Integrable and nonintegrable classical spin clusters

This study investigates the nonlinear dynamics of a pair of exchange-coupled spins with biaxial exchange and single-site anisotropy. It represents a Hamiltonian system with 2 degrees of freedom for which we have already established the (nontrivial) integrability criteria and constructed the integrals of the motion provided they exist. Here we present a comparative study of the phase-space trajectories for two specific models with the same symmetry properties, one of which (the XY model with exchange anisotropy) is integrable, and the other (the XY model with single-site anisotropy) nonintegrable. In the integrable model, the integrals of the motion (analytic invariants) can be reconstructed numerically by means of time averages of dynamical variables over all trajectories. In the nonintegrable model, such time averages over trajectories define nonanalytic invariants, where the nonanalyticities are associated with the presence of chaotic trajectories. A prominent feature in the nonintegrable model is the occurrence of very long time scales caused by the presence of low-flux cantori, which form sticky coats on the boundary between chaotic regions and regular islands or leaky walls between different chaotic regions. These cantori dominate the convergence properties of time averages and presumably determine the long-time asymptotic properties of dynamic correlation functions. Finally, we present a special class of integrable systems containing arbitrarily many spins coupled by general biaxial exchange anisotropy.     

Bound-bound state quantum wave packet dynamics in the intermediate coupling range: the A1Sigma+u (0+u) and b3IIu (0+u) system in Rb2

The real-time quantum dynamics of a wave packet confined to two coupled bound states at intermediate interaction strength is probed by experiment and calculations on the A1Sigma(+)(u) approximately b (3)II(u) system in Rb2 molecules. The general dynamics consists of rapid spreading out over the whole phase space. Particular conditions exist, however, under which the initial wave packet motion separates into two mesobatic trajectories with distinct frequencies. These diabatic or adiabatic hybrid trajectories are analogous to those responsible for longevity resonances in bound-unbound systems.     

The effects of self-fields on the electron trajectory in a two-stream free electron laser with a helical wiggler and an axial guiding magnetic field

A theory for the two-stream free-electron laser(TSFEL) with a helical wiggler and an axial guide magnetic field is developed.In the analysis,the effects of self-fields are taken into account.An analysis of the two-stream steady-state electron trajectories is given by solving the equation of motion.Numerical calculations show that there are seven groups of orbits in the presence of self-fields instead of two groups reported in the absence of self-fields.The stability of the trajectories is studied numerically.     

Triangular dynamics under pressure

Three planar classical particles interact via a potential proportional to the area of the triangle they form. This system is equivalent to two oscillators attached to the origin, the nearest being repelled by and the other being attracted to it (piecewise integrable Hamiltonian). Numerical simulations show two types of trajectories: those apparently escaping to infinity, and those in confined quasiperiodic orbits. Adiabatic theories lead to discrete recurrence relations and allow for the second type only. A general method allowing prediction of first return time of the slow motion as well as a short/long-period relation is presented. The issue of the possibly metastable nature of escaping trajectories is raised.Dedicated to Prof. Philippe Choquard.     

Critical behaviour in deterministic motion in a random environment

A numerical study of deterministic motion in a random environment in two dimensions is performed. All trajectories are localised except at two isolated points in the parameter space. The distribution of trajectory lengths shows “critical” behaviour as those points are approached. Contact with an analytic estimate, based on analogy with gelation theory, is made in one region of the parameter space. The spatial extent of the trajectories scales with arc length as if they were ideal random walks, surprisingly.     

Classical irreversibility and mapping of the isosceles triangle billiard

Mapping of the two-dimensional isosceles triangle billiard onto the circular one-dimensional motion of two mass points is described. The singular nature of trajectories directly incident on an acute vertex is discussed in the framework of the present mapping. For an obtuse-angled isosceles triangle, dynamical equations in two-particle space applied to an orbit along a hypotenuse incident on the obtuse vertex suggests irreversible behavior at the critical angle =2/3. Thus it is found that the nonsingular motion of a finite smooth-walled disk on this trajectory exhibits irreversibility. A finite spherical smooth-walled particle moving in a uniform right cylinder whose cross section includes this critical vertex angle likewise exhibits irreversibility. Each such example comprises an irreversible orbit for a single-particle Hamiltonian.     

Modeling and experimental validation of hydrodynamics in an ultrasonic batch reactor

Simulation of hydrodynamics in ultrasonic batch reactor containing immobilized enzymes as catalyst is done. A transducer with variable power and constant frequency (24 kHz) is taken as source of ultrasound (US). Simulation comprises two steps. In first step, acoustic pressure field is simulated and in second step effect of this field on particle trajectories is simulated. Simulation results are compared with experimentally determined particle trajectories using PIV Lab (particle image velocimetry). Effect of varying ultrasonic power, positioning and number of ultrasonic sources on particle trajectories is studied. It is observed that catalyst particles tend to orientate according to pattern of acoustic pressure field. An increase in ultrasonic power increases particle velocity and also brings more particles into motion. Simulation results are found to be in agreement with experimentally determined data.     

Paul阱中共线三离子体系的经典动力学

研究了在Paul阱囚禁场赝势作用下共线构形的三离子体系经典动力学特性.尽管这是一个非线性体系，但不存在混沌，即体系在任何能量下运动都是规则的，而相空间则由两个轨迹为对称和反对称周期(或准周期)轨道的KAM不变环面构成.体系的两条最简单的周期轨道S和A的周期随能量E的下降而增大，并在E趋于体系的最小值E_min=3.0时分别为反对称和对称谐振动. 关键词：     

Effects of self-fields on electron trajectory and gain in planar wiggler free-electron lasers with two-stream and ion-channel guiding

The effects of self-fields on electron trajectories and gain in planar wiggler free-electron lasers with two-stream and ion-channel guiding are investigated. An analysis of the two-stream quasi-steady-state electron trajectories is given by solving the equation of motion in the presence of ion-channel guiding and the planar wiggler. The electron trajectories and the gain are derived. The stability of the trajectories, the characteristics of the linear gain, and the normalized maximum gain are studied numerically. The numerical calculations show that there are eight group trajectories rather than the two groups reported in the absence of the self-fields. It is also shown that the normalized gain group seven （G7） decreases while the rest increases with the increase in normalized ion-channel frequency. The two-stream instability and the self-field lead to a decrease in the maximum gain, except for G7.     

On the absence of stable periodic orbits in domains of separatrix crossings in nonsymmetric slow-fast Hamiltonian systems

We consider a two degree of freedom Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. We assume that at frozen values of the slow variables there is a separatrix on the phase plane of the fast variables and there is a region in the phase space (the domain of separatrix crossings) where projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under rather general conditions, we prove that there are no stable periodic trajectories of any prescribed period inside the domain of separatrix crossings, except maybe for periodic trajectories passing anomalously close to the saddle point.     

Classification of symmetric periodic trajectories in ellipsoidal billiards

We classify nonsingular symmetric periodic trajectories (SPTs) of billiards inside ellipsoids of R(n+1) without any symmetry of revolution. SPTs are defined as periodic trajectories passing through some symmetry set. We prove that there are exactly 2(2n)(2(n+1)-1) classes of such trajectories. We have implemented an algorithm to find minimal SPTs of each of the 12 classes in the 2D case (R(2)) and each of the 112 classes in the 3D case (R(3)). They have periods 3, 4, or 6 in the 2D case and 4, 5, 6, 8, or 10 in the 3D case. We display a selection of 3D minimal SPTs. Some of them have properties that cannot take place in the 2D case.     

Theoretical investigation of the possibility of isotope separation during motion of charged particles in the electromagnetic field of a cylindrical capacitor and a linear current flowing along its axis

A characteristic feature of the trajectories of charges moving in constant axisymmetric radial electric and azimuthal magnetic fields, whose strengths are inversely proportional to the center from the symmetry axis is the exponential dependence of the turning points on the parameters of motion. This leads to a noticeable difference in the trajectories for isotope ions, which makes it possible to obtain a new method for their electromagnetic separation. The trajectories of isotopes being separated are studied theoretically. The conditions under which the trajectories are closed and form toroidal surfaces (storage vortex rings) have been determined earlier. These results are given for convenience in analyzing another problem, associated with the formation of such ionic toroidal vortex surfaces (stable in Wood’s sense) during a streak lightning discharge in the atmosphere (ball lightning model).     

利用高阶拉盖尔-高斯横模精确测量法布里-珀罗腔内原子的运动轨迹

研究了冷原子与法布里-珀罗腔内拉盖尔-高斯横模强耦合相互作用体系的透射光谱, 分析了透射光谱与原子在腔中运动轨迹的关系. 结果表明, 与厄米特-高斯横模相比, 拉盖尔-高斯横模的腔场与原子的最大耦合系数几乎不随阶数的增加而变化, 使得探测光谱的对比度受模式阶数的影响较小. 在拉盖尔-高斯横模场分布的圆环边缘附近, 原子运动轨迹的微小偏移会引起透射光谱的很大变化, 因此在这些位置可以实现原子运动轨迹的高精度探测.     

Measuring topological chaos

The orbits of fluid particles in two dimensions effectively act as topological obstacles to material lines. A spacetime plot of the orbits of such particles can be regarded as a braid whose properties reflect the underlying dynamics. For a chaotic flow, the braid generated by the motion of three or more fluid particles is computed. A "braiding exponent" is then defined to characterize the complexity of the braid. This exponent is proportional to the usual Lyapunov exponent of the flow, associated with separation of nearby trajectories. Measuring chaos in this manner has several advantages, especially from the experimental viewpoint, since neither nearby trajectories nor derivatives of the velocity field are needed.     

Relativistic precession of a gyroscope II. Ballistic motion

Schiff's effect is examined within the framework of the monadic method for defining the reference frame for the motion of a rotating body along ballistic trajectories in a Schwarzschild field. A general equation is obtained for calculating the precession angle in a comoving coordinate system. Examples of the motion of a gyroscope along parabolic, elliptic, and hyperbolic orbits are examined.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 44–47, April, 1981.The author is deeply grateful to the participants of the small gravitational seminar of the Gravitation Section of Minvuza SSSR for useful discussions.     

Regular and chaotic motions in applied dynamics of a rigid body

Periodic and regular motions, having a predictable functioning mode, play an important role in many problems of dynamics. The achievements of mathematics and mechanics (beginning with Poincare) have made it possible to establish that such motion modes, generally speaking, are local and form "islands" of regularity in a "chaotic sea" of essentially unpredictable trajectories. The development of computer techniques together with theoretical investigations makes it possible to study the global structure of the phase space of many problems having applied significance. A review of a number of such problems, considered by the authors in the past four or five years, is given in this paper. These include orientation and rotation problems of artificial and natural celestial bodies and the problem of controlling the motion of a locomotion robot. The structure of phase space is investigated for these problems. The phase trajectories of the motion are constructed by a numerical implementation of the Poincare point map method. Distinctions are made between regular (or resonance), quasiregular (or conditionally periodic), and chaotic trajectories. The evolution of the phase picture as the parameters are varied is investigated. A large number of "phase portraits" gives a notion of the arrangement and size of the stability islands in the "sea" of chaotic motions, about the appearance and disappearance of these islands as the parameters are varied, etc. (c) 1996 American Institute of Physics.     

Bipolarons in the intermediate coupling method

The properties of polarons and bipolarons are studied by the intermediate coupling method, which takes electron-lattice coupling into account. A Coulomb correlation in the electron motion is included. When the Fröhlich coupling constant is reduced, at α*=5.7 a bipolaron decays suddenly from a self-trapped state into two delocalized polarons. A phase diagram is constructed for the region where the stable bipolaron exists. These results are compared with those obtained by integration along trajectories.     

Flow properties of fast three-dimensional sequences for MR angiography

To reduce the scan time of time of flight or phase contrast angiography sequences, fast three-dimensional k-space trajectories can be employed. The best 3D trajectory depends on tolerable scan time, readout time, geometric flexibility, flow/motion properties and others. A formalism for flow/motion sensitivity comparison based on the velocity k-space behavior is presented. It consists in finding the velocity k-space position as a function of the spatial k-space position. The trajectories are compared graphically by their velocity k-space maps, with simulations and with an objective computed index. The flow/motion properties of various 3D trajectories (cones, spiral-pr hybrid, spherical stack of spirals, 3DFT, 3D echo-planar, and shells) were determined. In terms of flow/motion sensitivity the cones trajectory is the best, however, it is difficult to use it for anisotropic resolutions or fields of view. Tolerating more flow sensitivity, the stack of spirals trajectory offers more geometric flexibility.