On General Volume-Preserving Mechanical Systems via Cohomology

We present the general form of equations that generate a volume-preserving flow on a symplectic manifold (M,ω) via the highest Euler-Lagrange cohomology.It is shown that for every volume-preserving flow there are some 2-forms that play a similar role to the Hamiltonian in the Hamilton mechanics and the ordinary canonical equations with Hamiltonian H are included as a special case with a 2-form Hω/(n-1).     

General Volume-Preserving Mechanical Systems

We present the general form of equations that generate a volume-preserving flow on a symplectic manifold M, ) via the highest Euler–Lagrange cohomology. It is shown that for every volume-preserving flow there are some 2-forms that play a similar role to the Hamiltonian in Hamilton mechanics. The ordinary canonical equations are included as a special case with a 2-form 1/(n - 1)H, where H is the corresponding Hamiltonian.     

Coherent NMR Stark spectroscopy

We demonstrate phase-coherent Stark effects from a radiofrequency E field at twice the NMR frequency (2ω(0)) of (69)Ga in GaAs. The 2ω(0) phase (?(E)) selects component responses from the nuclear quadrupole Hamiltonian (H(Q)). This is possible by synchronizing few-μs 2ω(0) pulses with an NMR line-narrowing sequence, which averages the Stark interaction to dominate spectra on a background with 10(3)× enhanced resolution. Spectra vs ?(E) reveal relative sizes of tensorial factors in H(Q). Comparative modeling and numerical simulations evaluate spectral features unexplained by average Hamiltonian theory, and suggest improvements for quantitative calibration of individual response components. Application of this approach to bulk samples is of value to define Stark responses that may later be used to interrogate the internal electrostatics of structured samples.     

应用哈密顿公式研究傍轴光学系统

论述用Ｌ定义光学的哈密顿Ｈ，并给出哈密顿公式的基本方程式，利用哈密顿方程式，在旋转对称的光学系统里追迹光线，并利用哈密顿公式得到折射面和透镜的简单结果。最后通过举例说明这个方法是可行的。     

One- and Two-Mode Combination Squeezing Operator for Two Harmonic Oscillators with Coordinate-Momentum Coupling

By virtue of the technique of integration within an ordered product of operators, we derive the normal ordering expansion of a one- and two-mode combination squeezing operator for two harmonic oscillators with coordinate- momentum coupling. It turns out that this squeezing operator just diagonalizes the Hamiltonian H=p^21/2m1＋m1ω^21x^21/2＋p^222m2＋m2ω^22x^22/2-λx2p1 so its ground state is a one- and two-mode combination squeezed state. Quantum fluctuation in the ground state is calculated.     

Generic Dynamics of 4-Dimensional C 2 Hamiltonian Systems

We study the dynamical behaviour of Hamiltonian flows defined on 4-dimensional compact symplectic manifolds. We find the existence of a C ²-residual set of Hamiltonians for which there is an open mod 0 dense set of regular energy surfaces each either being Anosov or having zero Lyapunov exponents almost everywhere. This is in the spirit of the Bochi-Mañé dichotomy for area-preserving diffeomorphisms on compact surfaces [2] and its continuous-time version for 3-dimensional volume-preserving flows [1].     

Feldgleichungen der TREDERschen Gravitationstheorie,die aus einem Variationsprinzip ableitbar sind. I

In the frame work of TREDER 's gravitational theory we consider two classes of field equations which are derivable from two classes of LAGRANGE ian densities Ω⁽¹⁾(ω₁, ω₂), Ω⁽²⁾(s?₁, s?₂). ω₁, ω₂; s?₁, s?₂ are parameters. Ω⁽²⁾(ω₁, ω₂) gives us field equations which are up to the post-NEWTON ian approximation in the sense of NORDTVEDT , THORNE and WILL equivalent to the field equations given by BRANS and DICKE . For ω₂ = ?1 ?2ω₁ field equations follow from Ω⁽¹⁾(ω₁, ?1 ?2ω₁) which are in the above mentioned sense of post-NEWTON ian approximation equivalent to EINSTEIN 's equations. The field equations following from Ω⁽¹⁾(ω₁, ω₂) have a cosmological model with the well known cosmological singularities for T → ± ∞ in case that ω₁/(1 +3ω₁ +ω₂) ? γ > 0. For ω₁/(1 +3ω₁ +ω₂) ≤ 0 cosmological models with no cosmological singularities exist. From Ω⁽²⁾(s?₁, s?₂) we obtain field equations which at the best give us perihelion rotation 7% above EINSTEIN 's value and light deflection 7% below the corresponding EINSTEIN 's value. But in that case we are able to show the existence of a cosmological model without any cosmological singularity.     

双色激光场中一维氢分子的经典动力学性质研究

应用经典理论方法,采用辛算法,数值求解双色激光场作用下的一维共线氢分子的哈密顿正则方程,得到氢分子在激光场作用下的经典轨迹,并计算H₂⁺,H₂²⁺,H,H⁺等产物的产生概率随时间的演化,分析了双色场的光强、倍频、相位的变化对氢分子动力学性质的影响并给出相关的物理解释.     

Approach to the quantum evolution for underdamped, critically damped, and overdamped driven harmonic oscillators using unitary transformation

Exact solution of the Schrödinger equation is derived for underdamped, critically damped, and overdamped harmonic oscillators with a driving force. A unitary operator transforming Hamiltonian into a simple form is introduced. The transformed Hamiltonian, represented in terms of a modified frequency ω, is identical with the Hamiltonian of the standard harmonic oscillator for the underdamped oscillator, with the Hamiltonian of a free particle for the critically damped oscillator, and with the Hamiltonian of a system with a harmonic parabolic potential for the overdamped oscillator. The eigenvalues of underdamped oscillator are discrete while those of the critically damped and the overdamped oscillators are continuous.     

Effects on squeezing and sub-poissonian of light in fourth harmonic generation up to first-order Hamiltonian interaction

The effects on squeezing and sub-poissonian of light in fourth harmonic generation (FHG) are investigated based on the fully quantum mechanically up to the first order Hamiltonian interaction in gt, where g is the coupling constant between the modes per second and t is the interaction time between the waves during the process in a nonlinear medium. FHG is a process in which an incident laser beam of the fundamental frequency ω interacts with a nonlinear medium to produce the harmonic frequency at 4ω. The coupled Heisenberg equations of motion involving real and imaginary parts of the quadrature operators are established. The occurrence of amplitude squeezing effects in both the quadratures of the radiation field in the fundamental mode is investigated and found to be dependent on the selective phase values of the field amplitude. The photon statistics of the pump mode in this process have also been investigated and found to be sub-poissonian in nature. It is found that there is no possibility to produce squeezed light in the harmonic mode up to first-order interaction in gt. Further, we have found the case up to second-order Hamiltonian interaction in gt that the normal squeezing in the harmonic mode is directly depends upon the fourth-order squeezing of the initial pump field. This gives a method of converting higher-order (fourth-order) squeezing into normal squeezing in the harmonic mode and vice versa.     

一种精确求解三原子反应散射的传播子

The exact short time propagator, in a form similar to the Crank-Nicholson method but in the spirit of spectrally transformed Hamiltonian, was proposed to solve the triatomic reactive time-dependent schrödinger equation. This new propagator is exact and unconditionally convergent for calculating reactive scattering processes with large time step sizes. In order to improve the computational efficiency, the spectral difference method was applied. This resulted the Hamiltonian with elements confined in a narrow diagonal band. In contrast to our previous theoretical work, the discrete variable representation was applied and resulted in full Hamiltonian matrix. As examples, the collision energy-dependent probability of the triatomic H+H₂ and O+O₂ reaction are calculated. The numerical results demonstrate that this new propagator is numerically accurate and capable of propagating the wave packet with large time steps. However, the efficiency and accuracy of this new propagator strongly depend on the mathematical method for solving the involved linear equations and the choice of preconditioner.     

Coupled KdV Equations and Their Explicit Solutions Through Two-Dimensional Hamiltonian System with a Quartic Potential

Based on the second integrable ease of known two-dimensional Hamiltonian system with a quartie potentiM, we propose a 4 × 4 matrix speetrM problem and derive a hierarchy of coupled KdV equations and their Hamiltonian structures. It is shown that solutions of the coupled KdV equations in the hierarchy are reduced to solving two compatible systems of ordinary differentiM equations. As an application, quite a few explicit solutions of the coupled KdV equations are obtained via using separability for the second integrable ease of the two-dimensional Hamiltonian system.     

金刚石晶格上的对角无序与非对角无序非晶量子点

报道金刚石晶格上对角无序与非对角无序非晶量子点的理论研究，用简单的紧束缚哈密顿量描述模型的电子结构，用ｒｅｃｕｒｓｉｏｎ方法求解哈密顿方程，用边界条件对本征值的影响判断局域化，研究发现，带边为扩展态时，带宽的变化趋势与晶态量子点类似；带边为局域态时，尺寸超过某一临界长度后，带宽不变，但带边态密度随尺寸增大而增大。还研究了非晶量子点的介电函数虚部ε₂（?ω）。与扩展态对应的ε₂（?ω），对尺寸变化较敏感。与局域态对应的ε₂（?ω），当尺寸大于 关键词：     

Symmetries and regular behavior of Hamiltonian systems

The behavior of the phase trajectories of the Hamilton equations is commonly classified as regular and chaotic. Regularity is usually related to the condition for complete integrability, i.e., a Hamiltonian system with n degrees of freedom has n independent integrals in involution. If at the same time the simultaneous integral manifolds are compact, the solutions of the Hamilton equations are quasiperiodic. In particular, the entropy of the Hamiltonian phase flow of a completely integrable system is zero. It is found that there is a broader class of Hamiltonian systems that do not show signs of chaotic behavior. These are systems that allow n commuting "Lagrangian" vector fields, i.e., the symplectic 2-form on each pair of such fields is zero. They include, in particular, Hamiltonian systems with multivalued integrals. (c) 1996 American Institute of Physics.     

Higher Harmonics and Transport Coefficients of Plasmas in Circulary Polarized Magnetic Fields and Additional Electromagnetic Fields

With the methods of kinetic theory on the basis of the Boltzmann and Fokker-Planck kinetic equations the behaviour of a Lorentz plasma in a circularly polarized (rotating) magnetic field (rotation frequency ω), an alternating electric field (frequency ω′) and additional constant electromagnetic fields is investigated. By means of a generalized Fourier expansion it is shown that the above fields create in the plasma currents of the frequencies 0, ω′, ω–ω′,ω, ω+ω′, 2ω–ω′, 2ω, and 2ω+ω′. Transport coefficients are calculated explicitly and the validity of Onsager reciprocity relations and that of Kronig-Kramers relations is discussed. The special case of the electric field induced by the rotating magnetic field is treated separately. Finally, problems of plasma containment are discussed.     

Coupled scalar field equations for nonlinear wave modulations in dispersive media

A review of the generic features as well as the exact analytical solutions of a class of coupled scalar field equations governing nonlinear wave modulations in dispersive media like plasmas is presented. The equations are derivable from a Hamiltonian function which, in most cases, has the unusual property that the associated kinetic energy is not positive definite. To start with, a simplified derivation of the nonlinear Schrödinger equation for the coupling of an amplitude modulated high-frequency wave to a suitable low-frequency wave is discussed. Coupled sets of time-evolution equations like the Zakharov system, the Schrödinger-Boussinesq system and the Schrödinger-Korteweg-de Vries system are then introduced. For stationary propagation of the coupled waves, the latter two systems yield a generic system of a pair of coupled, ordinary differential equations with many free parameters. Different classes of exact analytical solutions of the generic system of equations are then reviewed. A comparison between the various sets of governing equations as well as between their exact analytical solutions is presented. Parameter regimes for the existence of different types of localized solutions are also discussed. The generic system of equations has a Hamiltonian structure, and is closely related to the well-known Hénon-Heiles system which has been extensively studied in the field of nonlinear dynamics. In fact, the associated generic Hamiltonian is identically the same as the generalized Hénon-Heiles Hamiltonian for the case of coupled waves in a magnetized plasma with negative group dispersion. When the group dispersion is positive, there exists a novel Hamiltonian which is structurally same as the generalized Hénon-Heiles Hamiltonian but with indefinite kinetic energy. The above correspondence between the two systems has been exploited to obtain the parameter regimes for the complete integrability of the coupled waves. There exists a direct one-to-one correspondence between the known integrable cases of the generic Hamiltonian and the stationary Hamiltonian flows associated with the only integrable nonlinear evolution equations (of polynomial and autonomous type) with a scale-weight of seven. The relevance of the generic system to other equations like the self-dual Yang-Mills equations, the complex Korteweg-de Vries equation and the complexified classical dynamical equations has also been discussed.     

Fractional hamilton formalism within caputo’s derivative

In this paper we develop a fractional Hamiltonian formulation for dynamic systems defined in terms of fractional Caputo derivatives. Expressions for fractional canonical momenta and fractional canonical Hamiltonian are given, and a set of fractional Hamiltonian equations are obtained. Using an example, it is shown that the canonical fractional Hamiltonian and the fractional Euler-Lagrange formulations lead to the same set of equations.     

离子与H碰撞中的电荷交换和电离

用经典轨道Monte-Carlo方法计算完全电离的离子与H原子的碰撞,通过求解系统的Hamilton方程获得碰撞过程的电荷交换截面和电离截面,并与ORNL的推荐数据作了比较,在中间能范围内,计算结果很好地与ORNL的推荐结果相符合,方法是可靠的,结果令人满意。     

Plane rotations and Hamilton—Dirac mechanics

Canonical formalism for SO(2) is developed. This group can be seen as a toy model of the Hamilton-Dirac mechanics with constraints. The Lagrangian and Hamiltonian are explicitly constructed and their physical interpretations are given. The Euler-Lagrange and Hamiltonian canonical equations coincide with the Lie equations. It is shown that the constraints satisfy CCR. Consistency of the constraints is checked. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.