Integrals Of Motion Research Articles

Basic Notions of Poisson and Symplectic Geometry in Local Coordinates, with Applications to Hamiltonian Systems

This work contains a brief and elementary exposition of the foundations of Poisson and symplectic geometries, with an emphasis on applications for Hamiltonian systems with second-class constraints. In particular, we clarify the geometric meaning of the Dirac bracket on a symplectic manifold and provide a proof of the Jacobi identity on a Poisson manifold. A number of applications of the Dirac bracket are described: applications for proof of the compatibility of a system consisting of differential and algebraic equations, as well as applications for the problem of the reduction of a Hamiltonian system with known integrals of motion.

Dynamical symmetry algebras of two superintegrable two-dimensional systems

A complete classification of 2D quadratically superintegrable systems with scalar potential on two-dimensional conformally flat spaces has been performed over the years and 58 models, divided into 12 equivalence classes, have been obtained. We will re-examine two pseudo-Hermitian quantum systems E 8 and E 10 from such a classification by a new approach based on extra sets of ladder operators. They correspond in fact to two of those equivalence classes. Those extra ladder operators are exploited to obtain the generating spectrum algebra and the dynamical symmetry one. We will relate the generators of the dynamical symmetry algebra to the Hamiltonian, thus demonstrating that the latter can be written in an algebraic form. We will also link them to the integrals of motion providing the superintegrability property. This demonstrates how the underlying dynamical symmetry algebra allows to write the integrals in terms of its generators and therefore explains the symmetries. Furthermore, we will exploit those algebraic constructions to generate extended sets of states and give the action of the ladder operators on them. We will present polynomials of the Hamiltonian and the integrals of motion that vanish on some of those states, then demonstrating that the sets of states not only contain eigenstates, but also generalized states which are beyond the well-known eigenstates of diagonalizable Hamiltonians and satisfy more complicated polynomial identities. Our approach provides a natural framework for such states.

Double Tisserand graphs for low-energy lunar transfer design

Tisserand graphs are a widespread tool for interplanetary trajectory and Moon tour design. They are based on the Jacobi constant being an integral of motion in the Circular Restricted Three Body Problem (CR3BP); as such, the classical Tisserand graph does not include the perturbation of bodies other than the flyby bodies. Low-energy transfers in the Earth-Moon system make use of the combined Earth, Moon, and Sun gravities, exploiting the third body perturbations to reach the Moon with a reduced transfer Δv. The paper, therefore, proposes a novel double Tisserand graph, where the level lines of both the Earth-Moon and the Sun-Earth CR3BPs are superimposed, portraying the complex 4-body dynamics into a single plot. Paths along such level lines correspond to trajectories utilizing the dynamical effect of the Moon or the Sun or a combination of both. We show how such a graph can be efficiently used for preliminary design of Weak Stability Boundary transfers, lunar resonance transfers, lunar flybys, weak lunar captures or any combination of them.

Spacelike trajectories in BTZ spacetime and comparison with timelike and lightlike trajectories

Spacelike trajectories or the motions of tachyons in (2 + 1) dimensional BTZ spacetime are discussed. A study of the effective potential in this case has also been done and it is shown that unlike that for tardyons or photons it does not have any local or global extremum. It rather possesses an inflection point that depends, along with the blackhole parameters, on the integrals of motion. Tachyons are, however, found to be sensitive, like photons and tardyons, to the usual frame dragging effect due to the rotational feature of the BTZ spacetime and thus pure radial motion cannot persist. It is further found that circular orbits exist nowhere except at the horizon. That the timelike object shows the same behaviour in this case is a remarkable feature of (2 + 1) dimensional gravity. Moreover, the trapping of tachyons at the horizon is also indicated. Comparison of the spacelike trajectories with the timelike and the lightlike trajectories and the corresponding effective potentials have been done analytically and graphically.

An Asymptotic Radius of Convergence for the Loewner Equation and Simulation of $$SLE_{\kappa }$$ Traces via Splitting

In this paper, we shall study the convergence of Taylor approximations for the backward Loewner differential equation (driven by Brownian motion) near the origin. More concretely, whenever the initial condition of the backward Loewner equation (which lies in the upper half plane) is small and has the form $Z_{0} = \varepsilon i$, we show these approximations exhibit an $O(\varepsilon)$ error provided the time horizon is $\varepsilon^{2+\delta}$ for $\delta > 0$. Statements of this theorem will be given using both rough path and $L^{2}(\mathbb{P})$ estimates. Furthermore, over the time horizon of $\varepsilon^{2-\delta}$, we shall see that "higher degree" terms within the Taylor expansion become larger than "lower degree" terms for small $\varepsilon$. In this sense, the time horizon on which approximations are accurate scales like $\varepsilon^{2}$. This scaling comes naturally from the Loewner equation when growing vector field derivatives are balanced against decaying iterated integrals of the Brownian motion. As well as being of theoretical interest, this scaling may be used as a guiding principle for developing adaptive step size strategies which perform efficiently near the origin. In addition, this result highlights the limitations of using stochastic Taylor methods (such as the Euler-Maruyama and Milstein methods) for approximating $SLE_{\kappa}$ traces. Due to the analytically tractable vector fields of the Loewner equation, we will show Ninomiya-Victoir (or Strang) splitting is particularly well suited for SLE simulation. As the singularity at the origin can lead to large numerical errors, we shall employ the adaptive step size proposed by Tom Kennedy to discretize $SLE_{\kappa}$ traces using this splitting. We believe that the Ninomiya-Victoir scheme is the first high order numerical method that has been successfully applied to $SLE_{\kappa}$ traces.

Effects of Mass Variation in the Collinear Perturbed Moulton-Copenhagen Configuration

The main idea of this paper is to investigate the motion properties of the smallest body under the gravitational forces of the three collinear spherical primaries. Here we place the three primaries on the same line where the masses of two primary bodies are taken equal and third primary body is having the solar radiation effect. The effects of Coriolis and centrifugal forces on the system is considered. Therefore this system is recognized as collinear perturbed Moulton-Copenhagen configuration. After determining the equations of motion and quasi-Jacobi integral, we numerically illustrate the locations of equilibrium points (in-plane and out-of-plane), regions of motion, Poincaré surfaces of section, basins of attraction and periodic orbits. And then the examination of stability for the equilibrium points lie either in-plane or out-of-plane are examined.

Substructure in the stellar halo near the Sun

Context. In an accompanying paper, we present a data-driven method for clustering in ‘integrals of motion’ space and apply it to a large sample of nearby halo stars with 6D phase-space information. The algorithm identified a large number of clusters, many of which could tentatively be merged into larger groups. Aims. The goal here is to establish the reality of the clusters and groups through a combined study of their stellar populations (average age, metallicity, and chemical and dynamical properties) to gain more insights into the accretion history of the Milky Way. Methods. To this end, we developed a procedure that quantifies the similarity of clusters based on the Kolmogorov–Smirnov test using their metallicity distribution functions, and an isochrone fitting method to determine their average age, which is also used to compare the distribution of stars in the colour–absolute magnitude diagram. Also taking into consideration how the clusters are distributed in integrals of motion space allows us to group clusters into substructures and to compare substructures with one another. Results. We find that the 67 clusters identified by our algorithm can be merged into 12 extended substructures and 8 small clusters that remain as such. The large substructures include the previously known Gaia-Enceladus, Helmi streams, Sequoia, and Thamnos 1 and 2. We identify a few over-densities that can be associated with the hot thick disc and host a small metal-poor population. Especially notable is the largest (by number of member stars) substructure in our sample which, although peaking at the metallicity characteristic of the thick disc, has a very well populated metal-poor component, and dynamics intermediate between the hot thick disc and the halo. We also identify additional debris in the region occupied by Sequoia with clearly distinct kinematics, likely remnants of three different accretion events with progenitors of similar masses. Although only a small subset of the stars in our sample have chemical abundance information, we are able to identify different trends of [Mg/Fe] versus [Fe/H] for the various substructures, confirming our dissection of the nearby halo. Conclusions. We find that at least 20% of the halo near the Sun is associated to substructures. When comparing their global properties, we note that those substructures on retrograde orbits are not only more metal-poor on average but are also older. We provide a table summarising the properties of the substructures, as well as a membership list that can be used for follow-up chemical abundance studies for example.

Substructure in the stellar halo near the Sun

Context.Merger debris is expected to populate the stellar haloes of galaxies. In the case of the Milky Way, this debris should be apparent as clumps in a space defined by the orbital integrals of motion of the stars.Aims.Our aim is to develop a data-driven and statistics-based method for finding these clumps in integrals-of-motion space for nearby halo stars and to evaluate their significance robustly.Methods.We used data fromGaiaEDR3, extended with radial velocities from ground-based spectroscopic surveys, to construct a sample of halo stars within 2.5 kpc from the Sun. We applied a hierarchical clustering method that makes exhaustive use of the single linkage algorithm in three-dimensional space defined by the commonly used integrals of motion energyE, together with two components of the angular momentum,LzandL⊥. To evaluate the statistical significance of the clusters, we compared the density within an ellipsoidal region centred on the cluster to that of random sets with similar global dynamical properties. By selecting the signal at the location of their maximum statistical significance in the hierarchical tree, we extracted a set of significant unique clusters. By describing these clusters with ellipsoids, we estimated the proximity of a star to the cluster centre using the Mahalanobis distance. Additionally, we applied the HDBSCAN clustering algorithm in velocity space to each cluster to extract subgroups representing debris with different orbital phases.Results.Our procedure identifies 67 highly significant clusters (> 3σ), containing 12% of the sources in our halo set, and 232 subgroups or individual streams in velocity space. In total, 13.8% of the stars in our data set can be confidently associated with a significant cluster based on their Mahalanobis distance. Inspection of the hierarchical tree describing our data set reveals a complex web of relations between the significant clusters, suggesting that they can be tentatively grouped into at least six main large structures, many of which can be associated with previously identified halo substructures, and a number of independent substructures. This preliminary conclusion is further explored in a companion paper, in which we also characterise the substructures in terms of their stellar populations.Conclusions.Our method allows us to systematically detect kinematic substructures in the Galactic stellar halo with a data-driven and interpretable algorithm. The list of the clusters and the associated star catalogue are provided in two tables available at the CDS.

A failed attempt to explain relative motion illusions via motion blur, and a new sparse version.

Visual patterns can evoke marked, even beautiful motion illusions even if they are static; eye movements in all likelihood serve as temporal modulators. This paper concentrates on Ouchi-type “relative” or “sliding” motion illusions. It outlines an eye-motion-evoked motion-blur hypothesis, which does not correctly predict the shift direction of maximal illusion. This failure led to a nearly new particularly simple stimulus: an arrangement of dashed lines that strongly evokes a relative motion illusion, the “orthogonal dotted lines sway.” The latter is well explained by motion integration.

Dynamical essence of the eccentric von Zeipel-Lidov-Kozai effect in restricted hierarchical planetary systems

Aims. The eccentric von Zeipel–Lidov–Kozai (ZLK) effect is widely used to explain dynamical phenomena in a variety of astrophysical systems. The purpose of this work is to clarify the dynamical essence of the eccentric ZLK effect by constructing an inherent connection between this effect and the dynamics of secular resonance in restricted hierarchical planetary systems. Methods. Dynamical structures of apsidal resonance were studied analytically by means of perturbative treatments. The resonant model was formulated by averaging the Hamiltonian (up to octupole order) over rotating ZLK cycles, producing an additional motion integral. The phase portraits under the resonant model can be used to analyse dynamical structures, including resonant centres, dynamical separatrices, and islands of libration. Results. By analysing phase portraits, five branches of libration centres and eight libration zones are found in eccentricity-inclination space. The analytical results of the libration zone and the numerical distributions of the resonant orbit agree very well, indicating that the resonant model for apsidal resonances is valid and applicable. Additionally, we found that in the test-particle limit, the distributions of flipping orbits are dominated by the apsidal resonances that are centred at an inclination of i = 90°. Conclusions. The eccentric ZLK effect is dynamically equivalent to the effect of apsidal resonance in restricted hierarchical planetary systems. The dynamical response of the eccentric ZLK effect (or of the effect of apsidal resonance) is to significantly excite the eccentricities and/or inclinations of test particles in the very long-term evolution.

Krylov complexity of many-body localization: Operator localization in Krylov basis

We study the operator growth problem and its complexity in the many-body localization (MBL) system from the Lanczos algorithm perspective. Using the Krylov basis, the operator growth problem can be viewed as a single-particle hopping problem on a semi-infinite chain with the hopping amplitudes given by the Lanczos coefficients. We find that, in the MBL systems, the Lanczos coefficients scale as \sim n/\ln(n)∼n/ln(n) asymptotically, same as in the ergodic systems, but with an additional even-odd alteration and an effective randomness. We use a simple linear extrapolation scheme as an attempt to extrapolate the Lanczos coefficients to the thermodynamic limit. With the original and extrapolated Lanczos coefficients, we study the properties of the emergent single-particle hopping problem via its spectral function, integrals of motion, Krylov complexity, wavefunction profile and return probability. Our numerical results of the above quantities suggest that the emergent single-particle hopping problem in the MBL system is localized when initialized on the first site. We also study the operator growth in the MBL phenomenological model, whose Lanczos coefficients also have an even-odd alteration, but approach constants asymptotically. The Krylov complexity grows linearly in time in this case.

Generalized para-Bose states

In this paper, we construct integrals of motion in a para-Bose formulation for a general time-dependent quadratic Hamiltonian, which, in its turn, commutes with the reflection operator. In this context, we obtain generalizations for the squeezed vacuum states (SVS) and coherent states (CS) in terms of the Wigner parameter. Furthermore, we show that there is a completeness relation for the generalized SVS owing to the Wigner parameter. In the study of the probability transition, we found that the displacement parameter acts as a transition parameter by allowing access to odd states, while the Wigner parameter controls the dispersion of the distribution. We show that the Wigner parameter is quantized by imposing that the vacuum state has even parity. We apply the general results to the case of the time-independent para-Bose oscillator and find that the mean values of the coordinate and momentum have an oscillatory behavior similarly to the simple harmonic oscillator, while the standard deviation presents corrections in terms of the squeeze, displacement, and Wigner parameters.

Conformal bridge transformation, mathcal{PT} - and supersymmetry

Supersymmetric extensions of the 1D and 2D Swanson models are investigated by applying the conformal bridge transformation (CBT) to the first order Berry-Keating Hamiltonian multiplied by i and its conformally neutral enlargements. The CBT plays the role of the Dyson map that transforms the models into supersymmetric generalizations of the 1D and 2D harmonic oscillator systems, allowing us to define pseudo-Hermitian conjugation and a suitable inner product. In the 1D case, we construct a mathcal{PT} -invariant supersymmetric model with N subsystems by using the conformal generators of supersymmetric free particle, and identify its complete set of the true bosonic and fermionic integrals of motion. We also investigate an exotic N = 2 supersymmetric generalization, in which the higher order supercharges generate nonlinear superalgebras. We generalize the construction for the 2D case to obtain the mathcal{PT} -invariant supersymmetric systems that transform into the spin-1/2 Landau problem with and without an additional Aharonov-Bohm flux, where in the latter case, the well-defined integrals of motion appear only when the flux is quantized. We also build a 2D supersymmetric Hamiltonian related to the “exotic rotational invariant harmonic oscillator” system governed by a dynamical parameter γ. The bosonic and fermionic hidden symmetries for this model are shown to exist for rational values of γ.

Structural properties of local integrals of motion across the many-body localization transition via a fast and efficient method for their construction

Many-body localization (MBL) is a novel prototype of ergodicity breaking due to the emergence of local integrals of motion (LIOMs) in a disordered interacting quantum system. To better understand the role played by the existence of such macroscopically LIOMs, we explore and study some of their structural properties across the MBL transition. We first, consider a one-dimensional XXZ spin chain in a disordered magnetic field and introduce and implement a non-perturbative, fast, and accurate method of constructing LIOMs. In contrast to already existing methods, our scheme allows obtaining LIOMs not only in the deep MBL phase but rather, near the transition point too. Then, we take the matrix representation of LIOM operators as an adjacency matrix of a directed graph whose elements describe the connectivity of ordered eigenbasis in the Hilbert-space. Our cluster size analysis for this graph shows that the MBL transition coincides with a percolation transition in the Hilbert-space. By performing finite-size scaling, we compare the critical disorder and correlation exponent $\nu$ both in the presence and absence of interaction. Finally, we also discuss how the distribution of diagonal elements of LIOM operators in a typical cluster signals the transition.

Detecting Order and Chaos by the mLE, SALI and GALI Methods in Three-Dimensional Nonlinear Yang–Mills System

The main objective of this study is to explore the nonlinear dynamics and chaos detecting in the three-dimensional (3D) generalized Hamiltonian Yang–Mills system as coupled quartic oscillators. We investigate its Liouvillian integrability. The integrable system is obtained and its associated first integral of motion is explicitly given. Also, a variety of nonlinear phenomena could be expressed using various numerical analysis, such as the construction of the system’s Poincaré Surface of Section (PSS), the maximum Lyapunov exponent (mLE), and the modern numerical methods like the Smaller (SALI) and the Generalized (GALI) Alignment Indexes. In this context, a series of numerical simulations are appropriate in order to explore ordered and chaotic motions of the system both in two (2D) and three dimensions, and one can observe chaos–order–chaos transition of the system when any parameters on which the system depends vary. Finally, several numerical schemes are shown to demonstrate the effectiveness and rapidity of these proposed methods for distinguishing chaos and order states of the system.

Transformer-based Cross Reference Network for video salient object detection

• We propose a Transformer-based network architecture for video salient object detection. • We propose a Gated Cross Reference (GCR) module to facilitate collaborative learning between appearance and motion cues. • The proposed Transformer-based method reaches excellent performance on five public benchmarks. Video salient object detection is a fundamental computer vision task aimed at highlighting the most conspicuous objects in a video sequence. There are two key challenges presented in video salient object detection: (1) how to extract effective feature representations from appearance and motion cues, and (2) how to combine both of them into robust saliency representation. To handle these challenges, in this paper, we propose a novel Transformer-based Cross Reference Network (TCRN), which fully exploits long-range context dependencies in both feature representation extraction and cross-modal (i.e., appearance and motion) integration. In contrast to existing CNN-based methods, our approach formulates video salient object detection as a sequence-to-sequence prediction task. In the proposed approach, the deep feature extraction is achieved by a pure vision transformer with multi-resolution token representations. Specifically, we design a Gated Cross Reference (GCR) module to effectively integrate appearance and motion into saliency representation. The GCR first propagates global context information between different modalities, and then perform cross-modal fusion by a gate mechanism. Extensive evaluations on five widely-used benchmarks show that the proposed Transformer-based method performs favorably against the existing state-of-the-art methods

Tackling multiple object tracking with complicated motions — Re-designing the integration of motion and appearance

Although numerous data association methods have been proposed for Multiple Object Tracking (MOT), how to integrate different features in the data association remains an open problem. For instance, over-relying on the motion feature may fail to do the necessary data association when object movements are complicated, while only depending on the appearance feature may lead to incorrect association results when intra-frame objects have similar appearances. To make an improved trade-off between the appearance feature and motion feature, we re-designed the integration of motion and appearance. In our online approach, the location and motion of each object are cast to adaptive searching windows, and within searching windows, matching is only related to the similarity of appearance features. In our offline approach, tracklets generated from our online approach are refined by forming the motion feature as spatiotemporal constraints and utilizing the appearance for clustering. We conduct experiments on multiple MOT datasets from diverse perspectives, including variant motion speed, illumination condition, object categories, etc. , and verify that our method can reach robust performance. Moreover, this method also demonstrates its effectiveness by further improving our previous 1 st place solutions in two CVPR 2020 MOT challenges. • Re-designing a novel integration of motion and appearance to tackle MOT with complicated motions • Reaching robust performance under variant motion speeds, illumination conditions, and object categories • Improving two 1st-place solutions of CVPR’20 WAD MOT Challenge and CVPR’20 MOTS challenge

Latency of Motion Onset Response as an Integrative Measure of Processing Sound Movement

—The characteristics of a specific event-related potential elicited by sound motion onset (motion-onset response, MOR) were estimated. Moving sound stimuli with delayed motion onset were presented to right-handed subjects in dichotic conditions. Sound motion was produced by linear changes of interaural time differences. The amplitude and latency of cN1 and cP2 components of evoked potentials were measured in a wide range of velocities. The latency of cN1 increased linearly with the increase in time of 1° azimuthal shift of a stimulus, whereas the cP2 latency remained nearly constant. The time interval of motion integration was about 140–180 ms. Thus, the latency of cN1 deflection may be considered as an electrophysiological correlate of integrative neuronal processes activated by sound motion presented as a separate auditory event.

Bath-induced delocalization in interacting disordered spin chains

Over a decade of work has culminated in the consensus that one-dimensional systems, subject to sufficiently large disorder, fail to thermalize and possess an extensive set of local integrals of motion. In this Letter, I will provide numerical evidence for the contrary. In particular, this work studies the dynamics of disordered spin chains which are weakly coupled to a Markovian bath. Within this approach, the critical disorder for stability to quantum avalanches exceeds ${W}^{*}\ensuremath{\gtrsim}20$ in the random field Heisenberg chain. In stark contrast to the Anderson insulator, the avalanche threshold drifts considerably with system size, with no evidence of saturation in the studied regime.

Milky Way mass with K giants and BHB stars using LAMOST, SDSS/SEGUE, and Gaia: 3D spherical Jeans equation and tracer mass estimator

ABSTRACT We measure the enclosed Milky Way mass profile to Galactocentric distances of ∼70 and ∼50 kpc using the smooth, diffuse stellar halo samples of Bird et al. The samples are Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST) and Sloan Digital Sky Survey/Sloan Extension for Galactic Understanding and Exploration (SDSS/SEGUE) K giants (KG) and SDSS/SEGUE blue horizontal branch (BHB) stars with accurate metallicities. The 3D kinematics are available through LAMOST and SDSS/SEGUE distances and radial velocities and Gaia DR2 proper motions. Two methods are used to estimate the enclosed mass: 3D spherical Jeans equation and Evans et al. tracer mass estimator (TME). We remove substructure via the Xue et al. method based on integrals of motion. We evaluate the uncertainties on our estimates due to random sampling noise, systematic distance errors, the adopted density profile, and non-virialization and non-spherical effects of the halo. The tracer density profile remains a limiting systematic in our mass estimates, although within these limits we find reasonable agreement across the different samples and the methods applied. Out to ∼70 and ∼50 kpc, the Jeans method yields total enclosed masses of 4.3 ± 0.95 (random) ±0.6 (systematic) × 1011 M⊙ and 4.1 ± 1.2 (random) ±0.6 (systematic) × 1011 M⊙ for the KG and BHB stars, respectively. For the KG and BHB samples, we find a dark matter virial mass of $M_{200}=0.55^{+0.15}_{-0.11}$ (random) ±0.083 (systematic) × 1012 M⊙ and $M_{200}=1.00^{+0.67}_{-0.33}$ (random) ±0.15 (systematic) × 1012 M⊙, respectively.