Integrals Of Motion Research Articles

Axisymmetric hybrid Vlasov equilibria with applications to tokamak plasmas

We derive axisymmetric equilibrium equations in the context of the hybrid Vlasov model with kinetic ions and massless fluid electrons, assuming isothermal electrons and deformed Maxwellian distribution functions for the kinetic ions. The equilibrium system comprises a Grad–Shafranov partial differential equation and an integral equation. These equations can be utilized to calculate the equilibrium magnetic field and ion distribution function, respectively, for given particle density or given ion and electron toroidal current density profiles. The resulting solutions describe states characterized by toroidal plasma rotation and toroidal electric current density. Additionally, due to the presence of fluid electrons, these equilibria also exhibit a poloidal current density component. This is in contrast to the fully kinetic Vlasov model, where axisymmetric Jeans equilibria can only accommodate toroidal currents and flows, given the absence of a third integral of the microscopic motion.

Temporal dynamics of ocular torsion and vertical vergence during visual, vestibular, and visuovestibular rotations.

Ocular torsion and vertical divergence reflect the brain's sensorimotor integration of motion through the vestibulo-ocular reflex (VOR) and the optokinetic reflex (OKR) to roll rotations. Torsion and vergence however express different response patterns depending on several motion variables, but research on their temporal dynamics remains limited. This study investigated the onset times of ocular torsion (OT) and vertical vergence (VV) during visual, vestibular, and visuovestibular motion, as well as their relative decay rates following prolonged optokinetic stimulations. Temporal characteristics were retrieved from three separate investigations where the level of visual clutter and acceleration were controlled. Video eye-tracking was used to retrieve the eye-movement parameters from a total of 41 healthy participants across all trials. Ocular torsion consistently initiated earlier than vertical vergence, particularly evident under intensified visual information density, and higher clutter levels were associated with more balanced decay rates. Additionally, stimulation modality and accelerations affected the onsets of both eye movements, with visuovestibular motion triggering earlier responses compared to vestibular motion, and increased accelerations leading to earlier onsets for both movements. The present study showed that joint visuovestibular responses produced more rapid onsets, indicating a synergetic sensorimotor process. It also showed that visual content acted as a fusional force during the decay period, and imposed greater influence over the torsional onset compared to vergence. Acceleration, by contrast, did not affect the temporal relationship between the two eye movements. Altogether, these findings provide insights into the sensorimotor integration of the vestibulo-ocular and optokinetic reflex arcs.

A Walk on the Retrograde Side (WRS) project

Relics of ancient accretion events experienced by the Milky Way are predominantly located within the stellar halo of our Galaxy. However, debris from different objects display overlapping distributions in dynamical spaces, making it extremely challenging to properly disentangle their contribution to the build-up of the Galaxy. To shed light on this chaotic context, we initiated a program aimed at the homogeneous chemical tagging of the local halo of the Milky Way, focusing on the component in retrograde motion, since this is expected to host a large fraction of stars accreted from past mergers. The A Walk on the Retrograde Side (WRS) project targets retrograde halo stars in the solar neighborhood with accurate 6D phase space information available, measuring the precise chemical abundance of several chemical elements from high-resolution spectroscopy. In this first paper, we present the project and the analysis of high-resolution spectra obtained with UVES at VLT and at LBT for 186 stars. We obtained accurate radial velocity and chemical abundances for several elements for all the target stars. In particular, we focus on the chemical composition of a specific subset of substructures that have been dynamically identified in the literature. Our study reveals that two among the more recently discovered structures in the retrograde halo, namely, Antaeus/L-RL64 and ED-3, have identical chemical patterns and similar integrals of motion, suggesting a common origin. In turn, the abundance patterns of this unified system differ from that of Gaia-Enceladus, confirming that it is an independent structure. Finally, Sequoia exhibits a different chemistry with respect to that of Gaia-Enceladus at [Fe/H] < −1.5 dex, showcasing an excess of stars with lower Mg and Ca in the common metallicity range.

Fractional diamagnetic Kepler problem and elastic collisions

In this work, we consider an application of fractional derivatives to realistic physical situations, namely the elastic collision of particles and the nonintegrable diamagnetic Kepler problem. The origin of fractional dynamics can be nonlocal interacting dynamics, memory effects, environments with fractal interacting properties, and relaxation processes, among others. In the case of collisions, considering identical and distinguishable particles, additional solutions appear compared to non-fractional dynamics. For specific velocities of one particle before the collision, several velocities of the other particle are allowed after the collision. Consequently, novel velocity distributions emerge. For the diamagnetic Kepler problem, the fractional dynamic strongly affects the regular and quasi-regular regimes of motion, while the completely chaotic motion remains essentially unaltered. Besides, we derive a fractional momentum-like integral of motion for the pure fractional Kepler problem.

Painlevé analysis, Prelle–Singer approach, symmetries and integrability of damped Hénon–Heiles system

We consider a modified damped version of Hénon–Heiles system and investigate its integrability. By extending the Painlevé analysis of ordinary differential equations we find that the modified Hénon–Heiles system possesses the Painlevé property for three distinct parametric restrictions. For each of the identified cases, we construct two independent integrals of motion using the well known Prelle–Singer method. We then derive a set of nontrivial non-point symmetries for each of the identified integrable cases of the modified Hénon–Heiles system. We infer that the modified Hénon–Heiles system is integrable for three distinct parametric restrictions. Exact solutions are given explicitly for two integrable cases.

Turbulence on open string worldsheets under non-integrable boundary conditions

We demonstrate the turbulent dynamics of the Nambu-Goto open string in the AdS3 spacetime. While the motion of a classical closed string in AdS is known to be integrable, the integrability of an open string motion depends on the boundary conditions at the string endpoints. We numerically solve the equations of motion of the open string under the boundary conditions where the endpoints are i) fixed to a finite radial coordinate in AdS, and ii) free. For i), we find turbulence on the string, that shows a cascade in the energy and angular momentum spectra. This result indicates the non-integrability of the open string with this type of boundary conditions. For ii), we find no turbulence. This is consistent with the integrability of the open string with the free boundary conditions.

Pure Decoherence of the Jaynes–Cummings Model: Initial Entanglement with the Environment, Spin Oscillations and Detection of Non-Orthogonal States

A model based on pure decoherence for the Jaynes–Cummings spin–boson system, coupled through its integral of motion to an infinite bosonic bath, is proposed and examined. The properties of the spin oscillation process suggest an initial entanglement between the environment and the spin–boson degrees of freedom. The study demonstrates that the potential applicability of the Jaynes–Cummings model in detecting non-orthogonal bosonic states is preserved in the presence of pure decoherence.

A new approach to integrals of discretizations by polarization

Recently, a family of unconventional integrators for ODEs with polynomial vector fields was proposed, based on the polarization of vector fields. The simplest instance is the by now famous Kahan discretization for quadratic vector fields. All these integrators seem to possess remarkable conservation properties. In particular, it has been proved that, when the underlying ODE is Hamiltonian, its polarization discretization possesses an integral of motion and an invariant volume form. In this note, we propose a new algebraic approach to derivation of the integrals of motion for polarization discretizations.

Lagrangian stochastic integrals of motion in isotropic random flows

A set of exact integrals of motion is found for systems driven by homogenous isotropic stochastic flow. The integrals of motion describe the evolution of (hyper-)surfaces of different dimensions transported by the flow and can be expressed in terms of local surface densities. The expression for the integrals is universal: it represents general geometric properties and does not depend on the statistics of the specific flow.

Quantum integrability and chaos in a periodic Toda lattice with balanced loss-gain.

We consider an equal-mass quantum Toda lattice with balanced loss-gain for two and three particles. The two-particle Toda lattice is integrable, and two integrals of motion that are in involution have been found. The bound-state energy and the corresponding eigenfunctions have been obtained numerically for a few low-lying states. The three-particle quantum Toda lattice with balanced loss-gain and velocity-mediated coupling admits mixed phases of integrability and chaos depending on the value of the loss-gain parameter. We have obtained analytic expressions for two integrals of motion that are in involution. Although an analytic expression for the third integral has not been found, the numerical investigation suggests integrability below a critical value of the loss-gain strength and chaos above this critical value. The level spacing distribution changes from the Wigner-Dyson to the Poisson distribution as the loss-gain parameter passes through this critical value and approaches zero. An identical behavior is seen in terms of the gap-ratio distribution of the energy levels. The existence of mixed phases of quantum integrability and chaos in the specified ranges of the loss-gain parameter has also been confirmed independently via the study of level repulsion and complexity in higher order excited states.

Stellar/BH population in AGN discs: direct binary formation from capture objects in nuclei clusters

ABSTRACT The active galactic nucleus (AGN) disc has been proposed as a potential channel for the merger of binary black holes. The population of massive stars and black holes in AGN discs captured from the nuclei cluster plays a crucial role in determining the efficiency of binary formation and final merger rate within the AGN discs. In this paper, we investigate the capture process using analytical and numerical approaches. We discover a new constant integral of motion for one object’s capture process. Applying this result to the whole population of the nuclei cluster captured by the AGN disc, we find that the population of captured objects depends on the angular density and eccentricity distribution of the nuclei clusters and is effectively independent of the radial density profile of the nuclei cluster and disc models. An isotropic nuclei cluster with thermal eccentricity distribution predicts a captured profile dN/dr ∝ r−1/4. The captured objects are found to be dynamically crowded within the disc. Direct binary formation right after the capture would be promising, especially for stars. The conventional migration traps that help pile up single objects in AGN discs for black hole mergers might not be required.

Relativistic Phase Space Diffusion of Compact Object Binaries in Stellar Clusters and Hierarchical Triples

The LIGO/Virgo detections of compact object mergers have posed a challenge for theories of binary evolution and coalescence. One promising avenue for producing mergers dynamically is through secular eccentricity oscillations driven by an external perturber, be it a tertiary companion (as in the Lidov–Kozai, LK, mechanism) or the tidal field of the stellar cluster in which the binary orbits. The simplest theoretical models of these oscillations use a “doubly averaged” (DA) approximation, averaging both over the binary’s internal Keplerian orbit and its “outer” barycentric orbit relative to the perturber. However, DA theories do not account for fluctuations of the perturbing torque on the outer orbital timescale, which are known to increase a binary’s eccentricity beyond the maximum DA value, potentially accelerating mergers. Here we reconsider the impact of these short-timescale fluctuations in the test-particle quadrupolar limit for binaries perturbed by arbitrary spherical cluster potentials (including LK as a special case), in particular including 1pN general relativistic (GR) apsidal precession of the internal orbit. Focusing on the behavior of the binary orbital elements around peak eccentricity, we discover a new effect, relativistic phase space diffusion (RPSD), in which a binary can jump to a completely new dynamical trajectory on an outer orbital timescale, violating the approximate conservation of DA integrals of motion. RPSD arises from an interplay between secular behavior at extremely high eccentricity, short-timescale fluctuations, and rapid GR precession, and can change the subsequent secular evolution dramatically. This effect occurs even in hierarchical triples, but has not been uncovered until now.

Local integrals of motion and the stability of many-body localization in Wannier-Stark potentials

Local integrals of motion and the stability of many-body localization in Wannier-Stark potentials

On the generalization of classical Zernike system

We generalize the results obtained recently (Blasco et al 2023 Nonlinearity 36 1143) by providing a very simple proof of the superintegrability of the Hamiltonian H=p⃗2+F(q⃗⋅p⃗) , q⃗,p⃗∈R2 , for any analytic function F. The additional integral of motion is constructed explicitly and shown to reduce to a polynomial in canonical variables for polynomial F. The generalization to the case q⃗,p⃗∈Rn is sketched.

Measuring out quasi-local integrals of motion from entanglement

Quasi-local integrals of motion are a key concept underpinning the modern understanding of many-body localisation, a phenomenon in which interactions and disorder come together. Despite the existence of several numerical ways to compute them—and in the light of the observation that much of the phenomenology of many properties can be derived from them—it is not obvious how to directly measure aspects of them in real quantum simulations; in fact, hard experimental evidence is still missing. In this work, we propose a way to extract the real-space properties of such quasi-local integrals of motion based on a spatially-resolved entanglement probe able to distinguish Anderson from many-body localisation from non-equilibrium dynamics. We complement these findings with a rigorous entanglement bound and compute the relevant quantities using tensor networks. We demonstrate that the entanglement gives rise to a well-defined length scale that can be measured in experiments.

Discrete gradients in short-range molecular dynamics simulations

AbstractDiscrete gradients (DG) or more exactly discrete gradient methods are time integration schemes that are custom-built to preserve first integrals or Lyapunov functions of a given ordinary differential equation (ODE). In conservative molecular dynamics (MD) simulations, the energy of the system is constant and therefore a first integral of motion. Hence, discrete gradient methods seem to be a natural choice as an integration scheme in conservative molecular dynamics simulations.

On rotation invariant integrable systems

The problem of finding the first integrals of the Newton equations in the $n$-dimensional Euclidean space is reduced to that of finding two integrals of motion on the Lie algebra $\mathrm{so}(4)$ which are invariant under $m\geq n-2$ rotation symmetry fields. As an example, we obtain several families of integrable and superintegrable systems with first, second, and fourth-degree integrals of motion in the momenta. The corresponding Hamilton-Jacobi equation does not admit separation variables in any of the known curvilinear orthogonal coordinate systems in the Euclidean space.

Classical and Quantum Descriptions of the Geometrodynamics of Black Holes

Analytical aspects of the classical and quantum geometrodynamics of charged black hole are considered. We start with a reduced action for a spherically symmetric of Maxwell-Einstein system written in characteristic variables. The feature of these configurations is that they admit two motion integrals, the total mass and charge. Momenta and constraint are introduced. Using the conservation laws and the Hamiltonian constraint, the momenta as functions of configuration variables are found. It turns out that the system of equations relating momenta and functional derivatives of an action on a configuration space (CS) is integrable. This allows us to obtain the action functional, as a solution of the Einstein-Hamilton-Jacobi equation in functional derivatives. Variations of the action functional with respect to mass and charge of the configuration lead to the motion trajectories in the CS. Further, the space-temporal action is transformed into an action in the configuration space similar to the Jacobi action of classical mechanics. This induces a metric in the CS. Thus, the metric on the CS is introduced and its geometry is studied. The field variables transformation is obtained which brings the CS metric to the "quasi-Lorentzian" form. On this basis, quantization is considered. Taking into account the structure of the CS, the momentum operators, DeWitt equations, mass and charge operators are constructed. Further, for comparison, consider the reduced CBH model defined in the T-region. In this simplified formulation, the T-model equations are integrated and lead to CBH with a continuous spectrum of mass and charge.

The Jacobi last multiplier, Lagrangian and Hamiltonian for Levinson–Smith type equations

We derive the Jacobi last multiplier for second-order ordinary differential equations of the Levinson–Smith type by using a combination of previous techniques employed for the Liénard-I and II classes of equations. This opens up the possibility for a Lagrangian or Hamiltonian description of the systems governed by the Levinson–Smith type of equations as well as simplifying the problem of finding first integrals of motion. The procedure has been illustrated by a number of suitable examples alongwith Kamke's equation with explicit time dependent coefficients.

Solutions of Maxwell equations for admissible electromagnetic fields, in spaces with simply transitive four-parameter groups of motions

All non-equivalent solutions of vacuum Maxwell equations are found for the case when space-time manifolds admit simply transitive four-parameter groups of motions [Formula: see text]. The potentials of the admissible electromagnetic fields admit the existence of the algebra of motion integrals of the Hamilton–Jacobi and Klein–Gordon–Fock equations which is isomorphic to the algebra of the group operators for the same group [Formula: see text].