The concept of integrability on cantor sets for Hamiltonian systems

Differentiable Hamiltonian systems close to nondegenerate, integrable Hamiltonian systems are shown to be integrable on a Cantor set in the sense that on some Cantor set, (i) the invariant KAM-tori form a smooth foliation, (ii) there exist smooth, independent integrals in involution, and (iii) there exists a complete solution of the Hamilton Jacobi equation. The complement of the Cantor set is shown to be small in measure.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.     

Analysis of the Chaotic Behaviour of Orbits Diffusing along the Arnold Web

In a previous work [Guzzo et al. DCDS B 5, 687–698 (2005)] we have provided numerical evidence of global diffusion occurring in slightly perturbed integrable Hamiltonian systems and symplectic maps. We have shown that even if a system is sufficiently close to be integrable, global diffusion occurs on a set with peculiar topology, the so-called Arnold web, and is qualitatively different from Chirikov diffusion, occurring in more perturbed systems. In the present work we study in more detail the chaotic behaviour of a set of 90 orbits which diffuse on the Arnold web. We find that the largest Lyapunov exponent does not seem to converge for the individual orbits while the mean Lyapunov exponent on the set of 90 orbits does converge. In other words, a kind of average mixing characterizes the diffusion. Moreover, the Local Lyapunov Characteristic Numbers (LLCNs), on individual orbits appear to reflect the different zones of the Arnold web revealed by the Fast Lyapunov Indicator. Finally, using the LLCNs we study the ergodicity of the chaotic part of the Arnold web.     

Construction of confidence regions in problem on probabilistic study into motion of minor bodies of the solar system

Different approaches to constructing regions of possible motions of minor bodies are examined. An economical approach to the minimization of the number of points of the initial region by means of defining this region with its boundary surface is offered, and the estimations of its efficiency are given. The methods for estimating the admissibility of the linear approach are considered. For this purpose, simple methods for calculating the nonlinear factors are offered, which makes it possible to classify a problem to be solved as either strongly or weakly nonlinear. Recommendations are given on the possibility of reducing the concrete estimation problem to a weakly nonlinear one, where the more economical linear approach can be used. The combined method of mapping the initial region in time is also offered which unites the linear and nonlinear approaches. By example of two asteroids, the area of applicability of linear mappings is estimated.     

Relations between integrable systems in plane and curved spaces

We consider trajectory isomorphisms between various integrable systems on an n-dimensional sphere S ⁿ and a Euclidean space . Some of the systems are classical integrable problems of Celestial Mechanics in plane and curved spaces. All the systems under consideration have an additional first integral quadratic in momentum and can be integrated analytically by using the separation of variables. We show that some integrable problems in constant curvature spaces are not essentially new from the viewpoint of the theory of integration, and they can be analyzed using known results of classical Celestial Mechanics.     

Evolution of peaks in weakly non-linear density field and dark halo profiles

Using the two-point Edgeworth series up to second order in the linear rms density fluctuation we construct the weakly non-linear conditional probability distribution function for the density field around an overdense region. This requires calculating the two-point analogues of the skewness parameter S ₃. We test the dependence of the two-point skewness on distance from the peak for scale-free power spectra and Gaussian smoothing. The statistical features of such a conditional distribution are given as the values obtained within linear theory corrected by the terms that arise as a result of weakly non-linear evolution. The expected density around the peak is found to be always below the linear prediction while its dispersion is always larger than in the linear case. For large enough overdensities the weakly non-linear corrections can be more significant than the peak constraint introduced by Bardeen et al. We apply these results to the spherical model of collapse as developed by Hoffman & Shaham and find that in general the effect of weakly non-linear interactions is to decrease the scale from which a peak gathers mass and therefore also the mass itself. In the case of an open universe this results in steepening of the final profile of the virialized proto-object.     

Global Secular Dynamics in the Planar Three-Body Problem

We use the global construction which was made in [6, 7] of the secular systems of the planar three-body problem, with regularized double inner collisions. These normal forms describe the slow deformations of the Keplerian ellipses which each of the bodies would describe if it underwent the universal attraction of only one fictitious other body. They are parametrized by the masses and the semi-major axes of the bodies and are completely integrable on a fixed transversally Cantor set of the parameter space. We study this global integrable dynamics reduced by the symmetry of rotation and determine its bifurcation diagram when the semi-major axes ratio is small enough. In particular it is shown that there are some new secular hyperbolic or elliptic singularities, some of which do not belong to the subset of aligned ellipses. The bifurcation diagram may be used to prove the existence of some new families of 2-, 3- or 4-frequency quasiperiodic motions in the planar three-body problem [7], as well as some drift orbits in the planar n-body problem [8].     

Complex statistics in Hamiltonian barred galaxy models

We use probability density functions (pdfs) of sums of orbit coordinates, over time intervals of the order of one Hubble time, to distinguish weakly from strongly chaotic orbits in a barred galaxy model. We find that, in the weakly chaotic case, quasi-stationary states arise, whose pdfs are well approximated by q-Gaussian functions (with 1 <?q < 3), while strong chaos is identified by pdfs which quickly tend to Gaussians (q =?1). Typical examples of weakly chaotic orbits are those that ??stick?? to islands of ordered motion. Their presence in rotating galaxy models has been investigated thoroughly in recent years due to their ability to support galaxy structures for relatively long time scales. In this paper, we demonstrate, on specific orbits of 2 and 3 degree of freedom barred galaxy models, that the proposed statistical approach can distinguish weakly from strongly chaotic motion accurately and efficiently, especially in cases where Lyapunov exponents and other local dynamic indicators appear to be inconclusive.     

On the measure of the structure around the last kam torus before and after its break-up

We review theorems for proving non-integrability of Hamiltonian dynamical systems, which are based on properties of the variational equations in real or complex time or on the destruction of the resonant tori of an integrable system under a perturbation.     

Resonant Absorption of Alfvén Waves in Steady Coronal Loops

The effect of equilibrium flow on linear Alfvén resonances in coronal loops is studied in the compressible viscous MHD model. By means of a finite element code, the full set of linearised driven MHD equations are solved for a one-dimensional equilibrium model in which the equilibrium quantities depend only on the radial coordinate. Computations of resonant absorption of Alfvén waves for two classes of coronal loop models show that the efficiency of the process of resonant absorption strongly depends on both the equilibrium parameters and the characteristics of the resonant wave. We find that a steady equilibrium shear flow can also significantly influence the resonant absorption of Alfvén waves in coronal magnetic flux tubes. The presence of an equilibrium flow may therefore be important for resonant Alfvén waves and coronal heating. A parametric analysis also shows that the resonant absorption can be strongly enhanced by the equilibrium flow, even up to total dissipation of the incoming wave.     

Control Of Chaos In Hamiltonian Systems

We present a technique to control chaos in Hamiltonian systems which are close to integrable. By adding a small and simple control term to the perturbation, the system becomes more regular than the original one. We apply this technique to a forced pendulum model and show numerically that the control is able to drastically reduce chaos.     

Non-Integrability and Structure of the Resonance Zones in a Class of Galactic Potentials

The structure of the resonance zone in nearly integrable Hamiltonian systems is studied by a more general method than the pendulum approximation. This method applies to the case of a non-degenerate integrable part in the Hamiltonian. This problem may be overcome in a class of galactic-type polynomial potentials, in the case where the higher-order term is by itself integrable. An illustrative example is worked out.     

On Einstein clusters as galactic dark matter haloes

We consider global and gravitational lensing properties of the recently suggested Einstein clusters of weakly interacting massive particles (WIMPs) as galactic dark matter haloes. Being tangential pressure dominated, Einstein clusters are strongly anisotropic systems which can describe any galactic rotation curve by specifying the anisotropy. Due to this property, Einstein clusters may be considered as dark matter candidates. We analyse the stability of the Einstein clusters against both radial and non-radial pulsations, and we show that the Einstein clusters are dynamically stable. With the use of the Buchdahl type inequalities for anisotropic bodies, we derive upper limits on the velocity of the particles defining the cluster. These limits are consistent with those obtained from stability considerations. The study of light deflection shows that the gravitational lensing effect is slightly smaller for the Einstein clusters as compared to the singular isothermal density sphere model for dark matter. Therefore, lensing observations may discriminate, at least, in principle, between Einstein cluster and the other dark matter models.     

Kinematics and Dynamics of Galaxies in Compact Groups for a sample of 31 groups

A long-term program based on Fabry-Perot Hα velocity field data of compact groups has been undertaken in order to analyse the kinematics of the compact group galaxies and the extent of their dark halos. The data are taken at the ESO and the CFH 3.6 m telescopes. The main goals of our project are: • To determine the evolutionary stages of the studied groups, • To search for tidal dwarf galaxy candidates in heavily interacting systems and • in combination with photometry available in the literature, to determine the Tully-Fisher relation for the group galaxies. The sample contains examples of groups at a variety of dynamical stages: from a false group that is in fact one single irregular galaxy with several star-forming blobs (e.g. H18) to a group whose members are strongly interacting and possibly forming tidal dwarf galaxies (e.g. H92) to a group in the final process of merging. We have so far identified five HCG classes: 1. merging groups2. strongly interacting, 3. weakly interacting, 4. non-groups, 5. single irregular galaxies. The ones which can be used in the TF studies are those in classes (2) and (3), for which comprehensive rotation curves can be derived. This revised version was published online in September 2006 with corrections to the Cover Date.     

Generalized Problem of Two and Four Newtonian Centers

We consider integrable spherical analog of the Darboux potential, which appear in the problem (and its generalizations) of the planar motion of a particle in the field of two and four fixed Newtonian centers. The obtained results can be useful when constructing a theory of motion of satellites in the field of an oblate spheroid in constant curvature spaces.     

Wave-trains in the solar wind

The main results of Whitham's averaged Lagrangian method for the treatment of linear wave-trains in a weakly inhomogeneous, moving medium are presented briefly. This method is then applied to an ideal, isotropic, one-fluid plasma which can be taken for the lowest order approximation for the interplanetary solar wind expansion.     

Regularizing Time Transformations in Symplectic and Composite Integration

We consider numerical integration of nearly integrable Hamiltonian systems. The emphasis is on perturbed Keplerian motion, such as certain cases of the problem of two fixed centres and the restricted three-body problem. We show that the presently known methods have useful generalizations which are explicit and have a variable physical timestep which adjusts to both the central and perturbing potentials. These methods make it possible to compute accurately fairly close encounters. In some cases we suggest the use of composite (instead of symplectic) alternatives which typically seem to have equally good energy conservation properties.This revised version was published online in October 2005 with corrections to the Cover Date.     

Predictions for the clustering properties of the Lyman-alpha forest – I. One-point statistics

We present predictions for the one-point probability distribution and cumulants of the transmitted QSO flux in the high redshift Lyman- α forest. We make use of the correlation between the Lyman- α optical depth and the underlying matter density predicted by gravitational instability theory and seen in numerical hydrodynamic simulations. We have modelled the growth of matter fluctuations using the non-linear shear‐free dynamics, an approximation which reproduces well the results of perturbation theory for the cumulants in the linear and weakly non-linear clustering regime. As high matter overdensities tend to saturate in spectra, the statistics of the flux distribution are dominated by weakly non-linear overdensities. As a result, our analytic approach can produce accurate predictions, when tested against N -body simulation results, even when the underlying matter field has root-mean-square fluctuations larger than unity. Our treatment can be applied to either Gaussian or non-Gaussian initial conditions. Here we concentrate on the former case, but also include a study of a specific non-Gaussian model. We discuss how the methods and predictions we present can be used as a tool to study the generic clustering properties of the Lyman- α forest at high redshift. With such an approach, rather than concentrating on simulating specific cosmological models, we may be in a position to directly test our assumptions for the Gaussian nature of the initial conditions, and the gravitational instability origin of structure itself. In a separate paper we present results for two-point statistics.     

A survey of weak stability boundaries in the Sun-Mars system

We investigate the weak stability boundary(WSB) for a new primary, Mars,in the framework of the planar circular restricted 3-body problem, and also in the planar bicircular restricted 4-body problem by including a perturbation due to Jupiter. For the sake of a simple stability/instability criterion, our computations have been done using the equations of motion in polar coordinates. It is found that the relative size of the weakly stable sets around Mars is much larger than that of the Earth-Moon and the Sun-Jupiter systems, as the mass ratio of the Sun-Mars system is significantly smaller.We propose that this difference could be scaled by the Hill radius. In an enlarged view of the domain close to Mars, the geometry of the WSB has been presented for various parameters and compared to previous works. Our results also show that Jupiter’s gravitational force would strongly affect the Martian stable regions and should be taken into account to design a ballistic capture trajectory.     

Normalization and the detection of integrability: The generalized Van Der Waals potential

Deprit and Miller have conjectured that normalization of integrable Hamiltonians may produce normal forms exhibiting degenerate equilibria to very high order. Several examples in the class of coupled elliptic oscillators are known. In order to test the utility of normalization as a detector of integrability we normalize, to high order, a perturbed Keplerian system known to have several integrable limits; the generalized van der Waals Hamiltonian for a hydrogen atom. While the separable limits give rise to high order degeneracy we find a non-separable, integrable limit for which the normal form does not exhibit degeneracy. We conclude that normalization may, in certain cases, indicate integrability but is not guaranteed to uncover all integrable limits.     

Weak stability boundary trajectories for the deployment of lunar spacecraft constellations

Suitable lunar constellation coverage can be obtained by separating the satellites in inclinations and node angles. It is shown in the paper that a relevant saving of velocity variation ΔV can be achieved using weak stability boundary trajectories. The weakly stable dynamics of such transfers allows the separation of the satellites from the nominal orbit to the required orbit planes with a small amount of ΔV. This paper also shows that only one different set of orbital parameters at Moon can be reached with the same ΔV manoeuvre starting from a nominal trajectory and ending at a fixed periselenium altitude. In fact, such a feature is proved to be common to other simpler dynamical systems, such as the two- and three-body problems.