Classification Of Orbits Research Articles

Stationary D = 4 black holes in supergravity: The issue of real nilpotent orbits

The complete classification of the nilpotent orbits of SO(2,2)2 in the representation (2,2,2,2), achieved in , is applied to the study of multi‐center, asymptotically flat, extremal black hole solutions to the STU model. These real orbits provide an intrinsic characterization of regular single‐center solutions, which is invariant with respect to the action of the global symmetry group SO(4,4), underlying the stationary solutions of the model, and provide stringent regularity constraints on multi‐centered solutions. The known almost‐BPS and composite non‐BPS solutions are revisited in this setting. We systematically provide, for the relevant SO(2,2)2‐nilpotent orbits of the global Noether charge matrix, regular representatives thereof. This analysis unveils a composition law of the orbits according to which those containing regular multi‐centered solutions can be obtained as combinations of specific single‐center orbits defining the constituent black holes. Some of the SO(2,2)2‐orbits of the total Noether charge matrix are characterized as “intrinsically singular” in that they cannot contain any regular solution.

Little string origin of surface defects

We derive a large class of codimension-two defects of 4d mathcal{N}=4 Super Yang-Mills (SYM) theory from the (2, 0) little string. The origin of the little string is type IIB theory compactified on an ADE singularity. The defects are D-branes wrapping the 2-cycles of the singularity. We use this construction to make contact with the description of SYM defects due to Gukov and Witten [1]. Furthermore, we provide a geometric perspective on the nilpotent orbit classification of codimension-two defects, and the connection to ADE-type Toda CFT. The only data needed to specify the defects is a set of weights of the algebra obeying certain constraints, which we give explicitly. We highlight the differences between the defect classification in the little string theory and its (2, 0) CFT limit.

General classification of charged test particle circular orbits in Reissner\u2013Nordstr\xf6m spacetime

We investigate charged particles’ circular motion in the gravitational field of a charged mass distribution described by the Reissner–Nordström spacetime. We introduce a set of independent parameters completely characterizing the different spatial regions in which circular motion is allowed. We provide a most complete classification of circular orbits for different sets of particle and source charge-to-mass ratios. We study both black holes and naked singularities and show that the behavior of charged particles depend drastically on the type of source. Our analysis shows in an alternative manner that the behavior of circular orbits can in principle be used to distinguish between black holes and naked singularities. From this analysis, special limiting values for the dimensionless charge of black hole and naked singularity emerge, namely, Q/M = 1/2, Q/M=sqrt{13}/5 and Q/M=sqrt{2/3} for the black hole case and Q/M = 1, Q/M=5/ (2 sqrt{6}), Q/M=3 sqrt{6}/7, and finally Q/M= sqrt{9/8} for the naked singularity case. Similarly and surprisingly, analogous limits emerge for the orbiting particles charge-to-mass ratio epsilon , for positive charges epsilon =1, epsilon =2 and epsilon =M/Q. These limits play an important role in the study of the coupled electromagnetic and gravitational interactions, and the investigation of the role of the charge in the gravitational collapse of compact objects.

Orbit classification in the Hill problem: I. The classical case

The case of the classical Hill problem is numerically investigated by performing a thorough and systematic classification of the initial conditions of the orbits. More precisely, the initial conditions of the orbits are classified into four categories: (i) non-escaping regular orbits; (ii) trapped chaotic orbits; (iii) escaping orbits; and (iv) collision orbits. In order to obtain a more general and complete view of the orbital structure of the dynamical system our exploration takes place in both planar (2D) and the spatial (3D) version of the Hill problem. For the 2D system we numerically integrate large sets of initial conditions in several types of planes, while for the system with three degrees of freedom, three-dimensional distributions of initial conditions of orbits are examined. For distinguishing between ordered and chaotic bounded motion the Smaller ALingment Index (SALI) method is used. We managed to locate the several bounded basins, as well as the basins of escape and collision and also to relate them with the corresponding escape and collision time of the orbits. Our numerical calculations indicate that the overall orbital dynamics of the Hamiltonian system is a complicated but highly interested problem. We hope our contribution to be useful for a further understanding of the orbital properties of the classical Hill problem.

Branes and the Kraft-Procesi transition

The Coulomb and Higgs branches of certain 3d N=4 gauge theories can be understood as closures of nilpotent orbits. Recently, a new theorem by Namikawa suggests that this is the simplest possible case, thus giving this class a special role. In this note we use branes to reproduce the mathematical work by Kraft and Procesi. It studies the classification of all nilpotent orbits for classical groups and it characterizes an inclusion relation via minimal singularities. We show how these minimal singularities arise naturally in the Type IIB superstring embedding of the 3d A-type theories. The Higgs mechanism can be used to remove the minimal singularity, corresponding to a transition in the brane configuration that induces a new effective 3d theory. This reproduces the Kraft-Procesi results, endowing the family of gauge theories with a new underlying structure. We provide an efficient procedure for computing such brane transitions.

On the number of topological orbits of complex germs in classes (xy, xa + yb)

We show that there exist an infinite number of topological orbits in classes of complex map germs from the plane to the plane that have a representative of type (xy, xa + yb), with (a, b) ≠ = (2, 3) or (2, 5). Our key tool to prove this existence is the existence (or not) of stems in the class; these germs are not -finitely determined and allow the determination of a non-finite number of topological orbits. We also show that the class (xy, x2 + y5) has two topological orbits.

THE ORBITAL STRUCTURE OF A TIDALLY INDUCED BAR

ABSTRACT Orbits are the key building blocks of any density distribution, and their study helps us understand the kinematical structure and the evolution of galaxies. Here, we investigate orbits in a tidally induced bar of a dwarf galaxy, using an N-body simulation of an initially disky dwarf galaxy orbiting a Milky Way-like host. After the first pericenter passage, a tidally induced bar forms in the stellar component of the dwarf. The bar evolution is different than in isolated galaxies and our analysis focuses on the period before it buckles. We study the orbits in terms of their dominant frequencies, which we calculate in a Cartesian coordinate frame rotating with the bar. Apart from the well-known x1 orbits, we find many other types, mostly with boxy shapes of various degree of elongation. Some of them are also near-periodic, admitting frequency ratios of 4/3, 3/2, and 5/3. The box orbits have various degrees of vertical thickness but only a relatively small fraction of those have banana (i.e., smile/frown) or infinity-symbol shapes in the edge-on view. In the very center we also find orbits known from the potential of triaxial ellipsoids. The elongation of the orbits grows with distance from the center of the bar in agreement with the variation of the shape of the density distribution. Our classification of orbits leads to the conclusion that more than 80% of them have boxy shapes, while only 8% have shapes of classical x1 orbits.

Integrable cocycles and global deformations of Lie algebra of type G2 in characteristic 2

ABSTRACTAll classes of integrable cocycles in H2(L,L) are obtained for Lie algebra of type G2 over an algebraically closed field of characteristic 2. It is proved that there exist only two orbits of classes of integrable cocycles with respect to automorphism group. The global deformation is shown to exist for any nontrivial class of integrable cocycles. These deformations are isomorphic to one of the two algebras of Cartan type, one of which being S(3:1,ω) while the other H(4:1,ω).

Orbital classification in anN-body bar

The dynamics and evolution of any galactic structure are strongly influenced by the properties of the orbits that constitute it. In this paper, we compare two orbit classification schemes, one by Laskar (NAFF) , and the other by Carpintero and Aguilar (CA), by applying both of them to orbits obtained by following individual particles in a numerical simulation of a barred galaxy. We find that, at least for our case and some provisos, the main frequencies calculated by the two methods are in good agreement: for $80\%$ of the orbits the difference between the results of the two methods is less than $5\%$ for all three main frequencies. However, it is difficult to evaluate the amount of regular or chaotic bar orbits in a given system. The fraction of regular orbits obtained by the NAFF method strongly depends on the critical frequency drift parameter, while in the CA method the number of fundamental frequencies strongly depends on the frequency difference parameter $L_{\rm r}$ and the maximum integer used for searching the linear independence of the fundamental frequencies. We also find that, for a given particle, in general the projection of its motion along the bar minor axis is more regular than the other two projections, while the projection along the intermediate axis is the least regular.

A classification of orbits admitting a unique invariant measure

We consider the space of countable structures with fixed underlying set in a given countable language. We show that the number of ergodic probability measures on this space that are S∞-invariant and concentrated on a single isomorphism class must be zero, or one, or continuum. Further, such an isomorphism class admits a unique S∞-invariant probability measure precisely when the structure is highly homogeneous; by a result of Peter J. Cameron, these are the structures that are interdefinable with one of the five reducts of the rational linear order (Q,<).

Nontrivial paths and periodic orbits of the T-fractal billiard table

We introduce and prove numerous new results about the orbits of the T-fractal billiard. Specifically, in section , we give a variety of sufficient conditions for the existence of a sequence of compatible periodic orbits. In section , we examine the limiting behavior of particular sequences of compatible periodic orbits. Additionally, sufficient conditions for the existence of particular nontrivial paths are given in section . The proofs of two results of Lapidus and Niemeyer (2013 The current state of fractal billiards Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II: Fractals in Applied Mathematics (Contemporary Mathematics vol 601) ed D Carfi et al (Providence, RI: American Mathematical Society) pp 251–88 (e-print: arXiv:math.DS.1210.0282v2, 2013) appear here for the first time, as well. In section , an orbit with an irrational initial direction reaches an elusive point in a way that yields a nontrivial path of finite length, yet, by our convention, constitutes a singular orbit of the fractal billiard table. The existence of such an orbit seems to indicate that the classification of orbits may not be so straightforward. A discussion of our results and directions for future research is then given in section .

Computer Aided Ballistic Orbit Classification Around Small Bodies

Orbital dynamics around small bodies are as varied as the shapes and dynamical states of these bodies. While various classes of orbits have been analyzed in detail, the global overview of relevant ballistic orbits at particular bodies is not easily computed or organized. Yet, correctly categorizing these orbits will ease their future use in the overall trajectory design process. This paper overviews methods that have been used to organize orbits, focusing on periodic orbits in particular, and introduces new methods based on clustering approaches.

Comparison of experimental observations in rotating machines with simple mathematical simulations

Through an earlier experimental study, an integrated fault classification approach was suggested for machines operating at different speeds. The suggested method was observed to separate faults adequately, which can lead to fault diagnosis. However, theoretical understanding of the proposed methods is important to further enhance the confidence of the earlier experimental study. Hence, simplified mathematical simulations of different rotor faults are carried out and used to test the earlier proposed classification technique in this paper. Observations in the orbit, spectra and fault classification diagrams are consistent with the previous experimental case.

A UNIFIED FRAMEWORK FOR THE ORBITAL STRUCTURE OF BARS AND TRIAXIAL ELLIPSOIDS

ABSTRACT We examine a large random sample of orbits in two self-consistent simulations of N-body bars. Orbits in these bars are classified both visually and with a new automated orbit classification method based on frequency analysis. The well-known prograde x1 orbit family originates from the same parent orbit as the box orbits in stationary and rotating triaxial ellipsoids. However, only a small fraction of bar orbits (∼4%) have predominately prograde motion like their periodic parent orbit. Most bar orbits arising from the x1 orbit have little net angular momentum in the bar frame, making them equivalent to box orbits in rotating triaxial potentials. In these simulations a small fraction of bar orbits (∼7%) are long-axis tubes that behave exactly like those in triaxial ellipsoids: they are tipped about the intermediate axis owing to the Coriolis force, with the sense of tipping determined by the sign of their angular momentum about the long axis. No orbits parented by prograde periodic x2 orbits are found in the pure bar model, but a tiny population (∼2%) of short-axis tube orbits parented by retrograde x4 orbits are found. When a central point mass representing a supermassive black hole (SMBH) is grown adiabatically at the center of the bar, those orbits that lie in the immediate vicinity of the SMBH are transformed into precessing Keplerian orbits that belong to the same major families (short-axis tubes, long-axis tubes and boxes) occupying the bar at larger radii. During the growth of an SMBH, the inflow of mass and outward transport of angular momentum transform some x1 and long-axis tube orbits into prograde short-axis tubes. This study has important implications for future attempts to constrain the masses of SMBHs in barred galaxies using orbit-based methods like the Schwarzschild orbit superposition scheme and for understanding the observed features in barred galaxies.

On the N = 1 ∗ $$ \mathcal{N}={1}^{\ast } $$ gauge theory on a circle and elliptic integrable systems

We continue our study of the N=1* supersymmetric gauge theory and its relation to elliptic integrable systems. Upon compactification on a circle, we show that the semi-classical analysis of the massless and massive vacua depends on the classification of nilpotent orbits, as well as on the conjugacy classes of the component group of their centralizer. We demonstrate that semi-classically massless vacua can be lifted by Wilson lines in unbroken discrete gauge groups. The pseudo-Levi subalgebras that play a classifying role in the nilpotent orbit theory are also key in defining generalized Inozemtsev limits of (twisted) elliptic integrable systems. We illustrate our analysis in the N=1* theories with gauge algebras su(3), su(4), so(5) and for the exceptional gauge algebra G2. We map out modular duality diagrams of the massive and massless vacua. Moreover, we provide an analytic description of the branches of massless vacua in the case of the su(3) and the so(5) theory. The description of these branches in terms of the complexified Wilson lines on the circle invokes the Eichler-Zagier technique for inverting the elliptic Weierstrass function. After fine-tuning the coupling to elliptic points of order three, we identify the Argyres-Douglas singularities of the su(3) theory.

On the structure of periodic complex Horadam orbits

Numerous geometric patterns identified in nature, art or science can be generated from recurrent sequences, such as for example certain fractals or Fermat’s spiral. Fibonacci numbers in particular have been used to design search techniques, pseudo random-number generators and data structures. Complex Horadam sequences are a natural extension of Fibonacci sequence to complex numbers, involving four parameters (two initial values and two in the defining recursion), therefore successive sequence terms can be visualized in the complex plane. Here, a classification of the periodic orbits is proposed, based on divisibility relations between orders of generators (roots of the characteristic polynomial). Regular star polygons, bipartite graphs and multi-symmetric patterns can be recovered for selected parameter values. Some applications are also suggested.

Three topologically nontrivial choreographic motions of three bodies

Choreographies are periodic orbits in which all bodies move on the same trajectory with equal time delay. The best known three-body choreography is figure-eight orbit. Here we introduce a search method specialized for choreographies and present three new orbits with vanishing angular momentum that are the first clear examples of choreographies that cannot be described as k-th powers of the figure-eight solution, according to the topological classification of orbits. We have also found 17 new “powers of the eight” choreographies. According to our numerical computation, one of two distinct \(k=7\) choreographies is linearly stable.

Reconstruction of chaotic saddles by classification of unstable periodic orbits: Kuramoto-Sivashinsky equation.

The unstable periodic orbits (UPOs) embedded in a chaotic attractor after an attractor merging crisis (MC) are classified into three subsets, and employed to reconstruct chaotic saddles in the Kuramoto-Sivashinsky equation. It is shown that in the post-MC regime, the two chaotic saddles evolved from the two coexisting chaotic attractors before crisis can be reconstructed from the UPOs embedded in the pre-MC chaotic attractors. The reconstruction also involves the detection of the mediating UPO responsible for the crisis, and the UPOs created after crisis that fill the gap regions of the chaotic saddles. We show that the gap UPOs originate from saddle-node, period-doubling, and pitchfork bifurcations inside the periodic windows in the post-MC chaotic region of the bifurcation diagram. The chaotic attractor in the post-MC regime is found to be the closure of gap UPOs.

Counting the massive vacua of N = 1 ∗ $$ \mathcal{N}={1}^{\ast } $$ super Yang-Mills theory

We compute the number of massive vacua of N=4 supersymmetric Yang-Mills theory mass-deformed to preserve N=1 supersymmetry, for any gauge group G. We use semi-classical techniques and efficiently reproduce the known counting for A,B and C-type gauge groups, present the generating function for both O(2n) and SO(2n), and compute the supersymmetric index for gauge groups of exceptional type. A crucial role is played by the classification of nilpotent orbits, as well as global properties of their centralizers. We give illustrative examples of new features of our analysis for the D-type algebras.

Regular subalgebras and nilpotent orbits of real graded Lie algebras

For a semisimple Lie algebra over the complex numbers, Dynkin (1952) developed an algorithm to classify the regular semisimple subalgebras, up to conjugacy by the inner automorphism group. For a graded semisimple Lie algebra over the complex numbers, Vinberg (1979) showed that a classification of a certain type of regular subalgebras (called carrier algebras) yields a classification of the nilpotent orbits in a homogeneous component of that Lie algebra. Here we consider these problems for (graded) semisimple Lie algebras over the real numbers. First, we describe an algorithm to classify the regular semisimple subalgebras of a real semisimple Lie algebra. This also yields an algorithm for listing, up to conjugacy, the carrier algebras in a real graded semisimple real algebra. We then discuss what needs to be done to obtain a classification of the nilpotent orbits from that; such classifications have applications in differential geometry and theoretical physics. Our algorithms are implemented in the language of the computer algebra system GAP, using our package CoReLG; we report on example computations.