Therefore, it is necessary to study numerical methods which preserve the symplectic structure of the Hamiltonian system.
因此，研究保持哈密尔顿系统的辛几何结构特征的数值方法是必然的。
In this part of the thesis, we investigated the ergodic properties of Hamiltonian systems with only a few degrees of freedom.
在这一部分，我们的任务是讨论少自由度Hamiltonian系统的遍历性质。
The main work of this thesis is to study the definitions and behaviors about equilibrium, nonequilibrium and some other thermodynamical variables in ergodic few-body Hamiltonian systems and extend the investigations of some hot topics, which have attracted much attention in the traditional thermodynamical systems with huge numbers of particles, to few-body systems.
本文的主要工作是对具有各态历经性质的少体哈密顿系统的平衡态、非平衡态和一些热力学量的定义和行为进行讨论，并把传统具有大量粒子的热力学系统中一些热点问题的研究推广到少体系统中。
We simplify the Hamiltonian of the system by Choosing the frequency detuning of laser fields and the transition frequency between the internal states of the ion, and furthermore, we describe the realization of a C-NOT gate, a swap gate and a phase gate.
通过适当选取激光场与离子内部跃迁频率的失谐量，简化了系统的哈密顿量，并进一步推导出受控非门、交换门与相位门的实现方法。
Result we get the Hamiltonian expressed by Hubbard operators of the spin-ladder model.
结果我们得到了用Hubbard算符表示的自旋梯模型的哈密顿量。
In this paper, the bound polaron in Quantum-dot Quantum well structures is investigated based on the continuous medium model and effective mass approximation, the effective Hamiltonian of the phonon and the polarization potential of the impurity are obtained by the first canonical transformation, the ground state of the electron is worked out by using the variational method after the second canonical transformation, the influence of electron-phonon interaction to the ground state of a bound polaron is also analysed.
本文采用连续介质模型和有效质量近似，研究了量子点量子阱结构中的束缚极化子。通过第一次幺正变换给出了声子的有效哈密顿量，从而求出杂质的极化势，通过第二次幺正变换后，利用变分方法得到电子的基态能量，分析了电子-声子相互作用对束缚极化子基态结合能的影响。
For a system, Hamiltonian is invariant under any translation of spatial coordinates.
对于一体系，哈密算符在任何空间坐标变换下是不变的。
We provide the general method of diagonalizationing this kind of Hamiltonian as standard form.
本文就提供了一般的将这类型哈密顿量对角化成标准形式的方法。
In the second part, we have considered the polariton system described by the model effective Hamiltonian involving single mode photon field interacting with a transverse optical phonon in a more realistic form. On this basis, we have discussed the dynamical evolution of the wave function, the dynamical variables and the non-classical properties for the system in a analytic form for the first time.
第二部分将极化激元系统约化成模型单模光子—横向声子有效相互作用系统，在此基础上以解析形式首次讨论了系统的波函数、力学量和压缩态、亚泊松分布等非经典效应的动力学演化行为。
Quasi Hamiltonian system; nonlinear stochastic optimal control; robustness; robust control; parametric uncertainty; uncertain disturbance; Bouc-Wen hysteretic system; Preisach hysteretic system; minimax optimal control; stochastic stabilization; stochastic averaging method; stochastic dynamical programming principle; stochastic differential game; maximal Lyapunov exponent
国家自然科学基金；拟Hamilton系统；非线性随机最优控制；鲁棒性；鲁棒控制；参数不确定性；不确定扰动； Bouc-Wen滞迟系统； Preisach滞迟系统；极小极大最优控制；随机稳定化；随机平均法；随机动态规划原理；随机微分对策；最大Lyapunov指数
For the more complicated case of orbital degenerate ground states, however, the theoretical studies on the spin Hamiltonian parameters are quite few.
同时由于该问题的复杂性，前人工作常常忽略一些因素的贡献，故难以对基态为轨道简并态时的自旋哈密顿参量作出合理的定量的解释。
In addition, in the previous studies, some contributions to these spin Hamiltonian parameters were often ignored or not well treated due to the complicity of the problem and so many experimental results of the spin Hamiltonian parameters of orbital degenerate ground states cannot be satisfactorily explained.
由于基态为轨道简并态时的自旋哈密顿参量涉及到许多具有重要应用价值的包含过渡及稀土离子的激光材料、发光材料、非线性光学和半导体材料以及生命物质等领域，上述研究有重要的理论和应用意义。
Since the spin Hamiltonian parameters for the orbital degenerate ground states are often related to various promising systems doped with transition-metal and rare earth ions, such as laser, luminescence and non-linear optical materials and life matter, the studies on the spin Hamiltonian parameters for orbital degenerate ground states are of theoretical and practical significance.
由于基态为轨道简并态时的自旋哈密顿参量涉及到许多具有重要应用价值的包含过渡及稀土离子的激光材料、发光材料、非线性光学和半导体材料以及生命物质等领域，上述研究有重要的理论和应用意义。
Analysis of tube-in-tube structure of variable cross-section in torsion; 2. By assuming the structure to be continuous, the Hamiltonian dual system of coordination analysis is established for tube-in-tube structure s, in which the shears and flexures of the inter and the outer tubes are considered as Timoshenko beams.
在结构连续化假定的基础上，将内筒和外筒视为铁摩辛柯梁，同时考虑内筒和外筒的剪切与弯曲变形，建立了筒中筒结构协同分析的哈密顿对偶体系，导出了相应的状态空间方程，其系统矩阵具有辛矩阵的特性，可用精细积分法求该体系的高精度数值解。
In this project, the calculated formulas and programs of the spin-Hamiltonian parameters (g and A factors, zero-field splitting) for transition metal ions in crystals (i. e., 3d2 and 3d7 ions in a tetrahedral site with cubic, trigonal or tetragonal symmetry, and 3d3 and 3d8 ions in a cubic octahedral site) are established from a two-mechanism model, in which not only the contribution due to the crystal-field mechanism, but also the contribution due to the charge-transfer mechanism (which is neglected in the previous papers) are included. From there formulas, the observed spin-Hamiltonian parameters in functional materials and compound semiconductors are calculated. It is found that the relative importance of charge-transfer mechanism increases with the increasing atomic number of ligand (e. g., for Co2 in GaP, InP and GaAs) or with the increasing valence state of 3dn ion (e. g., for Cr4, Mn5 and Fe63d2 ion series in oxygen tetrahedra). So, in these cases, the contributions of charge-transfer mechanism to spin-Hamiltonian parameters can not be neglected and the two-mechanism model should be used.
有鉴于此，本项目对固体中基态为轨道单态的过渡金属离子（具体为3d2和3d7离子在立方、三角和四角对称的四面体，3d3和3d8离子在立方八面体），建立了一套既包含晶场机制，又包括以前忽略的荷移机制的能处理自旋哈密顿参量g，A因子和零场分裂的双机制理论模型，计算公式和相应的计算程序，用以解释过渡金属离子在各种功能材料和化合物半导体中的自旋哈密顿参量实验数据，发现荷移机制对自旋哈密顿量的贡献随配体原子序数（如对Co2 在GaP，InP和GaAs）或过渡金属离子价态（如对等电子的3d2离子Cr4 ，Mn5 和Fe6 在氧四面体）的增加而增加；在这些情况，荷移机制的贡献必须考虑。
The course contains four sections as follows: mathematical logic (including basic concepts of propositional logic and predicate logic, propositional calculuses and inference theories), set theory (including set algebras, relations, functions and cardinal numbers), algebraic structure (including algebraic systems, semigroups and groups, rings and fields, lattices and Boolean algebras), graph theory (including basic concepts of graph, Euler graphs and Hamiltonian graphs, trees, planar graphs and coloring graphs, some special vertex subsets and edge subsets).
本课程包含四部分内容：数理逻辑（包含命题逻辑与一阶逻辑的基本概念、等值演算以及推理理论），集合论（包含集合代数、二元关系、函数和基数），代数结构（包含代数系统、半群与群、环与域、格与布尔代数），图论（包含图的基本概念、欧拉图与哈密顿图、树、平面图及图的着色、图的某些特殊的顶点子集与边子集）。
In the study of the stochastic chaos in simple pendulum, the random Melnikov process is derived and the mean-square criterion is used to determine the threshold amplitude of the bounded noise excitation for the onset of the chaos or random chaos in the system. For the coupled simple pendulum and harmonic oscillator, the Melnikov function is used to determine the condition for the onset of chaos in the case of Hamiltonian perturbations. In the case of non-Hamiltonian perturbation, the generalized random Melnikov process and mean-square criterion are used to determine the threshold amplitude of the bounded noise for the onset of random chaos.
在随机混沌研究中，应用随机Melnikov过程的均方准则研究了单摆在有界噪声激励下发生混沌或随机混沌时有界噪声激励的临界幅值；对于耦合的单摆-谐振子系统，先用Melnikov函数研究了在哈密顿扰动下发生混沌的必要条件；然后用推广的随机Melnikov过程方法研究了在非哈密顿扰动下发生随机混沌的必要条件；用最大Lyapunov指数及Poincaré截面方法结果与上述理论结果作了对比。
To construct canonical equation for the constrained Hamiltonian systems dynamics
在最后，介绍了一般的微分/代数方程组的一些重要概念，如指标等
Wave-current-bottom interactions serving as a general scientific background; emphasis on the inducement and effects of typical ambient current systems, and the richness of changeable bottoms; investigation and analysis of mass and momentum conservation models applied extensively in offshore and coastal engineering, and clear definition of the resulting hypotheses, approximating analytical processes and presenting characteristics; using the view and method of Hamiltonian dynamics and treating multiple-scale perturbation method as a main means, higher order evolution unified equation and generalized mild-slope equations thus derived; re-realization, extension and description of essence of basic concepts and laws such as mean currents, wave action and its conservation law, et al; determination and interpretation of nonlinear behavior on coastal dynamical field.
当今国际著名海岸流体力学家 Liu和Dingemans，曾提出一种弱非线性水波系统理论，即Liu-Dingemans三阶演化方程，在国际上颇具影响。经过仔细研讨、详细的推算和演证，发现存在若干个典型错误[1]，集中表现在概念表述和推导运算上，由此涉及到一大批方程，尤其是最后建立的三阶演化方程，在表述上的不确切性。对此，给予了明确指出、并作了修正，从而得到了一新三阶演化方程，在国际上产生积极影响。得到了Liu和Dingemans的承认，并关注着我们相关的系列工作[11，13，16，17]。
Where is a single-particle Hamiltonian of noninteracting fermions.
其中，是非作用费米子的单粒子哈密顿量。