Dynamical Symmetry,by Carl E. Wulfman

The contemporary view of stellar movements is that the motion is orbital motion in the attracting field of the galaxy as a whole. The orbits can be classified according to their eccentricity and inclination to the plane of the galaxy. Orbits of low eccentricity and inclination are characteristic of very young stars. Most stars describe orbits of moderate eccentricity and moderate inclination, but some orbits are known to occur with very high eccentricity and high inclination to the plane of the galaxy, these orbits being characteristic of very old stars, and in particular RR Lyrae Variable stars. The article sketches the technique by which velocities relative to the Sun are found and gives a brief account of the mathematics by means of which these velocities are shown to correspond to galactic orbits, and the article concludes with a sketch of the interpretation of the statistics of galactic orbits in terms of the history of the galaxy.     

Stability of circular orbits of spinning particles in Schwarzschild-like space–times

Circular orbits of spinning test particles and their stability in Schwarzschild-like backgrounds are investigated. For these space–times the equations of motion admit solutions representing circular orbits with particles spins being constant and normal to the plane of orbits. For the de Sitter background the orbits are always stable with particle velocity and momentum being co-linear along them. The world-line deviation equations for particles of the same spin-to-mass ratios are solved and the resulting deviation vectors are used to study the stability of orbits. It is shown that the orbits are stable against radial perturbations. The general criterion for stability against normal perturbations is obtained. Explicit calculations are performed in the case of the Schwarzschild space–time leading to the conclusion that the orbits are stable.     

Efficient topological chaos embedded in the blinking vortex system

We consider the particle mixing in the plane by two vortex points appearing one after the other, called the blinking vortex system. Mathematical and numerical studies of the system reveal that the chaotic particle mixing, i.e., the chaotic advection, is observed due to the homoclinic chaos, but the mixing region is restricted locally in the neighborhood of the vortex points. The present article shows that it is possible to realize a global and efficient chaotic advection in the blinking vortex system with the help of the Thurston-Nielsen theory, which classifies periodic orbits for homeomorphisms in the plane into three types: periodic, reducible, and pseudo-Anosov (pA). It is mathematically shown that periodic orbits of pA type generate a complicated dynamics, which is called topological chaos. We show that the combination of the local chaotic mixing due to the topological chaos and the dipole-like return orbits realize an efficient and global particle mixing in the blinking vortex system.     

Semiclassical interpretation of the gross-shell structure in deformed nuclei

An analysis of the classical closed orbits in the infinitely deep elongated ellipsoidal well is given with respect to their relevance to the gross-shell structure of the single-particle spectra in deformed nuclei. It is shown that the deformed-shape gross-shells responsible for the ground state deformations, as well as those responsible for the fission intermediate states, find a reasonable explanation in terms of the semiclassical three-dimensional quantization involving the planar orbits in the axis-of-symmetry plane and the simplest three-dimensional orbits which appear at large distortions. The former are found important for the ground state shapes. The results are also compared with the harmonic oscillator model.     

Surface states of a semi-infinite diamond crystal limited by the (100) plane

Two kinds of surface states with energies lying in the gap between the valence and the conductive zone, were found for a semi-infinite diamond crystal, limited by the (100) plane, using the MO LCAO method, in which the molecular orbits are expressed as linear combinations ofsp ³ hybrid orbits of carbon atoms. One of these zones of surface states is completely occupied, the other completely unoccupied. The nature of these surface states is discussed and it is shown that the (100) surface of the diamond behaves similarily as if each surface atom had a free electron pair.     

Spherical Photon Orbits Around a Kerr Black Hole

Two circular photon orbits are known to exist in the equatorial plane of the Kerr black hole. In this paper, we investigate so-called spherical photon orbits—orbits with constant coordinate radii that are not confined to the equatorial plane. A one-parameter class of solutions is found, which includes the circular orbits as special cases. The properties of these spherical orbits are then analyzed, with the aim of classifying them by qualitative differences in their behavior. Finally, representative orbits from each class are plotted out, including a zero-angular momentum photon orbit and one with non-fixed azimuthal direction.     

Chemical potential shift due to coupled orbit conduction

The hexagonal, compensated metals (e.g. Be, Mg, Zn) exhibit large transverse resistivity oscillations when a magnetic field perpendicular to the ΓKM plane coherently couples hole and electron orbits near that plane. These orbits form a narrow slab of states which may be sufficiently isolated from the remaining Fermi surface that it is advantageous for their chemical potential to change significantly in sympathy with the resistivity as a function of the current. We show how this happens and review possible experimental observation of it.     

Equatorial Plane Circular Orbits in the Taub-NUT Spacetime

Accelerated circular orbits in the equatorial plane of the Taub-NUT spacetime are analyzed to investigate the effects of its gravitomagnetic monopole source. The effect of a small gravitomagnetic monopole on these orbits is compared to the corresponding orbits pushed slightly off the equatorial plane in the absence of the monopole.     

高斯束激光摇摆场中的电子运动轨道分析

对高斯束激光摇摆场中的电子运动轨道进行了分析．推导出轴向导引磁场与反向传播的激光摇摆场作用下的单粒子电子运动轨道，并对轨道的稳定性进行了分析．发现当瑞利长度较大时，得到的结果与平面电磁波摇摆场的结果相一致．与静磁摇摆场的自由电子激光相似，在激光摇摆场中的电子轨道也存在两类分立轨道，但是，与静磁摇摆场不同的是，这两类轨道都是稳定的 关键词：     

Complex Classical Mechanics of a QES Potential

We study a combined parity(P) and time reversal(T) invariant non-Hermitian quasi-exactly solvable(QES) potential, which exhibits PT phase transition, in the complex plane classically to demonstrate different quantum effects. The particle with real energy makes closed orbits around one of the periodic wells of the complex potential depending on the initial condition. However interestingly the particle escapes to an open orbits even with real energy if it is placed beyond a certain distance from the center of the well. On the other hand when the particle energy is complex the trajectory is open and the particle tunnels back and forth between two wells which are separated by a classically forbidden path. The tunneling time is calculated for different pair of wells and is shown to vary inversely with the imaginary component of energy. Our study reveals that spontaneous PT symmetry breaking does not affect the qualitative features of the particle trajectories in the analogous complex classical model.     

Complex Classical Mechanics of a QES Potential

We study a combined parity (P) and time reversal (T) invariant non-Hermitian quasi-exactly solvable (QES) potential, which exhibits PT phase transition, in the complex plane classically to demonstrate different quantum effects. The particle with real energy makes closed orbits around one of the periodic wells of the complex potential depending on the initial condition. However interestingly the particle escapes to an open orbits even with real energy if it is placed beyond a certain distance from the center of the well. On the other hand when the particle energy is complex the trajectory is open and the particle tunnels back and forth between two wells which are separated by a classically forbidden path. The tunneling time is calculated for different pair of wells and is shown to vary inversely with the imaginary component of energy. Our study reveals that spontaneous PT symmetry breaking does not affect the qualitative features of the particle trajectories in the analogous complex classical model.     

Bistability,basins of attraction and predictability in a forced mass-reaction model

A simple mass-reaction model with periodically forced contact rate is considered. It is shown that the forced differential equation exhibits at least two distinct periodic orbits for several ranges of forcing amplitude. Basins of attraction are computed for co-existing stable periodic orbits and are shown to be intertwined in a complex manner. Dimension (capacity) of the basin is computed, and it is shown how the basin structure affects final state predictability.     

Winding number formula for Maslov indices

A formula for the Maslov index of closed curves on Lagrangian manifolds is derived. The index is expressed as the number of times the plane tangent to the curve winds around it. Applications include unstable periodic orbits, in which case the Lagrangian manifolds are the stable or unstable manifolds of the orbits, and cycles on the invariant tori of integrable systems, in which case the manifolds are the tori themselves.     

Periodic orbits in a two-variable coupled map

Periodic orbits are calculated for a linear transformation composed of two coupled tent maps using a symbolic dynamics defined as the direct product of the single-map symbols {0,1,2}. As the coupling strength is increased orbits are pruned and a crossover to one-dimensional behavior is observed. The disallowed binary orbits containing only symbols {0,1} form a connected region in a binary symbol plane. Stable orbits may appear for strong couplings.     

Homoclinic (Heteroclinic) Orbit of Complex Dynamical System and Spiral Structure

Starting from iterated systems, it is shown that the homoclinic (heteroclinic) orbit is a kind of spiral structure. The emphasis is laid to show that there are homoclinic or heteroclinic orbits in complex discrete and continuous systems, and these homoclinic or heteroclinic orbits are some kind of spiral structure.     

Magnetic levels in superlattices

A magnetic field parallel to the layers of a GaAs---GaAlAs superlattice leads to a quantisation of the subband dispersion relation. The discrete energy levels are calculated with a semiclassical quantisation scheme and it is shown that within the energy width of the subbands, closed orbits, and in the superlattice minigap open orbits, are formed. Experimentally this behaviour is observed as sharp peaks in the interband Landau level absorption for energies within the subband width (closed orbits) and the disappearance of these peaks at higher energies (where no closed orbits exist).     

Chaotic spectroscopy

The spectra of quantized chaotic billiards from the point of view of scattering theory are discussed. It is shown how the spectral and resonance density functions both fluctuate about a common mean. A semiclassical treatment explains this in terms of classical scattering trajectories and periodic orbits of the Poincare scattering map. It is shown that this formalism provides an alternative derivation and a new interpretation of Gutzwiller's periodic orbits sum for the spectral density. Moreover, it is a convenient starting point for a derivation of a Riemann-Siegel "look alike" expression for the secular equation in terms of periodic orbits of finite length.     

Ergodic and topological properties of coulombic periodic potentials

The motion of a classical pointlike particle in a two-dimensional periodic potential with negative coulombic singularities is examined. This motion is shown to be Bernoullian for many potentials and high enough energies. Then the motion on the plane is a diffusion process. All such motions are topologically conjugate and the periodic orbits can be analysed with the help of a group.This work is part of a thesis submitted to Freie Universität Berlin