边界不相关可积系统中无穷多运动积分的研究(Ⅰ)

借助于经典可积系统中的零曲率条件,得到了边界不相关条件下二维可积系统的运动积分生成函数,及其边界K矩阵的求解方程;而可积边界条件将由K_±矩阵的求解过程中得出.本文给出的边界不相关可积系统的哈密顿表述,可用于比E.KSklyanin方式更广的范围.     

均匀恒定磁场中带电狄拉克粒子的超对称性和本征能谱

基于狄拉克方程中γ矩阵有结构、可分解的观点, 把γ矩阵分解为自旋空间和正反粒子空间的算子的直积, 确定了均匀恒定磁场中带电狄拉克粒子的哈密顿量的动力学超对称性和可互易的完备的物理量算子集及其量子数集, 求得了用上述量子数完全集标志的该哈密顿量的解析的本征解,讨论了系统的哈密顿H的动力学超对称性中自旋对称性和正反粒子对称性破缺的不同情况,确定了自旋剩余超对称性导致的自旋简并子空间的超对称性变换群算子.     

用U(4)群研究氘代乙炔分子的振动激发态能谱

氘代乙炔分子的动力学对称群是三个U(4)群的直积群,找到一个适当群链,其中各子群的卡塞米尔算子组成分子的哈密顿,组合系数由该分子的实验光谱数据确定.利用这个代数哈密顿计算的能谱与实验值相符.实验上尚未观测的光谱也可以计算出来.     

哈密顿Ermakov系统的形式不变性

把形式不变性的方法用于研究哈密顿Ermakov系统，从哈密顿Ermakov系统的形式不变性出发，运用比较系数法得到与形式不变性相应的点对称变换生成元的表达式及势能所满足的偏微分方程.结果表明，在点对称变换下，只有自治的哈密顿Ermakov系统才具有形式不变性. 关键词： 哈密顿Ermakov系统 拉格朗日函数 点对称变换 形式不变性     

用U（4）群研究氘代乙炔分子的振动激发态能谱

氘代乙炔分子的动力学对称群是三个U(4)群的直积群，找到一个适当群链，其中各子群的卡塞米尔算子组成分子的哈密顿，组合系数由该分子的实验光谱数据确定。利用这个代数哈密顿计算的能谱与实验值相符。实验上尚未观测的光谱也可以计算出来。     

ZnSe: Fe2+中的动态Jahn-Teller效应和远红外光谱的研究

推导了3d^4/3d^6离子基态5D在立方晶体场，自旋－轨道耦合和动态Jahn-Teller效应作用下的哈密顿矩阵，并用对角化该哈密顿矩阵的方法研究了Fe^2+在ZnSe中的远红外光谱，理论计算与实验符合得好，研究表明，在ZnSe:Fe^2+中，比晶体场理论分析多出的分裂谱线是Fe^2+与ZnSe晶格间的动态Jahn-Teller效应引起的，还预测了其它Jahn-Teller效应分裂谱，所推导的哈密顿矩阵对研究3d^4/3d^6离子在立方晶体中的精细光谱，电子顺磁共振谱和动态Jahn-Teller分裂都是有用的。     

LC振荡电路的辛分析法

将哈密顿体系辛方法拓展到LC电路,研究LC振荡电路的辛分析法.由以电量q为变量的拉格朗日函数出发,引出对偶变量磁通链ф,将电量q与ф组成状态参量,把LC电路问题导向辛体系.对辛表述下的对偶方程利用分离变量法进行求解,问题转化成了辛本征问题.只要求出系统的哈密顿矩阵H,便可通过求解相应的本征值方程得到LC电路的振荡规律.算例验证了本文方法的有效性和正确性.将LC电路问题导入辛体系,为LC电路提供了一种新的分析方法.     

ZnSe∶Fe~(2+)中的动态Jahn-Teller效应和远红外光谱的研究

推导了 3d4 / 3d6离子基态 5D在立方晶体场、自旋 轨道耦合和动态Jahn Teller效应作用下的哈密顿矩阵 ,并用对角化该哈密顿矩阵的方法研究了Fe2 + 在ZnSe中的远红外光谱 ,理论计算与实验符合得好 .研究表明 ,在ZnSe∶Fe2 + 中 ,比晶体场理论分析多出的分裂谱线是Fe2 + 与ZnSe晶格间的动态Jahn Teller效应引起的 .还预测了其它Jahn Teller效应分裂谱 .所推导的哈密顿矩阵对研究 3d4 / 3d6离子在立方晶体中的精细光谱、电子顺磁共振谱和动态Jahn Teller分裂都是有用的 .     

非周期系统束流包络分析

本文分析了非周期交变聚焦系统的束流包络。直接求解包络函数β的非线性方程得到了β函数的精确分析表达式。利用这些分析表达式,讨论了束流包络极大值和极小值存在的判别条件,导出了计算包络极大值和极小值及其位置的精确公式。这是不同于常用的矩阵计算方法的另一种方法。二者可以相互校核。     

一个一维周期驱动哈密顿系统的实例及混沌控制

提出一种新的周期驱动非线性不可积哈密顿系统模型,并对其特性进行了讨论.通过简单的非反馈控制装置对这一系统进行混沌控制,将其混沌轨道分别控制在周期,准周期及指定混沌轨道上.与以往的控制方法不同的是,控制项仅是一结构简单、可调节的限位装置.为保守系统混沌控制的实际应用提供可供选择的途径     

Completely Integrable Hamiltonian Systems Generated by Poisson Structures in R^3

The completely integrable Hamiltonian systems have been applied to physics and mechanics intensively. We generate a family of completely integrable Hamiltonian systems from some kinds of exact Poisson structures in R^3 by the realization of the Poisson algebra. Moreover, we prove that there is a Poisson algebra which cannot be realized by an exact Poisson structure.     

r-Matrix Structure for a Restricted Flow with Bargmann Constraint

This paper deals with the integrability of a finite-dimensional Hamiltonian system linked with the generalized coupled KdV hierarchy. For this purpose the associated Lax representation is presented after an elementary calculation. It is shown that the Lax representation enjoys a dynamical r-matrix formula instead of a classical one in the Poisson bracket on R2N. Consequently the resulting system is proved to be completely integrable in view of its r-matrix structure.     

r-Matrix Structure for a Restricted Flow with Bargmann Constraint

This paper deals with the integrability of a finite-dimensional Hamiltonian system linked with the generalized coupled KdV hierarchy. For this purpose the associated Lax representation is presented after an elementary calculation. It is shown that the Lax representation enjoys a dynamical r-matrix formula instead of a classical one in the Poisson bracket on R^2N. Consequently the resulting system is proved to be completely integrable in view of its r-matrix structure.     

Symplectic Foliation Structures of Non-Equilibrium Thermodynamics as Dissipation Model: Application to Metriplectic Nonlinear Lindblad Quantum Master Equation

The idea of a canonical ensemble from Gibbs has been extended by Jean-Marie Souriau for a symplectic manifold where a Lie group has a Hamiltonian action. A novel symplectic thermodynamics and information geometry known as “Lie group thermodynamics” then explains foliation structures of thermodynamics. We then infer a geometric structure for heat equation from this archetypal model, and we have discovered a pure geometric structure of entropy, which characterizes entropy in coadjoint representation as an invariant Casimir function. The coadjoint orbits form the level sets on the entropy. By using the KKS 2-form in the affine case via Souriau’s cocycle, the method also enables the Fisher metric from information geometry for Lie groups. The fact that transverse dynamics to these symplectic leaves is dissipative, whilst dynamics along these symplectic leaves characterize non-dissipative phenomenon, can be used to interpret this Lie group thermodynamics within the context of an open system out of thermodynamics equilibrium. In the following section, we will discuss the dissipative symplectic model of heat and information through the Poisson transverse structure to the symplectic leaf of coadjoint orbits, which is based on the metriplectic bracket, which guarantees conservation of energy and non-decrease of entropy. Baptiste Coquinot recently developed a new foundation theory for dissipative brackets by taking a broad perspective from non-equilibrium thermodynamics. He did this by first considering more natural variables for building the bracket used in metriplectic flow and then by presenting a methodical approach to the development of the theory. By deriving a generic dissipative bracket from fundamental thermodynamic first principles, Baptiste Coquinot demonstrates that brackets for the dissipative part are entirely natural, just as Poisson brackets for the non-dissipative part are canonical for Hamiltonian dynamics. We shall investigate how the theory of dissipative brackets introduced by Paul Dirac for limited Hamiltonian systems relates to transverse structure. We shall investigate an alternative method to the metriplectic method based on Michel Saint Germain’s PhD research on the transverse Poisson structure. We will examine an alternative method to the metriplectic method based on the transverse Poisson structure, which Michel Saint-Germain studied for his PhD and was motivated by the key works of Fokko du Cloux. In continuation of Saint-Germain’s works, Hervé Sabourin highlights the, for transverse Poisson structures, polynomial nature to nilpotent adjoint orbits and demonstrated that the Casimir functions of the transverse Poisson structure that result from restriction to the Lie–Poisson structure transverse slice are Casimir functions independent of the transverse Poisson structure. He also demonstrated that, on the transverse slice, two polynomial Poisson structures to the symplectic leaf appear that have Casimir functions. The dissipative equation introduced by Lindblad, from the Hamiltonian Liouville equation operating on the quantum density matrix, will be applied to illustrate these previous models. For the Lindblad operator, the dissipative component has been described as the relative entropy gradient and the maximum entropy principle by Öttinger. It has been observed then that the Lindblad equation is a linear approximation of the metriplectic equation.     

An Approach to Loop Quantum Cosmology Through Integrable Discrete Heisenberg Spin Chains

The quantum evolution equation of Loop Quantum Cosmology (LQC)—the quantum Hamiltonian constraint—is a difference equation. We relate the LQC constraint equation in vacuum Bianchi I separable (locally rotationally symmetric) models with an integrable differential-difference nonlinear Schrödinger type equation, which in turn is known to be associated with integrable, discrete Heisenberg spin chain models in condensed matter physics. We illustrate the similarity between both systems with a simple constraint in the linear regime.     

Hamiltonian systems on fibered manifolds

We offer a new geometric theory of Hamiltonian systems with an infinite number of degrees of freedom in which the Hamiltonian operators are nonlinear differential operators on fields. The Poisson bracket is carried into the vertical bracket by the mapping between functionals and Hamitonian operators which is established by a Hamiltonian structure.     

Incomplete Dirac reduction of constrained Hamiltonian systems

First-class constraints constitute a potential obstacle to the computation of a Poisson bracket in Dirac’s theory of constrained Hamiltonian systems. Using the pseudoinverse instead of the inverse of the matrix defined by the Poisson brackets between the constraints, we show that a Dirac–Poisson bracket can be constructed, even if it corresponds to an incomplete reduction of the original Hamiltonian system. The uniqueness of Dirac brackets is discussed. The relevance of this procedure for infinite dimensional Hamiltonian systems is exemplified.     

The Hamiltonian description of incompressible fluid ellipsoids

We construct the noncanonical Poisson bracket associated with the phase space of first order moments of the velocity field and quadratic moments of the density of a fluid with a free-boundary, constrained by the condition of incompressibility. Two methods are used to obtain the bracket, both based on Dirac’s procedure for incorporating constraints. First, the Poisson bracket of moments of the unconstrained Euler equations is used to construct a Dirac bracket, with Casimir invariants corresponding to volume preservation and incompressibility. Second, the Dirac procedure is applied directly to the continuum, noncanonical Poisson bracket that describes the compressible Euler equations, and the moment reduction is applied to this bracket. When the Hamiltonian can be expressed exactly in terms of these moments, a closure is achieved and the resulting finite-dimensional Hamiltonian system provides exact solutions of Euler’s equations. This is shown to be the case for the classical, incompressible Riemann ellipsoids, which have velocities that vary linearly with position and have constant density within an ellipsoidal boundary. The incompressible, noncanonical Poisson bracket differs from its counterpart for the compressible case in that it is not of Lie-Poisson form.     

Vlasov moments, integrable systems and singular solutions

The Vlasov equation governs the evolution of the single-particle probability distribution function (PDF) for a system of particles interacting without dissipation. Its singular solutions correspond to the individual particle motions. The operation of taking the moments of the Vlasov equation is a Poisson map. The resulting Lie-Poisson Hamiltonian dynamics of the Vlasov moments is found to be integrable is several cases. For example, the dynamics for coasting beams in particle accelerators is associated by a hodograph transformation to the known integrable Benney shallow-water equation. After setting the context, the Letter focuses on geodesic Vlasov moment equations. Continuum closures of these equations at two different orders are found to be integrable systems whose singular solutions characterize the geodesic motion of the individual particles.