Four-dimensional Phase Space Research Articles

Transverse linear dynamics in an axisymmetric ionization cooling channel

This paper outlines a formalism for the description of the linear transverse dynamics of charged particles in an axisymmetric ionization cooling channel. The particle trajectories in the absence of Coulomb scattering are described in terms of lattice functions \`a la Courant and Snyder, which depend only on the electric and magnetic fields in the channel. The process of multiple Coulomb scattering, which introduces stochastic terms into the particle equations of motion, is treated (in Gaussian approximation) by obtaining the distribution function in phase space as a solution of a Fokker-Planck equation. The distribution function is then used to obtain moment equations for the transverse variables and for combinations of variables such as the emittance and angular momentum. The distribution function is also used to obtain an expression for the peak four-dimensional phase space density and for the fraction of the beam that is within a certain area in phase space. The special case of a periodic channel is then considered and expressions for the asymptotic rms emittance and peak phase space density are obtained. Finally, the application of the general formalism to a numerical example, based on the reported design of a cooling channel for a neutrino source, is considered, and comparisons are made with numerical simulations of that channel.

Conservative Numerical Schemes for the Vlasov Equation

A new scheme for solving the Vlasov equation using a phase space grid is proposed. The algorithm is based on the conservation of the flux of particles, and the distribution function is reconstructed using various techniques that allow control of spurious oscillations or preservation of the positivity. Several numerical results are presented in two- and four-dimensional phase space and the scheme is compared with the semiLagrangian method. This method is almost as accurate as the semiLagrangian one, and the local reconstruction technique is well suited for parallel computation.

Nonlinear Construction of Informational Sets in an Aircraft Tracking Problem Under Inertial Control

The paper deals with a tracking problem of an aircraft motion in the horizontal plane under constant trajectory velocity and inertial control of the heading velocity. Measurements of the aircraft position come at some instants. Measurement errors obey the geometric constraints. Any probability characteristics of the errors are absent. An algorithm and nonlinear procedures are suggested for constructing informational sets in the four-dimensional phase space. The informational set describes the totality of all phase states consistent with the sequence of received measurements. Simulation results are represented. Comparison with exact calculations is implemented.

Anomalous Exponents and Dipole Solutions for the Thin Film Equation

We investigate similarity solutions of the thin film equation. In particular we look at solutions on the half-line $x\ge0$ with compact support and zero contact angle boundary conditions in x=0. Such dipole solutions feature an anomalous exponent and are therefore called similarity solutions of the second kind. Using a combination of phase space analysis and numerical simulations, we numerically construct trajectories representing these solutions, at the same time obtaining broader insight into the nature of the four-dimensional phase space. Additional asymptotic analysis provides further information concerning the evolution to self-similarity.

Optical assessment of nonimaging concentrators

An optical measurement method for nonimaging radiation concentrators is proposed. A Lambertian light source is placed in the exit aperture of the concentrator. Looking into the concentrator's entrance aperture from a remote position, one can photograph the transmission patterns. The patterns show the transmission of radiation through the concentrator with the full resolution of the four-dimensional phase space of geometric optics. By matching ray-tracing simulations to the measurement, one can achieve detailed and accurate information about the geometry of the concentrator. This is a remote, noncontact measurement and can be performed in situ for installed concentrators. Additional information regarding small-scale reflector waviness and surface reflectivity can also be obtained from the same measurement with additional analysis.

Four-Dimensional Informational Sets in a Problem of Aircraft Tracking

The paper deals with a method of constructing the upper approximation of the informational sets that describe position, heading and velocity of aircraft motion in the horizontal plane. The informational sets are constructed in the four-dimensional phase space. The construction is implemented on the basis of measurements of aircraft geometric position. The constraints on measurement errors are known. Results of computer simulation are presented.

A Database ofCOBE‐normalized Cold Dark Matter Simulations

We have simulated the formation and evolution of large-scale structure in the universe, for 68 different COBE-normalized cosmological models. For each cosmological model, we have performed between one and three simulations, for a total of 160 simulations. This constitutes the largest database of cosmological simulations ever assembled, and the largest cosmological parameter space ever covered by such simulations. We are making this database available to the astronomical community. We provide instructions for accessing the database and for converting the data from computational units to physical units. The database includes tilted cold dark matter (TCDM) models, tilted open cold dark matter (TOCDM) models, and tilted Λ cold dark matter (TΛCDM) models. (For several simulations, the primordial exponent n of the power spectrum is near unity, hence these simulations can be considered as untilted.) The simulations cover a four-dimensional cosmological parameter phase space, the parameters being the present density parameter Ω0, cosmological constant λ0, and Hubble constant H0, and the rms density fluctuation σ8 at scale 8 h-1 Mpc. All simulations were performed using a P3M algorithm with 643 particles on a 1283 mesh, in a cubic volume of comoving size 128 Mpc. Each simulation starts at a redshift of 24 and is carried up to the present. More simulations will be added to the database in the future. We have performed a limited amount of data reduction and analysis of the final states of the simulations. We computed the rms density fluctuation, the two-point correlation function, the velocity moments, and the properties of clusters. Our results are the following: 1. The numerical value σ of the rms density fluctuation differs from the value σ obtained by integrating the power spectrum at early times and extrapolating linearly up the present. This results from the combined effects of discreteness in the numerical representation of the power spectrum, the presence of a Gaussian factor in the initial conditions, and late-time nonlinear evolution. The first of these three effects is negligible. The second and third are comparable, and can both modify the value of σ8 by up to 10%. Nonlinear effects, however, are important only for models with σ8 > 0.6, and can result in either an increase or a decrease in σ8. 2. The observed galaxy two-point correlation function is well reproduced (assuming an unbiased relation between galaxies and mass) by models with σ8 ~ 0.8, nearly independently of the values of the other parameters, Ω0, λ0, and H0. For models with σ8 > 0.8, the correlation function is too large and its slope is too steep. For models with σ8 0.9, it has a U shape, which is probably a numerical artifact caused by the finite number of particles used in the simulations. For all models, clusters have densities in the range 100-1000 times the mean background density, the spin parameters λ are in the range 0.008-0.2, with the median near 0.05, and about of the clusters are prolate. Rotationally supported disks do not form in these simulations.

Quantum chaos ofSU3observables

Hamiltonians built of SU 3 generators are presented which allow for qualitatively different classical limits. The generators of the group SU 3 give eight independent observables. In the classical limit two invariant functions of these observables, the so-called Casimir functions c 2 and c 3 , take fixed values in the ranges ½ c 2 (2/3) and -(2/9) c 3 (2/9) (an SU 2 system, in contrast, has just one such Casimir function, the squared total angular momentum). Generically, this leads to a six-dimensional phase space. However, for two special (`degenerate') pairs of values c 2 = (2/3),c 3 = -(2/9) and c 2 = (2/3),c 3 = (2/9) the classical dynamics is only four-dimensional. One and the same Hamilton function of the generators may be integrable in the four-dimensional phase space but generate chaos in the six-dimensional case. We give two examples of Hamiltonians for which that alternative does arise; one of these is closely related to the Lipkin Hamiltonian familiar from nuclear shell models. The transition from integrable to chaotic classical behaviour is accompanied by the usual quantum transition from level clustering to level repulsion.

Classical and quantum mechanical two body problem with central interaction on simply connected spaces of constant sectional curvature

The classical and quantum problems of two particles with central interaction on simply connected spaces of a constant curvature are considered. The reduction of classical problem to the dynamic systems in four-dimensional phase space are carried out using Marsden-Weinstein method. The classification of reduced dynamic systems was given. By using isometry group, in quantum case, systems of ordinary differential equations for energy levels are obtained. For the sphere in the case of equal masses some energy levels are calculated.

Symplectic topology of integrable dynamical systems. Rough topological classification of classical cases of integrability in the dynamics of a heavy rigid body

Physical and mechanical systems with four-dimensional phase space are considered. The classification of nondegenerate integral systems is studied. A “physical zone,’ i.e., the systems connected with real physical applications, is determined. Bibliography: 27 titles.

Reduction of the two-body problem with central interaction on simply connected spaces of constant sectional curvature

The problem of two particles with central interaction on simply connected spaces of a constant curvature was considered. Due to the absence of the Galilei transformation in this case the reduction to the dynamic problem in four-dimensional phase space was carried out using the Marsden-Weinstein method. Canonically conjugate coordinates were found. The classification of reduced dynamic systems was given. For some of them conditions of the existence global solutions for dynamic equations with attractive potentials were found. The comparison of the structure of obtained Hamiltonians with integrals of the one-particle problem with Bertrand's potentials was carried out.

Quantum-mechanical model of the Reissner-Nordström black hole

We consider a Hamiltonian quantum theory of spherically symmetric, asymptotically flat electrovacuum spacetimes. The physical phase space of such spacetimes is spanned by the mass and the charge parameters $M$ and $Q$ of the Reissner-Nordstr\"{o}m black hole, together with the corresponding canonical momenta. In this four-dimensional phase space, we perform a canonical transformation such that the resulting configuration variables describe the dynamical properties of Reissner-Nordstr\"{o}m black holes in a natural manner. The classical Hamiltonian written in terms of these variables and their conjugate momenta is replaced by the corresponding self-adjoint Hamiltonian operator, and an eigenvalue equation for the ADM mass of the hole, from the point of view of a distant observer at rest, is obtained. Our eigenvalue equation implies that the ADM mass and the electric charge spectra of the hole are discrete, and the mass spectrum is bounded below. Moreover, the spectrum of the quantity $M^2-Q^2$ is strictly positive when an appropriate self-adjoint extension is chosen. The WKB analysis yields the result that the large eigenvalues of the quantity $\sqrt{M^2-Q^2}$ are of the form $\sqrt{2n}$, where $n$ is an integer. It turns out that this result is closely related to Bekenstein's proposal on the discrete horizon area spectrum of black holes.

Influence of Initial Conditions on the Postcritical Behavior of a Nonlinear Aeroelastic System

The behavior of a nonlinear, non-Hamiltonian system in the postcritical (flutter) domain is studied with special attention to the influence of initial conditions on the properties of attractors situated at a certain point of the control parameter space. As a prototype system, an elastic panel is considered that is subjected to a combination of supersonic gas flow and quasistatic loading in the middle surface. A two natural modes approximation, resulting in a four-dimensional phase space and several control parameters is considered in detail. For two fixed points in the control parameter space, several plane sections of the four-dimensional space of initial conditions are presented and the asymptotic behavior of the final stationary responses are identified. Amongst the latter there are stable periodic orbits, both symmetric and asymmetric with respect to the origin, as well as chaotic attractors. The mosaic structure of the attraction basins is observed. In particular, it is shown that even for neighboring initial conditions can result in distinctly different nonstationary responses asymptotically approach quite different types of attractors. A number of closely neighboring periodic attractors are observed, separated by Hopf bifurcations. Periodic attractors also are observed under special initial conditions in the domains where chaotic behavior is usually expected.

The structure of infinitesimal symmetries of geodesic flows on a two-dimensional torus

The problem of geodesic lines on a two-dimensional torus is considered. One-parameter symmetry groups in the four-dimensional phase space that are generated by vector fields commuting with the initial Hamiltonian vector field are studied. As proved by Kozlov and Bolotin, a geodesic flow on a two-dimensional torus admitting a non-trivial infinitesimal symmetry of degree n has a many-valued integral that is a polynomial of degree at most n in the momentum variables. Kozlov and the present author proved earlier that first- and second-order infinitesimal symmetries are related to hidden cyclic coordinates and separated variables. In the present paper the structure of polynomial infinitesimal symmetries of degree at most four is described under the assumption that these symmetry fields are non-Hamiltonian.

Chaotic Hamiltonian dynamics of particle's horizontal motion in the atmosphere

A non-separable, near-integrable, two degrees-of-freedom (d.o.f.) Hamiltonian system arising in the context of particle's dynamics on a geopotential of the atmosphere is studied. In the unperturbed system, homoclinic orbits to periodic motion in the longitude angle exist, giving rise to several types of homoclinic chaos in the perturbed system; a new type of homoclinic chaos is found, exhibiting chaos which is non-uniformly distributed in the longitude angle along the homoclinic loop while it is topologically uniformly distributed in this angle in a neighborhood of the hyperbolic periodic orbit, i.e. the return map to different values of the longitude angle are topologically conjugate only near the hyperbolic periodic orbit. The concept of colored energy surfaces and its corresponding energy-momentum map are developed as analytical tools for delineating the regions in the four-dimensional phase space where the various types of homoclinic chaos prevail. Geometrical interpretation of the Melnikov analysis is offered for detecting this non-uniformity in angle. Applying the results of the analysis to the planetary atmosphere with an infinite wavelength perturbation of amplitude ϵ, we find that for eastward going particles with initial velocities {(ifu 0, v 0) = (ū, 0)} the chaotic band occupies two narrow (O(ϵ)) bands on both sides of the equator nea latitudes φ = ± arccos(1 − 2 u) . By contrast, for westward going particles the chaotic zone is thicker ( O( ϵ; ) ) and is centered on the equator. Moreover, the dependence of the chaotic zone's thickness on the perturbation frequency is much more sensitive to the value of the initial speed for eastward going particles than for those going westward. Qualitative and quantitative differences between the distribution of the homoclinic chaotic motion in the longitude angle for westward and eastward travelling wave perturbations are predicted and numerically confirmed.

The activation energy for {111} cross-slip in L12ordered alloys

Abstract The activation energy for octahedral cross-slip of superdislocations in L12 ordered alloys from a (111) glide plane into a (111) cross-slip plane is calculated in a three-dimensional treatment. By approximating the bow-outs in the two superpartials by piecewise straight segments, the problem can be treated in a four-dimensional phase space. For sufficiently low and high stresses the activation path can be identified directly. For intermediate stresses a more sophisticated treatment is necessary. For numerical evaluation, the material parameters of Ni3Al are used. It turns out that the saddle-point energy is far beyond the regime to allow homogeneous thermal activation.

Local analysis of formal stability and existence of fixed points in 4d symplectic mappings

Symplectic mappings in a four-dimensional phase space are analysed; in the neighbourhood of an elliptic fixed point whose eigenvalues are close to satisfy a double resonance condition, the perturbative tools of resonant normal forms are outlined. The analysis of the interpolating Hamiltonian of the normal form allows one to determine the existence and the stability of the fixed points which are due to the double resonance: only some combinations of stabilities are allowed. Due to the asymptotic character of the series, the results are formal, but supported by numerical computations.

Sectional and projectional emittance measurementsa)

For many applications of ion sources the quality of the generated ion beam plays an increasingly important role. Ion sources consist of the plasma generator and the extraction system. Both parts can, due to high ion temperature or various aberrations, contribute to a degrading of the beam quality. Though the beam quality is determined by many more factors, the transverse motion of the particles certainly is one of the important parameters. Knowledge of it can be obtained by an emittance measurement. This is best done in a four-dimensional phase space, yielding a density distribution as function of the transverse spatial and momentum coordinates. Often, however, due to practical considerations, only two dimensions of the four-dimensional ‘‘trace space’’ are being measured. This two-dimensional data can be obtained as a section or as a projection of the four-dimensional trace space, where both methods have their merits. Projectional emittance measurements can usually be performed much easier and quicker, but by the projection of the four-dimensional space onto a two-dimensional plane information is lost. And although not all the particles of the beam are represented in a section of trace space, here aberrative distortions of the emittance can be seen most clearly and allow an easier comparison to numerically obtained data. The advantages and disadvantages of both methods are discussed. Numerical and experimental examples are presented.

Static and time-dependent perturbations of the classical elliptical billiard

The elliptical billiard problem defines a two-dimensional integrable discrete dynamical system. Integrability not being a robust property, we study some static and time-dependent perturbations of this problem. For the static case, we observe the transition from integrability to chaos, on some perturbations of the ellipse. Then we study time-dependent perturbations, supposing that the boundary deforms periodically with the time, remaining always an ellipse. We investigate numerically the now four-dimensional phase space, looking mainly at the question of whether or not the velocity of a given trajectory may increase indefinitely.

Sectional and projectional emittance measurements (abstract)a)

For many applications of ion sources the quality of the generated ion beam plays an increasingly important role. Ion sources consist of the plasma generator and the extraction system. Both parts can, due to high ion temperature or various aberrations, contribute to a degrading of the beam quality. Though the beam quality is determined by many more factors, the transverse motion of the particles certainly is one of the important parameters. Knowledge of it can be obtained by an emittance measurement. This is best done in a four-dimensional phase space, yielding a density distribution as function of the transverse spatial and momentum coordinates. Often, however, due to practical considerations, only two dimensions of the four-dimensional ‘‘trace space’’ are being measured. This two-dimensional data can be obtained as a section or as a projection of the four-dimensional trace space, where both methods have their merits. Projectional emittance measurements can usually be performed much easier and quicker, but by the projection of the four-dimensional space onto a two-dimensional plane information is lost. And although not all the particles of the beam are represented in a section of trace space, here aberrative distortions of the emittance can be seen most clearly and allow an easier comparison to numerically obtained data. The advantages and disadvantages of both methods are discussed. Numerical and experimental examples are presented.