Non-linear diffusion in Hamiltonian systems exhibiting chaotic motion

The exact evolution equation for the angle averaged phase space density in action-angle space is derived from the Liouville equation using projection operator techniques. This equation involves a correlation function of the initial value of the phase space density with the angle dependent part of the Hamiltonian and a correlation function of the angle dependent part of the Hamiltonian and a correlation function of the angle dependent part of the Hamiltonian with itself. Each of these correlation functions develops in time with angle projected dynamics. We show their relation to the correlation functions which develop in time with usual Hamiltonian dynamics. These correlation functions are then studied in the standard model of Chirikov, and we conclude that they behave as $\text{e}{}^{\mathrm{-\sigma t}}\text{cos}\text{(Ωt + φ)}$ in regions of irregular motion. We conjecture that angle averaged correlation functions behave this way in general, and we give an argument based on the mixing property of the Hamiltonian system. Our argument goes beyond the usual mixing, so we regard it as a quasi-mixing hypothesis. Under this hypothesis the equation for the angle averaged phase space density becomes a diffusion equation which incorporates much of the non-linear dynamics of Hamiltonian systems exhibiting chaotic motion.     

广义Hamilton系统的Mei对称性导致的Mei守恒量

研究广义Hamilton系统的Mei对称性导致的守恒量. 首先,在群的一般无限小变换下给出广义Hamilton系统的Mei对称性的定义、判据和确定方程;其次,研究系统的Mei守恒量存在的条件和形式,得到Mei对称性直接导致的Mei守恒量; 而后,进一步给出带附加项的广义Hamilton系统Mei守恒量的存在定理; 最后,研究一类新的三维广义Hamilton系统,并研究三体问题中3个涡旋的平面运动. 关键词： 广义Hamilton系统 Mei对称性 Mei守恒量 三体问题     

一维减幅-增幅谐振子的守恒量与对称性

从一维减幅-增幅谐振子的运动微分方程出发得到系统的运动积分常数,从而得到系统的Lagrange函数和Hamilton函数,再根据Hamilton函数的形式假定守恒量的形式,由Poisson括号的性质得到了系统的三个守恒量,并讨论与三个守恒量相应的无限小变换的Noether对称性与Lie对称性.还对守恒量与对称性的物理意义作了合理的解释. 关键词： 一维减幅-增幅谐振子 守恒量 Noether对称性 Lie对称性     

Chaotic ray dynamics in slowly varying two-dimensional photonic crystals

We use Hamiltonian optics to investigate chaotic ray dynamics in a photonic crystal whose lattice parameters vary slowly with position. The ray dynamics are chaotic even in regimes where only stable motion has been found in previous studies of energy band transport. Stable ray paths provide dynamical barriers that localize chaotic motion to certain regions of the photonic crystal.     

Singular densities,test particles and generalized Hamiltonian systems in general relativity

We derive the motion equations and the structure equations of neutral and charged test particles, starting from the gravitational field equations. The method consists in the application of conservation laws to singular tensor densities, which represent regions of strong matter concentration. Moreover, a Hamiltonian formulation of the particle equations is given, in the form of implicit differential equations generated by Hamiltonian Morse families.     

On the dynamics of internal waves interacting with the equatorial undercurrent

The interaction of the nonlinear internal waves with a nonuniform current with a speci?c form, characteristic for the equatorial undercurrent, is studied. The current has no vorticity in the layer, where the internal wave motion takes place. We show that the nonzero vorticity that might be occuring in other layers of the current does not affect the wave motion. The equations of motion are formulated as a Hamiltonian system.     

Three Electrons in a Two-Dimensional Parabolic Trap: The Relative Motion Solution

In agreement with the Kohn theorem the relative motion (rel) of three electrons in a two-dimensional parabolic trap separates from the centre-of-mass (CM) motion. By introducing new coordinates the Hamiltonian for relative motion in the approximation of non-interacting electrons can be taken to the normal form. The eigenstates of the normalized Hamiltonian are products of the Fock-Darwin states for normal modes. The energy levels for relative motion are obtained by diagonalizing the exact Hamiltonian in the eigenbasis for the non-interacting case. In this basis the interaction matrix elements can be obtained in the analytical form. Since the rank of the Hamiltonian matrix is significantly reduced, the calculations are faster and more accurate than those for the full (CM + rel) motion. This advantage is especially important for the calculations of excited states and the analysis of energy spectra.     

Quantum and classical dissipation of charged particles

A Hamiltonian approach is presented to study the two dimensional motion of damped electric charges in time dependent electromagnetic fields. The classical and the corresponding quantum mechanical problems are solved for particular cases using canonical transformations applied to Hamiltonians for a particle with variable mass. Green’s function is constructed and, from it, the motion of a Gaussian wave packet is studied in detail.     

Electrons in a curvilinear quantum wire

The adiabatic motion of electrons in curvilinear quantum wires was studied. It was assumed that the cross section of a wire was constant along its length. The potential that limited electron motion across a wire and the shape of the cross section of the wire were considered arbitrary, while the curvature and the torsion (defined as the derivative of the cross section rotation angle with respect to the length) were assumed to be small. An effective nonrelativistic Hamiltonian for the motion of electrons along a wire with the conservation of transverse quantum numbers was obtained. The spin-orbit coupling Hamiltonian related to the curvature and torsion of a wire was found. Particular cases of a rectilinear twisted quantum wire with a noncircular cross section and a curvilinear quantum wire on a plane were studied. Various transverse potential models limiting the motion of electrons were considered. In particular, the coefficients of the effective Hamiltonian for quantum wires with rectangular and circular cross sections and hard walls and for wires with a parabolic potential were found.     

Dynamics of homogeneous magnetizations in strong transverse driving fields

Spatially homogeneous solutions of the Landau-Lifshitz-Gilbert equation are analysed. The different cases of conservative as well as dissipative motion are considered separately. For the linearly polarized driven Hamiltonian system we apply a global perturbation theory to uncover the main resonances as well as the phase space structure. The case of circularly polarized driven dissipative motion is studied in detail. We present the complete bifurcation diagram including bifurcations up to codimension three.     

Dynamics of homogeneous magnetizations in strong transverse driving fields

Spatially homogeneous solutions of the Landau-Lifshitz-Gilbert equation are analysed. The different cases of conservative as well as dissipative motion are considered separately. For the linearly polarized driven Hamiltonian system we apply a global perturbation theory to uncover the main resonances as well as the phase space structure. The case of circularly polarized driven dissipative motion is studied in detail. We present the complete bifurcation diagram including bifurcations up to codimension three.     

Bi-Orbital States in Hyperbolic Traps

We consider a charge in a general electromagnetic trap near a hyperbolic stationary point. The two-dimensional trap Hamiltonian is the sum of a hyperbolic harmonic part and higher order anharmonic corrections. We suppose that two frequencies of the harmonic part are under the resonance 1 : (?1). In this case, anharmonic terms define the dynamics and an effective Hamiltonian on the space of motion constants of the ideal harmonic operator. We show that if the anharmonic part is symmetric, then the effective Hamiltonian has unstable equilibriums and separatrix, which define distinct classically allowed regions in the space of motion constants of the ideal trap. The corresponding stationary states of the trapped charge can form bi-orbital states, i.e., a state localized on two distinct classical trajectories. We obtain semiclassical asymptotics of the energy splitting corresponding to the charge tunneling between these two trajectories in the phase space and express it in terms of complex instantons.     

A solution of the motion of a charged particle in magnetic mirror systems

The paper solves the motion of a charged particle in an axially symmetrical, magnetostatic field, which forms magnetic mirror systems. The solution is based on the Hamiltonian for the motion of a charged particle in such a field. The whole problem is solved in orthogonal curvilinear coordinates, the coordinate curves being formed by the lines of force and the curves perpendicular to them. By applying the perturbation method of multi-periodical systems to the Hamiltonian in these coordinates, a Hamiltonian is obtained for the motion of the guiding centre which contains only the coordinate of motion along the lines of force, the others being cyclic. Thus, in addition to the energy two other first integrals of the equations of motion are obtained (since the impulses corresponding to the cyclic coordinates are constant), from which the conditions for confinement of motion in a magnetic field are immediately obtained. Since the Hamiltonian obtained in this way contains only one coordinate, the whole problem is solved by two quadratures, which define the dependence of the time and azimuthal angle on the line-of-force coordinate, while the other quantities are constant.In conclusion, the author thanks Dr. M. Seidl for valuable remarks and discussion on this work.     

Investigation of Constrained Systems with Singular Higher-Order Lagrangians

Constrained Hamiltonian systems with singular higher-order Lagrangians are investigated by using two methods: the Dirac's and the Hamilton-Jacobi methods. Three examples are studied and it is shown that the equations of motion which are obtained by these two methods are in exact agreement.     

原子间相互作用对光场和原子激光压缩性质的影响

对文献中给出的光场与二能级原子玻色-爱因斯坦凝聚体(BEC)相互作用系统的哈密顿量进行分析，表明文献中对原子间相互作用部分的处理有不合理之处，文献中的处理过高估计了原子间相互作用的贡献，从而对该哈密顿量作出了改进.用改进的哈密顿量解析地求解了非旋波近似下光子和原子算符的运动方程，并结合BEC的有关实验条件对哈密顿量中的有关参数作出了估计，研究了光场与原子玻色-爱因斯坦凝聚体相互作用系统中，光场和耦合输出相干原子束的压缩性质.结果表明：光场两正交分量的涨落均随时间按余弦规律周期性地变化，原子激光的两正交分量 关键词： 玻色-爱因斯坦凝聚 压缩相干态 光场的正交压缩 压缩原子激光     

Infinite Number of Integrals of Motion in Classically Integrable System With Boundary (Ⅱ)

In Affine Toda field theory, links among three generating functions for integrals of motion derived from P. (Ⅰ) are studied, and some classically integrable boundary conditions are obtained. An infinite number of integrals of motion are calculated in ZMS model with quasi-periodic condition. We find the classically integrable boundary conditions and K± matrices of ZMS model with independent boundary conditions on each end. It is identified that an infinite number of integrals of motion does exist and one of them is the Hamiltonian, so this system is completely integrable.     

Geometry of quadrilateral nets: Second Hamiltonian form

Discrete Darboux–Manakov–Zakharov systems possess two distinct Hamiltonian forms (by this term we mean that equations of motion are discrete time extensions of Hamiltonian equations of motion). In the framework of discrete-differential geometry one Hamiltonian form appears in a geometry of a circular net. In this paper a geometry of the second form is identified.     

Motion and gravitational radiation of a binary system consisting of an oscillating and rotating coplanar dusty disk and a point-like object

A binary system composed of an oscillating and rotating coplanar dusty disk and a point mass is considered. The conservative dynamics is treated on the Newtonian level. The effects of gravitational radiation reaction and wave emission are studied to leading quadrupole order. The related waveforms are given. The dynamical evolution of the system is determined semi-analytically exploiting the Hamiltonian equations of motion which comprise the effects both of the Newtonian tidal interaction and the radiation reaction on the motion of the binary system in elliptic orbits. Tidal resonance effects between orbital and oscillatory motions are considered in the presence of radiation damping.     

有限体系能量耗散运动的功率谱分析

研究了有限的多自由度耦合哈密顿系统能量耗散过程的动力学问题. 通过数值模拟，实现有关自由度的能量耗散过程，采用功率谱方法研究有关系统运动中的坐标q(t)或动量p(t)的演化过程. 结果发现：在有关系统的能量耗散过程中，有关自由度运动的频率成分向高频和低频方向都有明显展宽，随着无关系统自由度增多，展宽明显增加；低频成分比较稳定，高频成分随着时间的演变逐渐减少. 关键词： 有限系统 能量耗散 有关自由度 功率谱