservation of the first integrals is proved with the help of L-matrix identities. The proposed numerical scheme permits explicit implementation.     

微扰Kepler系统轨道微分方程的近似Lie对称性与近似不变量

把极角θ视为独立变量,得到Kepler系统的轨道微分方程.首先讨论Kepler系统轨道微分方程的Lie对称性和不变量,微扰Kepler系统轨道微分方程的精确Lie对称性和精确不变量,其次讨论微扰Kepler系统轨道微分方程的近似Lie对称性和近似不变量,并得到了微扰Kepler系统的9个一阶近似Lie对称性和6个一阶近似不变量,其中1个实为精确不变量,而其余5个分别等于微扰系数ε乘以Kepler系统相应的5个不变量。     

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.     

Relativistic wave equation for a boson-fermion system

We present a two-body relativistic wave equation for a system composed of a boson and a fermion. One-body equations such as the Dirac and the Klein-Gordon equations are often used as an approximate equation for relativistic two-body systems. However, when the masses of two particles are not very different, the use of one-body equations comes into question. We use the Feshbach-Villars formalism for the boson so that the wave equation can be given in the form of an eigenvalue equation for the Hamiltonian. Differences between our equation and the one-body equations are examined and illustrated in a numerical example of a two-body system with scalar and vector potentials.Communicated by: W. Weise