Generic Orbits Research Articles

On some geometric characteristics of the orbit foliations of the co-adjoint action of some 5-dimensional solvable Lie groups

In this paper, we discribe some geometric charateristics of the so-called MD(5,3C)-foliations and MD(5,4)- foliations, i.e., the foliations formed by the generic orbits of co-adjoint action of MD(5,3C)-groups and MD(5,4)-groups.

New method for constructing semi-invariants and integrals of the full symmetric $\mathfrak{s}\mathfrak{l}_n $ Toda lattice

We consider the full symmetric representation of the Lax operator matrix of the Toda lattice, which is known as the full symmetric Toda lattice. The phase space of this system is the generic orbit of the coadjoint action of the Borel subgroup B n + of SL n (ℝ). This system is integrable. We propose a new method for constructing semi-invariants and integrals of the full symmetric Toda lattice. Using only the equations of motion for the Lax eigenvector matrix, we prove the existence of the semi-invariants that are Plucker coordinates in the corresponding projective spaces. We use these semi-invariants to construct the integrals. Our new approach provides simple exact formulas for the full set of independent semi-invariants and integrals expressed in terms of the Lax matrix and also in terms of its eigenvector and eigenvalue matrices of the full symmetric Toda lattice without using the chopping and Kostant procedures. We describe the structure of the additional integrals of motion as functions on the flag space modulo the Toda flows and show how the Plucker coordinates of different projective spaces define different families of the additional integrals.

Self-force from reconstructed metric perturbations: Numerical implementation in Schwarzschild spacetime

We present a first numerical implementation of a new scheme by Pound et al. [1] that enables the calculation of the gravitational self-force in Kerr spacetime from a reconstructed metric-perturbation in a radiation gauge. The numerical task of the metric reconstruction essentially reduces to solving the fully separable Teukolsky equation, rather than having to tackle the linearized Einstein's equations themselves in the Lorenz gauge, which are not separable in Kerr. The method offers significant computational saving compared to existing methods in the Lorenz gauge, and we expect it to become a main workhorse for precision self-force calculations in the future. Here we implement the method for circular orbits on a Schwarzschild background, in order to illustrate its efficacy and accuracy. We use two independent methods for solving the Teukolsky equation, one based on a direct numerical integration, and the other on the analytical approach of Mano, Suzuki, and Takasugi. The relative accuracy of the output self-force is at least $1{0}^{\ensuremath{-}7}$ using the first method, and at least $1{0}^{\ensuremath{-}9}$ using the second; the two methods agree to within the error bars of the first. We comment on the relation to a related approach by Shah et al. [2], and discuss foreseeable applications to more generic orbits in Kerr spacetime.

Oscillating Statistics of Transitive Dynamics

We prove that topologically generic orbits of C0 transitive and non-uniquely ergodic dynamical systems, exhibit an extremely oscillating asymptotical statistics. Precisely, the minimum weak* compact set of invariant probabilities, that describes the asymptotical statistics of each orbit of a residual set, contains all the ergodic probabilities. If besides f is ergodic with respect to the Lebesgue measure, then also Lebesgue-almost all the orbits exhibit that kind of extremely oscillating statistics.

Topological dynamics of unordered Ramsey structures

We investigate the connections between Ramsey properties of Fraisse classes K and the universal minimal flow M(GK) of the automorphism group GK of their Fraisse limits. As an extension of a result of Kechris, Pestov and Todorcevic (2005) we show that if the class K has finite Ramsey degree for embeddings, then this degree equals the size of M(GK). We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if K is a relational Ramsey class and GK is amenable, then M(GK) admits a unique invariant Borel probability measure that is concentrated on a unique generic orbit.

Contractions of Lie algebras with 2-dimensional generic coadjoint orbits

We determine all the contractions within the class of finite-dimensional real Lie algebras whose coadjoint orbits have dimensions ≤2.

Orbit Approach to Separation of Variables in $${\mathfrak{sl}}$$ sl (3)-Related Integrable Systems

Using the orbit method we attempt to reveal geometric and algebraic meaning of separation of variables for the integrable systems on coadjoint orbits in an $\mathfrak{sl}(3)$ loop algebra. We consider two types of generic orbits embedded into a common manifold, endowed with two nonsingular Lie-Poisson brackets. We prove that separation of variables on orbits of both types is realized by the same variables of separation. We also construct the integrable systems on these orbits: a coupled 3-component nonlinear Schr\"{o}dinger equation and an isotropic SU(3) Landau-Lifshitz equation.

Automorphic Lie algebras with dihedral symmetry

The concept of automorphic Lie algebras arises in the context of reduction groups introduced in the early 1980s in the field of integrable systems. automorphic Lie algebras are obtained by imposing a discrete group symmetry on a current algebra of Krichever–Novikov type. Past work shows remarkable uniformity between algebras associated to different reduction groups. For example, if the base Lie algebra is and the poles of the automorphic Lie algebra are restricted to an exceptional orbit of the symmetry group, changing the reduction group does not affect the Lie algebra structure. In this research we fix the reduction group to be the dihedral group and vary the orbit of poles as well as the group action on the base Lie algebra. We find a uniform description of automorphic Lie algebras with dihedral symmetry, valid for poles at exceptional and generic orbits.

Character and multiplicity formulas for compact Hamiltonian G-spaces

Let K ⊂ G be compact connected Lie groups with common maximal torus T . Let ( M , ω ) be a prequantisable compact connected symplectic manifold with a Hamiltonian G -action. Geometric quantisation gives a virtual representation of G ; we give a formula for the character χ of this virtual representation as a quotient of virtual characters of K . When M is a generic coadjoint orbit our formula agrees with the Gross–Kostant–Ramond–Sternberg formula. We then derive a generalisation of the Guillemin–Prato multiplicity formula which, for λ a dominant integral weight of K , gives the multiplicity in χ of the irreducible representation of K of highest weight λ .

Order and chaos in the rotation and revolution of two massive line segments.

As a generalization of Newton's two body problem, we explore the dynamics of two massive line segments interacting gravitationally. The extension of each line segment or slash (/) provides extra degrees of freedom that enable the interplay between rotation and revolution in an especially simple example. This slash-slash (//) body problem can thereby elucidate the dynamics of nonspherical space structures, from asteroids to space stations. Fortunately, as we show, Newton's laws imply exact algebraic expressions for the force and torque between the slashes, and this greatly facilitates analysis. The diverse dynamics include a stable synchronous orbit, families of unstable periodic orbits, generic chaotic orbits, and spin-orbit coupling that can unbind the slashes. In particular, retrograde orbits where the slashes spin opposite to their orbits are stable, with regular dynamics and smooth parameter spaces, while prograde orbits are unstable, with chaotic dynamics and fractal parameter spaces.

A Global Bifurcation Analysis of the Subcritical Hopf Normal Form Subject to Pyragas Time-Delayed Feedback Control

Unstable periodic orbits occur naturally in many nonlinear dynamical systems. They can generally not be observed directly, but a number of control schemes have been suggested to stabilize them. One such scheme is that by Pyragas [Phys. Lett. A, 170 (1992), pp. 421--428], which uses time-delayed feedback to target a specific unstable periodic orbit of a given period and stabilize it. This paper considers the global effect of applying Pyragas control to a nonlinear dynamical system. Specifically, we consider the standard example of the subcritical Hopf normal form subject to Pyragas control, which is a delay differential equation (DDE) that models how a generic unstable periodic orbit is stabilized. Our aim is to study how this DDE model depends on its different parameters, including the phase of the feedback and the imaginary part of the cubic coefficient, over their entire ranges. We show that the delayed feedback control induces infinitely many curves of Hopf bifurcations, from which emanate infinitely m...

Classification of Lie algebras with generic orbits of dimension 2 in the coadjoint representation

We provide a complete list of real Lie algebras with generic orbits of dimension 2 in the coadjoint representation.Bibliography: 20 titles.

Four-qubit pure states as fermionic states

The embedding of the $n$-qubit space into the $n$-fermion space with $2n$ modes is a widely used method in studying various aspects of these systems. This simple mapping raises a crucial question: Does the embedding preserve the entanglement structure? It is known that the answer is affirmative for $n=2$ and $n=3$. That is, under either local unitary (LU) operations or with respect to stochastic local operations and classical communication (SLOCC), there is a one-to-one correspondence between the two- (or three)-qubit orbits and the two- (or three)-fermion orbits with four (or six) modes. However, these results do not generalize as the mapping from the $n$-qubit orbits to the $n$-fermion orbits with $2n$ modes is no longer surjective for $n>3$. Here we consider the case of $n=4$. We show that, surprisingly, the mapping of orbits from qubits to fermions is injective under SLOCC, and a similar result holds under LU for generic orbits. As a by-product, we obtain a complete answer to the problem of SLOCC equivalence of pure four-qubit states.

Coadjoint Orbits of Certain Motion Groups and Their Coherent States

Let ℍn, n ≥ 1, be the (2n+1)-dimensional Heisenberg group and let K be a closed connected subgroup of the unitary group U(n) acting on ℍn by automorphisms. Using the moment map, we provide in this paper a dequantization procedure for all generic admissible coadjoint orbits of the semidirect product G = K ⋉ ℍn. In the opposite direction, we show that Gilmore-Perelomov's coherent states define “pure state quantizations” of such orbits.

Topological transport in spin-orbit coupled bosonic Mott insulators.

We investigate topological transport in spin-orbit coupled bosonic Mott insulators. We show that interactions can lead to anomalous quasiparticle dynamics even when the spin-orbit coupling is Abelian. To illustrate the latter, we consider the spin-orbit coupling realized in the experiment of Lin etal. [Nature (London) 471, 83 (2011)]. For this spin-orbit coupling, we compute the quasiparticle dispersions and spectral weights, the interaction-induced momentum-space Berry curvature, and the momentum-space distribution of spin density, and propose experimental signatures. Furthermore, we find that, for a generic spin orbit coupling, in our approximation for the single-particle propagator, the ground state can in principle support an integer Hall conductivity if the sum of the Chern numbers of the hole bands is nonzero.

High-order expansions of the Detweiler-Whiting singular field in Schwarzschild spacetime

The self field of a charged particle has a component that diverges at the particle. We use both coordinate and covariant approaches to compute an expansion of this singular field for generic geodesic orbits in Schwarzschild spacetime for scalar, electromagnetic and graviational cases. We check agreement of both approaches and give, as an application, the calculation of previously unknown regularisation parameters. In this so-called "mode-sum regularization" approach, each mode of the field is finite, while their sum diverges. The sum may be rendered finite and convergent by the subtraction of "regularization parameters". Higher order parameters lead to faster convergence in the mode-sum. As a second example application, we compute high order expressions for the effective source approach to self-force calculations.

ESTIMATING HYPERBOLICITY OF CHAOTIC BIDIMENSIONAL MAPS

We apply to bidimensional chaotic maps the numerical method proposed by Ginelli et al. [2007] to approximate the associated Oseledets splitting, i.e. the set of linear subspaces spanned by the so-called covariant Lyapunov vectors (CLV) and corresponding to the Lyapunov spectrum. These subspaces are the analog of linearized invariant manifolds for nonperiodic points, so the angles between them can be used to quantify the degree of hyperbolicity of generic orbits; however, such splitting being noninvariant under smooth transformations of phase space, it is interesting to investigate the properties of transversality when coordinates change, e.g. to study it in distinct dynamical systems. To illustrate this issue on the Chirikov–Taylor standard map, we compare the probability densities of transversality for two different coordinate systems; these are connected by a linear transformation that deforms splitting angles through phase space, changing also the probability density of almost-zero angles although complete tangencies are in fact invariant. This is completely due to the PDF transformation law and strongly suggests that any statistical inference from such distributions must be generally taken with care.

Weak quadratic overgroups for a class of Lie groups

Let \(G\) be a connected and simply connected Lie group with Lie algebra \(\mathfrak g \). We say that a subset \(X\) in the set \(\mathfrak g ^\star / G\) of coadjoint orbits is convex hull separable when the convex hulls differ for any pair of distinct coadjoint orbits in \(X\). In this paper, we define a class of solvable Lie groups, and we give an explicit construction of an overgroup \(G^+\) and a quadratic map \(\varphi \) sending each generic orbit in \(\mathfrak g ^\star \) to a \(G^+\)-orbit in \(\mathfrak{g ^+}^\star \), in such a manner that the set \(\varphi (\mathfrak g ^\star _{gen}){/ G^+}\) is convex hull separable. We then call \(G^+\) a weak quadratic overgroup for \(G\). Thanks to this construction, we prove that any nilpotent Lie group, with dimension at most 7 admits such a weak quadratic overgroup. Finally, we produce different examples of solvable Lie groups, having weak quadratic overgroups, but which are not in our class of Lie groups and for which usual constructions fail to hold.

Integrability of supergravity black holes and new tensor classifiers of regular and nilpotent orbits

In this paper we apply in a systematic way a previously developed integration algorithm of the relevant Lax equation to the construction of spherical symmetric, asymptotically flat black hole solutions of $ \mathcal{N} = 2 $ supergravities with symmetric Special Geometry. Our main goal is the classification of these black-holes according to the H⋆-orbits in which the space of possible Lax operators decomposes. By H⋆ one denotes the isotropy group of the coset UD=3/H⋆ which appears in the time-like dimensional reduction of supergravity from D = 4 to D = 3 dimensions. The main result of our investigation is the construction of three universal tensors, extracted from quadratic and quartic powers of the Lax operator, that are capable of classifying both regular and nilpotent H⋆-orbits of Lax operators. Our tensor based classification is compared, in the case of the simple one-field model S 3, to the algebraic classification of nilpotent orbits and it is shown to provide a simple discriminating method. In particular we present a detailed analysis of the S 3 model, constructing explicitly its solutions and discussing the Liouville integrability of the corresponding dynamical system. By means of the Kostant-representation of a generic Lie algebra element, we were able to develop an algorithm which produces the necessary number of hamiltonians in involution required by Liouville integrability of generic orbits. The degenerate orbits correspond to extremal black-holes and are nilpotent. We present an in depth discussion of their identification and of the construction of the corresponding supergravity solutions. We dwell on the relation between H⋆ orbits and critical points of the geodesic potential showing that there is correspondence yet not one-to-one.

Harmonic structure of generic Kerr orbits

Generic Kerr orbits exhibit intricate three-dimensional motion. We offer a classification scheme for these intricate orbits in terms of periodic orbits. The crucial insight is that for a given effective angular momentum $L$ and angle of inclination $\ensuremath{\iota}$, there exists a discrete set of orbits that are geometrically $n$-leaf clovers in a precessing orbital plane. When viewed in the full three dimensions, these orbits are periodic in $r\ensuremath{-}\ensuremath{\theta}$. Each $n$-leaf clover is associated with a rational number, $1+{q}_{r\ensuremath{\theta}}={\ensuremath{\omega}}_{\ensuremath{\theta}}/{\ensuremath{\omega}}_{r}$, that measures the degree of perihelion precession in the precessing orbital plane. The rational number ${q}_{r\ensuremath{\theta}}$ varies monotonically with the orbital energy and with the orbital eccentricity. Since any bound orbit can be approximated as near one of these periodic $n$-leaf clovers, this special set offers a skeleton that illuminates the structure of all bound Kerr orbits, in or out of the equatorial plane.