Runge-Lenz Research Articles

[formula omitted] symmetry in hydrogen-atom-like model with spin

As the simplest atom in nature, the hydrogen atom has been explored thoroughly from the perspective of non-relativistic quantum mechanics to relativistic quantum mechanics. Among the research on hydrogen atom, its energy level is the most basic, which can be obtained more conveniently predicated on the SO(4) symmetry than the wave-equation resolution. Moreover, “spin” is another indispensable topic in quantum mechanics, appearing as an intrinsic degree of freedom. In this work, we generalize the quantum Runge-Lenz vector to a spin-dependent one, and then extract a novel Hamiltonian of hydrogen-atom-like model with spin based on the requirement of SO(4) symmetry. Furthermore, the energy spectrum of hydrogen-atom-like model with spin potentials is also determined by the remarkable approach of SO(4) symmetry. Our findings extend the ground of hydrogen atom, and may contribute to other complicated models based on hydrogen atom.

Close Binary Stars Modeled by Two Prolate Ellipsoids in Synchronous Rotation

The presence of tidal deformations in close binary stars has already been confirmed by astronomical observations. The present paper aims to simply address an astronomy problem, studying the relative movement of close binaries disturbed by their mutual deformation through some basic concepts and tools of celestial mechanics. For this purpose, the tidal effect is modeled by considering that each star is an elongated revolution ellipsoid in such a way that axes of revolution are coincident, and their largest axes point toward each other along the motion. The potential for mutual attraction is then obtained, resulting in a perturbed Keplerian system with perturbation proportional to the inverse of the cubic distance between the stars, thus being a one-degree-of-freedom problem and, therefore, integrable. The effective potential, the integrals of energy and angular momentum, and the Laplace vector are used to obtain qualitative information about the dynamics before integrating it. The motion describes a rosette-like orbit with periodic osculating elements, or a circle when the energy is a local minimum. Finally, an analytical solution is presented in terms of elliptic functions by using a regularizing and linearizing function.

Analog of a Laplace–Runge–Lenz vector for particle orbits (time-like geodesics) in Schwarzschild spacetime

In Schwarzschild spacetime, time-like geodesic equations, which define particle orbits, have a well-known formulation as a dynamical system in coordinates adapted to the time-like hypersurface containing the geodesic. For equatorial geodesics, the resulting dynamical system is shown to possess a conserved angular quantity and two conserved temporal quantities, whose properties and physical meaning are analogs of the conserved Laplace–Runge–Lenz vector, and its variant known as Hamilton’s vector, in Newtonian gravity. When a particle orbit is projected into the spatial equatorial plane, the angular quantity yields the coordinate angle at which the orbit has either a turning point (where the radial velocity is zero) or a centripetal point (where the radial acceleration is zero). This is the same property as the angle of the respective Laplace–Runge–Lenz and Hamilton vectors in the plane of motion in Newtonian gravity. The temporal quantities yield the coordinate time and the proper time at which those points are reached on the orbit. In general, for orbits that have a single turning point, the three quantities are globally constant, and for orbits that possess more than one turning point, the temporal quantities are just locally constant as they jump at every successive turning point, while the angular quantity similarly jumps only if an orbit is precessing. This is analogous to the properties of a generalized Laplace–Runge–Lenz vector and generalized Hamilton vector which are known to exist for precessing orbits in post-Newtonian gravity. The angular conserved quantity is used to define a direct analog of these vectors at spatial infinity.

New vistas on the Laplace–Runge–Lenz vector

New vistas on the Laplace–Runge–Lenz vector

Runge–Lenz vector as a 3d projection of SO(4) moment map in phase space

We show, using the methods of geometric algebra, that Runge–Lenz vector in the Kepler problem is a 3-dimensional projection of SO(4) moment map that acts on the phase space of 4-dimensional particle motion. Thus, Runge–Lenz vector is a consequence of geometric symmetry of phase space.

A Novel Potential Featuring Off-Center Circular Orbits

In Book 1, Proposition 7, Problem 2 of his 1687 Philosophiae Naturalis Principia Mathematica, Isaac Newton poses and answers the following question: Let the orbit of a particle moving in a central force field be an off-center circle. How does the magnitude of the force depend on the position of the particle onthat circle? In this article, we identify a potential that can produce such a force, only at zero energy. We further map the zero-energy orbits in this potential to finite-energy free motion orbits on a sphere; such a duality is a particular instance of a general result by Goursat, from 1887. The map itself is an inverse stereographic projection, and this fact explains the circularity of the zero-energy orbits in the system of interest. Finally, we identify an additional integral of motion - an analogue of the Runge-Lenz vector in the Coulomb problem - that is responsible for the closeness of the zero-energy orbits in our problem.

Comparative study of Mercury’s perihelion advance

Mercury’s motion has been studied using numerical methods in the framework of a model including only the non-relativistic Newtonian gravitational interactions of the solar system: eight major planets and Pluto in translation around the Sun. Since the true trajectory of Mercury is an open, non-planar curve, special attention to the exact definition of the advance of Mercury’s perihelion has been given. For this purpose, the concepts of an extended and a geometrical perihelion have been introduced. In addition, for each orbital period, a mean ellipse was fitted to the trajectory of Mercury. I have shown that the perihelion advance of Mercury deduced from the behavior of the Laplace–Runge–Lenz vector, as well as from the extended and geometrical perihelion advance depend on the fitting time interval and for intervals of the order of 1 000 years converge to a value of 532.1 $$^{\prime \prime }$$ per century. The behavior of the perihelia, either extended or geometrical, is strongly impacted by the influence of Jupiter. The advance of the extended perihelion depends on the time step used in the calculations, while the advance of the geometrical perihelion and that deduced by the rotation of the Laplace–Runge–Lenz vector depends only slightly on it.

Algebra of Dunkl Laplace–Runge–Lenz vector

We introduce the Dunkl version of the Laplace–Runge–Lenz vector associated with a finite Coxeter group W acting geometrically in mathbb R^N and with a multiplicity function g. This vector generalizes the usual Laplace–Runge–Lenz vector and its components commute with the Dunkl–Coulomb Hamiltonian given as the Dunkl Laplacian with an additional Coulomb potential gamma /r. We study the resulting symmetry algebra R_{g, gamma }(W) and show that it has the Poincaré–Birkhoff–Witt property. In the absence of a Coulomb potential, this symmetry algebra R_{g,0}(W) is a subalgebra of the rational Cherednik algebra H_g(W). We show that a central quotient of the algebra R_{g, gamma }(W) is a quadratic algebra isomorphic to a central quotient of the corresponding Dunkl angular momenta algebra H_g^{so(N+1)}(W). This gives an interpretation of the algebra H_g^{so(N+1)}(W) as the hidden symmetry algebra of the Dunkl–Coulomb problem in mathbb {R}^N. By specialising R_{g, gamma }(W) to g=0, we recover a quotient of the universal enveloping algebra U(so(N+1)) as the hidden symmetry algebra of the Coulomb problem in {mathbb R}^N. We also apply the Dunkl Laplace–Runge–Lenz vector to establish the maximal superintegrability of the generalised Calogero–Moser systems.

Review of the Dynamics of Atomic and Molecular Systems of Higher than Geometric Symmetry—Part I: One-Electron Rydberg Quasimolecules

The review covers the dynamics of different kinds of one electron Rydberg quasimolecules in various environments, such as being subjected to electric and/or magnetic fields or to a plasma environment. The higher than geometrical symmetry of these systems is due to the existence of an additional conserved quantity: the projection of the supergeneralized Runge–Lenz vector on the internuclear axis. The review emphasizes the fundamental and practical importance of the results concerning the dynamics of these systems.

Monodromy in prolate spheroidal harmonics

AbstractWe show that spheroidal wave functions viewed as the essential part of the joint eigenfunctions of two commuting operators onhave a defect in the joint spectrum that makes a global labeling of the joint eigenfunctions by quantum numbers impossible. To our knowledge, this is the first explicit demonstration that quantum monodromy exists in a class of classically known special functions. Using an analog of the Laplace–Runge–Lenz vector we show that the corresponding classical Liouville integrable system is symplectically equivalent to the C. Neumann system. To prove the existence of this defect, we construct a classical integrable system that is the semiclassical limit of the quantum integrable system of commuting operators. We show that this is a generalized semitoric system with a nondegenerate focus–focus point, such that there is monodromy in the classical and the quantum systems.

A shadow of the repulsive Rutherford scattering and Hamilton vector

The fact that repulsive Rutherford scattering casts a paraboloidal shadow is rarely exploited in introductory mechanics textbooks. Another rarely used construction in such textbooks is the Hamilton vector, a cousin of the more famous Laplace–Runge–Lenz vector. We will show how the latter (Hamilton’s vector) can be used to explain and clarify the former (paraboloidal shadow).

Algebraic approach to the Dunkl–Coulomb problem and Dunkl oscillator in arbitrary dimensions

The Dunkl–Coulomb and the Dunkl oscillator models in arbitrary space-dimensions are introduced. These models are shown to be maximally superintegrable and exactly solvable. The energy spectrum and the wave functions of both systems are obtained using different realizations of the Lie algebra $$\text{ so }(1,2).$$ The N-dimensional Dunkl oscillator admits separation of variables in both Cartesian and polar coordinates and the corresponding separated solutions are expressed in terms of the generalized Hermite, Laguerre and Gegenbauer polynomials. The N-dimensional Dunkl–Coulomb Hamiltonian admits separation of variables in polar coordinates and the separated wave functions are expressed in terms of the generalized Laguerre and Gegenbauer polynomials. The symmetry operators generalizing the Runge–Lenz vector operator are given. Together with the Dunkl angular momentum operators and reflection operators they generate the symmetry algebra of the N-dimensional Dunkl–Coulomb Hamiltonian which is a deformation of $$\text{ so }(N+1)$$ by reflections for bound states and is a deformation of $$\text{ so }(N,1)$$ by reflections for positive energy states. The symmetry operators of the N-dimensional Dunkl oscillator are obtained by the Schwinger construction and generate its invariance algebra which is a deformation of $${\textit{su}}(N)$$ by reflections.

Reduced 4D oscillators and orbital elements in Keplerian systems: Cushman–Deprit coordinates

We study the reduction and regularization processes of perturbed Keplerian systems from an astronomical point of view. Our approach connects axially symmetric perturbed 4-DOF oscillators with Keplerian systems, including the case of rectilinear solutions. This is done through a preliminary reduction recently studied by the authors. Then, the reduction program continues by removing the Keplerian energy. For each value of the semi-major axis, we explain the astronomical meaning of the sextuples defining the orbit space $$\mathbb {S}^2\times \mathbb {S}^2$$ and its connection with the orbital elements. More precisely, we present alternative sextuple coordinates for the set of bounded Keplerian orbits that ‘separate’ the node of the orbital plane from the argument of perigee giving the Laplace vector in that plane. Still, the reduction of the axial symmetry defined by the third component of the angular momentum is performed. For the thrice reduced space $$\varGamma _{0,L,H}$$ we propose the Cushman–Deprit coordinates, a variant to the set given by Cushman. The main feature of these variables is that they are all with the same dimensions, which is convenient for the normalization procedure. As an application of the proposed scheme, we study the spatial lunar problem.

Three body first post-Newtonian effects on the secular dynamics of a compact binary near a spinning supermassive black hole

The binary black holes (BBHs) formed near the supermassive black holes (SMBHs) in the galactic nuclei would undergo eccentricity excitation due to the gravitational perturbations from the SMBH and therefore merger more efficiently. In this paper, we study the coupling of the three body 1st post-Newtonian (PN) effects with the spin effects from the SMBH in the hierarchical triple system. We extend previous work by including the coupling between the de Sitter precession and the Lense-Thirring precession from the SMBH spin. This coupling includes both the precessions of the inner orbit angular momentum and the Runge-Lenz vector around the outer orbit angular momentum in a general reference frame. We find the change of the (maximal) eccentricity in the neighboring Kozai-Lidov cycles due to spin effects is detectable by LISA in the future. Our general argument on the coupling of the three body 1PN effects in three body systems could be extended to any other situation as long as the outer orbital plane evolves.

A new correction method for quasi-Keplerian orbits

A pure two-body problem has seven integrals including the Kepler energy, the Laplace vector and the angular momentum vector. However, only five of them are independent. When the five independent integrals are preserved, the two other dependent integrals are naturally preserved from a theoretical viewpoint; but they may not necessarily be from a numerical computational viewpoint. Because of this, we use seven scale factors to adjust the integrated positions and velocities so that the adjusted solutions strictly satisfy the seven constraints. Noticing the existence of the two dependent integrals, we adopt the Newton iterative method combined with singular value decomposition to calculate these factors. This correction scheme can be applied to perturbed two-body and N-body problems in the solar system. In this case, the seven quantities associated with each planet slowly vary with time. More accurate values can be given to the seven slowly-varying quantities by integrating the integral invariant relations of these quantities and the equations of motion. They should be satisfied with the adjusted solutions. Numerical tests show that the new method can significantly reduce the rapid growth of numerical errors for all orbital elements.

Privacy-preserving quality prediction for edge-based IoT services

Quality of Service (QoS) prediction and privacy preservation are two key challenges in service recommendation. Nevertheless, the existing QoS prediction methods cannot be directly utilized in edge computing networks (ECNs) due to the user mobility and distribution nature in these networks. To address this problem, the DEQP2 model, a distributed edge QoS prediction model with privacy-preserving, is proposed in this paper. In the DEQP2 model, the Laplace vector mechanism is first employed for distributed privacy protection processing at the edge. Then, the distributed edge differential privacy (DEDP) QoS prediction algorithm is proposed, to enable a comprehensive consideration of the influence of user preferences and the edge environment. Finally, we conduct experiments on the EdgeQoS dataset, which is an integrated dataset derived from the WSDream and Telecom real-world datasets. The experimental results show that the proposed DEQP2 model provides measurable privacy preservation without significantly reducing the accuracy.

The use of Kepler solver in numerical integrations of quasi-Keplerian orbits

ABSTRACT A Kepler solver is an analytical method used to solve a two-body problem. In this paper, we propose a new correction method by slightly modifying the Kepler solver. The only change to the analytical solutions is that the obtainment of the eccentric anomaly relies on the true anomaly that is associated with a unit radial vector calculated by an integrator. This scheme rigorously conserves all integrals and orbital elements except the mean longitude. However, the Kepler energy, angular momentum vector, and Laplace–Runge–Lenz vector for perturbed Kepler problems are slowly varying quantities. However, their integral invariant relations give the quantities high-precision values that directly govern five slowly varying orbital elements. These elements combined with the eccentric anomaly determine the desired numerical solutions. The newly proposed method can considerably reduce various errors for a post-Newtonian two-body problem compared with an uncorrected integrator, making it suitable for a dissipative two-body problem. Spurious secular changes of some elements or quasi-integrals in the outer Solar system may be caused by short integration times of the fourth-order Runge–Kutta algorithm. However, they can be eliminated in a long integration time of 108 yr by the proposed method, similar to Wisdom–Holman second-order symplectic integrator. The proposed method has an advantage over the symplectic algorithm in the accuracy but gives a larger slope to the phase error growth.

Conserved quantities in parity-time symmetric systems

Conserved quantities such as energy or the electric charge of a closed system, or the Runge-Lenz vector in Kepler dynamics are determined by its global, local, or accidental symmetries. They were instrumental to advances such as the prediction of neutrinos in the (inverse) beta decay process and the development of self-consistent approximate methods for isolated or thermal many-body systems. In contrast, little is known about conservation laws and their consequences in open systems. Recently, a special class of these systems, called parity-time ($\mathcal{PT}$) symmetric systems, has been intensely explored for their remarkable properties that are absent in their closed counterparts. A complete characterization and observation of conserved quantities in these systems and their consequences is still lacking. Here we present a complete set of conserved observables for a broad class of $\mathcal{PT}$-symmetric Hamiltonians and experimentally demonstrate their properties using single-photon linear optical circuit. By simulating the dynamics of a four-site system across a fourth-order exceptional point, we measure its four conserved quantities and demonstrate their consequences. Our results spell out non-local conservation laws in non-unitary dynamics and provide key elements that will underpin self-consistent analysis of non-Hermitian quantum many-body systems that are forthcoming.

The classical Lenz vector and the two-dimensional quantum harmonic oscillator

Both the two-dimensional harmonic oscillator and the Newton potential allow particular solutions for the orbits which are ellipses with center of attraction in the center, in the first case, and in one focus, in the second. The same complex map which allows to go from Kepler’s to Hooke’s orbits, and back, is used to transform the Lenz vector, defined for the Kepler orbit, into two conserved quantities for the harmonic motion. Upon quantization, the resulting operators, together with the angular momentum Lz, are found to correspond to the generators of the SU(2) internal symmetry of the two-dimensional quantum oscillator and the connection to the Schwinger model of angular momentum is made apparent. We give a self-contained new look on this topic.

A dissipative Kepler problem with a family of singular drags

In this work, we consider the Kepler problem with a family of singular dissipations of the form $$-\frac{k}{|x|^\beta }\dot{x},\quad k>0, \beta >0.$$ We present some results about the qualitative dynamics as $$\beta $$ increases from zero (linear drag) to infinity. In particular, we detect some threshold values of $$\beta $$, for which qualitative changes in the global dynamics occur. In the case $$\beta =2$$, we refine some results obtained by Diacu and prove that, unlike for the case of the linear drag, the asymptotic Runge–Lenz vector is discontinuous.