讲授力学原理统一性——以弹簧振子为例

对一个系统进行力学分析可以运用不同的力学原理而得到相同结果。本文以弹簧振子为例,详细阐述了牛顿第二定律、机械能守恒定理、拉格朗日方程、哈密顿正则方程、哈密顿变分原理这五种力学原理之间的相互关系。其中牛顿第二定律是从受力角度进行分析,而机械能守恒定理、拉格朗日方程、哈密顿正则方程则是从能量角度出发。哈密顿变分原理则为更为普遍的力学原理,通过对其变分可以推导出拉格朗日方程、哈密顿正则方程以及运动微分方程。弄清楚这五种力学原理的统一性,有助于我们更加合理地选择运用它们去解决工程问题。     

理想、完整约束系统的哈密顿正则方程

引入了广义动能概念,利用勒让德变换导出了方程中的偏导数是关于广义动能偏导数的理想完整约束系统哈密顿正则方程,据此讨论了有关守恒律.     

应用辛格式求解一维定态Schr(o)dinger方程

将一维定态Schr(o)dinger方程转化为等价的哈密顿正则方程,并采用辛格式计算了7Li2分子的振动能级.     

用分析力学方法论证电子在核的库仑场中必在一平面内绕核运动

用哈密顿正则方程讨论电子在核的库仑场中的运动，是《理论力学教程》分析力学部份的一个重要例子．本文提出了一种较为合理的解释．     

约束哈密顿系统在相空间中的精确不变量与绝热不变量

研究小干扰力作用下约束哈密顿系统对称性的摄动问题.建立了非保守约束哈密顿系统的正则方程,在增广相空间中研究了系统的对称性与精确不变量.基于力学系统的高阶绝热不变量的概念,给出了系统的各阶绝热不变量的形式及存在条件,并建立了绝热不变量与对称变换之间的对应关系 关键词： 约束哈密顿系统 对称性 摄动 不变量     

应用辛格式求解一维定态Schrdinger方程

将一维定态Schr dinger方程转化为等价的哈密顿正则方程 ,并采用辛格式计算了7Li2 分子的振动能级     

应用辛格式求解一维定态Schr?dinger方程

将一维定态Schrdinger方程转化为等价的哈密顿正则方程,并采用辛 格式计算了7Li2分子的振动能级。     

一个强场模型的哈密顿形式

强场中的量了系统可能激发到高激发态，构造正则方程的本征函数展开法一般不是适用。这一个１维强场模型问题用对称差商代替哈密顿算符中的偏导数构造正则方程，再用平方守恒，辛格式示数值解，结果与理论分析一致，表明提出的方法是成功的，有可能克服传统差分法求强场量了系统时间演化遇到的困难。     

相对运动能量积分

在相对运动拉格朗日方程的基础上作勒让德变换得到动力学方程的正则形式，利用正则方程讨论了作相对运动的系统存在能量积分的条件。若约束是稳定的，主动力又是稳定的保守力，则系统的哈密顿函数就是总机械能，最后给出了相对运动机械能守恒定律。     

高等教育自学考试物理专业理论力学(分析力学部分)课程考试大纲

总说明 (l)分析力学是由拉格朗日、哈密顿等人建立并完善起来的经典力学理论.它用更普遍、更一般的方法处理力学体系的运动问题,这不仅深刻揭示出宏观机械运动的统一规律,而且还揭示出力学规律与其他物理规律之间的统一性.在从经典物理向近代物理的过渡中分析力学起了重要作用. 本课程只要求考生对分析力学的基础内容,即虚功原理、拉格朗日方程、哈密顿正则方程、哈密顿原理等有所掌握或了解,从而对力学规律有更深入的认识,并能掌握运用这些理论去处理力学问题的方法,为学习后继理论物理课程提供一个较好的基础. (2)分析力学所揭示的规律具…     

Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems

This paper focuses on studying Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems. Firstly, the discrete generalized Hamiltonian canonical equations and discrete energy equation of nonholonomic Hamiltonian systems are derived from discrete Hamiltonian action. Secondly, the determining equations and structure equation of Lie symmetry of the system are obtained. Thirdly, the Lie theorems and the conservation quantities are given for the discrete nonholonomic Hamiltonian systems. Finally, an example is discussed to illustrate the application of the results.     

Solutions of Two-Mode Bosonic and Transformed Hamiltonians

We have constructed the quasi-exactly-solvable two-mode bosonic realization of SU(2). Two-mode boson Hamiltonian is defined through a differential equation which is solved by quantum Hamilton-Jacobi formalism. The squeezed states of two-mode boson systems are characterized through canonical transformation. The illustrated concept of squeezed boson systems has been applied two-mode bosonic Hamiltonian which is a squeezed one and is determined through a differential equation. This differential equation is solved and energy eigenvalues are found approximately.     

De Broglie-Bohm Interpretation of Canonical Quantum Cosmology Implies Early Stages of the Universe Dominated by Radiation

The Einstein's general relativity is formulated in the Hamiltonian form for a spatially flat, isotropic and homogeneous universe. Subsequently, we perform the canonical quantization procedure to the Hamiltonian to obtain the Wheeler-DeWitt equation. Solving the Wheeler-DeWitt equation and employing the de Broglie-Bohm interpretation to the wave function of the universe, we obtain a new version of spatially flat Friedmann equation for the early universe where the scale factor of the universe is taken to be sufficiently small.     

Canonical transformations and exact invariants for time‐dependent Hamiltonian systems

An exact invariant is derived for n‐degree‐of‐freedom non‐relativistic Hamiltonian systems with general time‐dependent potentials. To work out the invariant, an infinitesimalcanonical transformation is performed in the framework of the extended phase‐space. We apply this approach to derive the invariant for a specific class of Hamiltonian systems. For the considered class of Hamiltonian systems, the invariant is obtained equivalently performing in the extended phase‐space a finitecanonical transformation of the initially time‐dependent Hamiltonian to a time‐independent one. It is furthermore shown that the invariant can be expressed as an integral of an energy balance equation. The invariant itself contains a time‐dependent auxiliary function ξ (t) that represents a solution of a linear third‐order differential equation, referred to as the auxiliary equation. The coefficients of the auxiliary equation depend in general on the explicitly known configuration space trajectory defined by the system's time evolution. This complexity of the auxiliary equation reflects the generally involved phase‐space symmetry associated with the conserved quantity of a time‐dependent non‐linear Hamiltonian system. Our results are applied to three examples of time‐dependent damped and undamped oscillators. The known invariants for time‐dependent and time‐independent harmonic oscillators are shown to follow directly from our generalized formulation.     

Imprints of the Quantum World in Classical Mechanics

The imprints left by quantum mechanics in classical (Hamiltonian) mechanics are much more numerous than is usually believed. We show that the Schrödinger equation for a nonrelativistic spinless particle is a classical equation which is equivalent to Hamilton’s equations. Our discussion is quite general, and incorporates time-dependent systems. This gives us the opportunity of discussing the group of Hamiltonian canonical transformations which is a non-linear variant of the usual symplectic group.     

Hamiltonian Formalism of the Landau–Lifschitz Equation for a Spin Chain with an Easy Plane

With a suitable gauge transformation, the Hamiltonian formalism of the Landau–Lifschitz equation for a spin chain with an easy plane is established by standard procedure. Action-angle variables are obtained and the canonical equation is given.     

高阶微商场论中奇异拉氏量系统的量子正则对称性

给出了高阶徽商场论中奇异拉氏量系统规范生成元的构成.从相空间中Green函数的生成泛函出发,导出了约束Hamilton系统正则形式的Ward恒等式.指出该系统的量子正则方程与由Dirac猜想得到的经典正则方程不同.给出了与Chern-Simons理论等价的一个广义动力学系统的量子化.将正则Ward恒等式初步应用于该系统,不作出对正则动量的路径积分,也可导出场的传播子与正规顶角之间的某些关系.     

构造哈密尔顿系统jet辛差分格式的生成函数法

考虑哈密尔顿系统的保结构算法,在经典哈密尔顿系统的jet辛算法的基础上,给出了一般哈密尔顿系统的jet辛差分格式的定义.并利用带有变系数辛矩阵的一般哈密尔顿系统中的构造辛差分格式的生成函数法的思想,来建立由一般的反对称矩阵所确定的微分二形式与生成函数的关系,再利用哈密尔顿-雅可比方程来构造jet辛的差分格式.     

Canonical algorithms for numerical integration of charged particle motion equations

A technique for numerically integrating the equation of charged particle motion in a magnetic field is considered. It is based on the canonical transformations of the phase space in Hamiltonian mechanics. The canonical transformations make the integration process stable against counting error accumulation. The integration algorithms contain a minimum possible amount of arithmetics and can be used to design accelerators and devices of electron and ion optics.     

机电动力系统的动量依赖对称性和非Noether守恒量

研究了Lagrange-Maxwell机电动力系统的Hamilton正则方程及动量依赖对称性的定义、判据、结构方程和守恒量的形式.研究表明，结构方程中的函数ψ只需是对称群的不变量.得到求解机电动力系统守恒量的新方法，并给出了应用实例. 关键词： 机电动力系统 Lie群分析 对称性 守恒量