Kepler Problem Research Articles

Kuryshkin-Wodkiewicz quantum measurement model for alkaline metal atoms

The constructive form of the Kuryshkin-Wodkiewicz model of quantum measurements was earlier developed in detail for the quantum Kepler problem. For more complex quantum objects, such a construction is unknown. At the same time, the standard (non-constructive) model of Holevo-Helstrom quantum measurements is suitable for any quantum object. In this work, the constructive model of quantum measurements is generalized to a wider class of quantum objects, i.e., the optical spectrum of atoms and ions with one valence electron. The analysis is based on experimental data on the energy ordering of electrons in an atom according to the Klechkovsky-Madelung rule and on the substantiation of a single-particle potential model for describing the energy spectrum of optical electrons in alkali metal atoms. A representation of the perturbation of a single-particle potential in the form of a convolution of the potential of an electron in a hydrogen atom with the Wigner function of a certain effective state of the core in an alkali metal atom representation allows reducing all calculation algorithms for alkali metals to the corresponding algorithms for the hydrogen atom.

Fractional time-scales Noether theorem with Caputo [formula omitted] derivatives for Hamiltonian systems

This paper examines a new Noether theorem for Hamiltonian systems with Caputo Δ derivatives based on fractional time-scales calculus, which overcomes the difficulties unified to study the Noether theorems of fractional continuous systems and fractional discrete systems. To begin with, the fractional time-scales definitions and properties with Caputo Δ derivatives are introduced. Next, the fractional time-scales Hamilton canonical equations with Caputo Δ derivatives are formulated. For the fractional time-scales Hamiltonian system, the definitions and criteria of Noether symmetries without transforming time and with transforming time are given, respectively. Furthermore, the corresponding Noether theorem without transforming time and its Noether theorem with transforming time are obtained. The latter one can reduce to the time-scales Noether theorem with Δ derivatives or the fractional Noether theorem with Caputo derivatives for Hamiltonian systems. Finally, the fractional time-scales damped oscillator and Kepler problem are taken as examples to verify the correctness of the results.

Machine learning and serving of discrete field theories

A method for machine learning and serving of discrete field theories in physics is developed. The learning algorithm trains a discrete field theory from a set of observational data on a spacetime lattice, and the serving algorithm uses the learned discrete field theory to predict new observations of the field for new boundary and initial conditions. The approach of learning discrete field theories overcomes the difficulties associated with learning continuous theories by artificial intelligence. The serving algorithm of discrete field theories belongs to the family of structure-preserving geometric algorithms, which have been proven to be superior to the conventional algorithms based on discretization of differential equations. The effectiveness of the method and algorithms developed is demonstrated using the examples of nonlinear oscillations and the Kepler problem. In particular, the learning algorithm learns a discrete field theory from a set of data of planetary orbits similar to what Kepler inherited from Tycho Brahe in 1601, and the serving algorithm correctly predicts other planetary orbits, including parabolic and hyperbolic escaping orbits, of the solar system without learning or knowing Newton’s laws of motion and universal gravitation. The proposed algorithms are expected to be applicable when the effects of special relativity and general relativity are important.

Accuracy of Symmetric Multi-Step Methods for the Numerical Modelling of Satellite Motion

Stability of high-order linear multistep St¨ormer-Cowell and symmetric methods are discussed in detail in this paper. Efficient algorithms for obtaining intervals of absolute stability and periodicity are given for these methods. To demonstrate the accuracy of numerical integration of the orbit over an interval about one year two model problems are considered. First problem is the 3D Kepler problem. Second one is a specially designed 3D model problem that has the exact solution and simulates the Earth-Moon-satellite system

First kind symmetric periodic solutions and their stability for the Kepler problem and anisotropic Kepler problem plus generalized anisotropic perturbation

Motivated by some problems in Celestial Mechanics that combines quasihomogeneous potential in the anisotropic space, we investigate the existence of several families of first kind symmetric periodic solutions for a family of planar perturbed Kepler problem. In addition, we give sufficient conditions for the existence of first kind periodic solutions and also we characterize its type of stability. As an application of this general situation, we discuss the existence of symmetric periodic solutions for the anisotropic Kepler problem plus a generalized anisotropic perturbation, (shortly, p-AKPQ problem) and for the Kepler problem plus a generalized anisotropic perturbation (shortly, p-KPQ problem), as continuation of circular orbits of the two-dimensional Kepler problem. To get this objective, we consider different types of perturbations and then we apply our main result.

On the Orbital Elements of the Two-body Problem with Slowly Decreasing Mass: The Gyldén–Mestchersky Cases

Abstract We study the dynamics of two-body problems with gravitational parameters μ = μ(t) decreasing with the time. In particular we focus our attention on the behavior of the orbital elements in the first two Gyldén–Mestchersky cases: in the first case (GM-I) , where μ 0 is the constant initial value of the parameter and α a small positive constant. In the second case (GM-II) . In the GM-I problem, using a suitable transformation of the radius vector and the physical time, which reduces the original problem to a pure Kepler problem, we are able to obtain explicit analytical expressions of the orbital elements that hold true for all and therefore derive their asymptotic behavior. Moreover using the alternative scaled time these expressions show clearly their periodic and secular terms. For the GM-II problem a similar transformation to the GM-I problem leads to a perturbed Kepler problem with a radial direction perturbation. Although exact solutions in terms of elliptic functions may be obtained, the use of some integrals leads to analytical expressions of orbital elements that allow us to study their behavior. Finally, some comparisons between the time evolution of elements in both problems are also included.

Square Roots of Kepler Ellipses, Electrons and Rydberg Atoms in a Magnetic Field

Young Kepler’s daring ideas on the structure of the Solar system are applied to the analysis of planetary distances in the exoplanetary system HD 10180. Using Zhukovsky’s transformation, the essence of the spinor regularization of Kepler’s problem is explained as extracting the square root of an ellipse and using a Kepler eccentric anomaly instead of the usual time. The achievements of Kharkiv radio astronomers in the search for radio recombination lines of Rydberg carbon atoms at the UTR-2 radio telescope are considered. A generalized spinor regularization of the Kepler problem is used to analyze the energy spectra of Rydberg hydrogen atoms in a magnetic field.

Perturbed (2n−1)-Dimensional Kepler Problem and the Nilpotent Adjoint Orbits of U(n,n)

We study the regularized $(2n-1)$-Kepler problem and other Hamiltonian systems which are related to the nilpotent coadjoint orbits of $U(n,n)$. The Kustaanheimo-Stiefel and Cayley regularization procedures are discussed and their equivalence is shown. Some integrable generalization (perturbation) of $(2n-1)$-Kepler problem is proposed.

The dynamics of the relativistic Kepler problem

We deal with the Hamiltonian system (HS) associated to the Hamiltonian in polar coordinates H=12pr2+pϕ2r2-1r-∊2r2, where ∊ is a small parameter. This Hamiltonain comes from the correction given by the special relativity to the motion of the two-body problem, or by the first order correction to the two-body problem coming from the general relativity. This Hamiltonian system is completely integrable with the angular momentum C and the Hamiltonian H. We have two objectives.First we describe the global dynamics of the Hamiltonian system (HS) in the following sense. Let Sh and Sc are the subset of the phase space where H=h and C=c, respectively. Since C and H are first integrals, the sets Sc,Sh and Shc=Sh∩Sc are invariant by the action of the flow of the Hamiltonian system (HS). We determine the global dynamics on those sets when the values of h and c vary.Second recently Tudoran (2017) provided a criterion which detects when a non-degenerate equilibrium point of a completely integrable system is Lyapunov stable. Every equilibrium point q of the completely integrable Hamiltonian system (HS) is degenerate and has zero angular momentum, so the mentioned criterion cannot be applied to it. But we will show that this criterion is also satisfied when it is applied to the Hamiltonian system (HS) restricted to zero angular momentum.

Magnetized Kepler manifolds and their quantization

Let k≥1 be an integer and μ be the half of a nonzero integer. It is reported here that the phase space of the (2k+1)-dimensional magnetized Kepler problem with magnetic charge μ can be identified with the elliptic co-adjoint orbit of the real Lie algebra so(2,2k+2) that corresponds to the dominant weight (|μ|,…,|μ|︸k+1,μ).As a corollary, one has the following result: the aforementioned elliptic co-adjoint orbit admits a natural quantization in which the Hilbert space of quantum states is the space of square integrable sections of a Hermitian vector bundle over the punctured Euclidean space in dimension 2k+1; moreover, this Hilbert space provides a geometric realization for the unitary highest weight so(2,2k+2)-module with highest weight (−k−|μ|,|μ|,…,|μ|︸k,μ).

Testing a New Conservative Method for Solving the Cauchy Problem for Hamiltonian Systems on Test Problems

A new numerical method for solving the Cauchy problem for Hamiltonian systems is tested in detail as applied to two benchmark problems: the one-dimensional motion of a point particle in a cubic potential field and the Kepler problem. The global properties of the resulting approximate solutions, such as symplecticity, time reversibility, total energy conservation, and the accuracy of numerical solutions to the Kepler problem are investigated. The proposed numerical method is compared with three-stage symmetric symplectic Runge–Kutta methods, the discrete gradient method, and nested implicit Runge–Kutta methods.

Kepler Problem in Space with Deformed Lorentz-Covariant Poisson Brackets

We propose a Lorentz-covariant deformed algebra describing a (3 + 1)-dimensional quantized spacetime, which in the nonrelativistic limit leads to undeformed one. The deformed Poincare transformations leaving the algebra invariant are identified. In the classical limit the Lorentz-covariant deformed algebra yields the deformed Lorentz-covariant Poisson brackets. Kepler problem with the deformed Lorentz-covariant Poisson brackets is studied. We obtain that the precession angle of an orbit of the relativistic particle in the gravitational field depends on the mass of the particle, i.e. equivalence principle is violated. We propose a condition for the recovery of the equivalence principle in the space with the deformed Poisson brackets. Comparing our analytical result with the experimental data for the precession angle of Mercury’s orbit we provide an estimation of minimal length.

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.

Periodic orbits of the two fixed centers problem with a variational gravitational field

We prove the existence of periodic orbits of the two fixed centers problem bifurcating from the Kepler problem. We provide the analytical expressions of these periodic orbits when the mass parameter of the system is sufficiently small.

Bifurcations of symmetric periodic orbits via Floer homology

We give criteria for the existence of bifurcations of symmetric periodic orbits in reversible Hamiltonian systems in terms of local equivariant Lagrangian Rabinowitz Floer homology. As an example, we consider the family of the direct circular orbits in the rotating Kepler problem and observe bifurcations of torus-type orbits. Our setup is motivated by numerical work of H\'enon on Hill's lunar problem.

The Central Force Problem in n Dimensions

The motion of a particle moving under the influence of a central force is a fundamental paradigm in dynamics. The problem of planetary motion, specifically the derivation of Kepler’s laws motivated Newton’s monumental work, Principia Mathematica, effectively signalling the start of modern physics. Today, the central force problem stands as a basic lesson in dynamics. In this article, we discuss the classical central force problem in a general number of spatial dimensions n, as an instructive illustration of important aspects such as integrability, super-integrability and dynamical symmetry. The investigation is also in line with the realisation that it is useful to treat the number of dimensions as a variable parameter in physical problems. The dependence of various quantities on the spatial dimensionality leads to a proper perspective of the problems concerned. We consider, first, the orbital angular momentum (AM) in n dimensions, and discuss in some detail the role it plays in the integrability of the central force problem. We then consider an important super-integrable case, the Kepler problem, in n dimensions. The existence of an additional vector constant of the motion (COM) over and above the AM makes this problem maximally super-integrable. We discuss the significance of these COMs as generators of the dynamical symmetry group of the Hamiltonian. This group, the rotation group in n + 1 dimensions, is larger than the kinematical symmetry group for a general central force, namely, the rotation group in n dimensions.

О гипотезе Мищенко — Фоменко для обобщённого осциллятора и системы Кеплера

Рассматриваются деформации задачи Кеплера и гармонического осциллятора, для которых дополнительные интегралы движения являются координатами приведённого дивизора, согласно теореме Римана — Роха. Для этого семейства некоммутативно интегрируемых систем обсуждается справедливость гипотезы Мищенко — Фоменко о существовании интегралов движения из единого функционального класса, в данном случае полиномиальных интегралов движения.

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.

Exact intrinsic half angular momentum from the Schrödinger equation

The presence of (exact) intrinsic half angular moment (IHAM) in the Schrodinger equation has been explicitly shown through the spherical coordinates $$(r,\theta ,\varphi )$$ following a self-adjoint transformation. The special self-adjoint equations (SSAEs) were obtained by this self-adjoint transformation on the usual wavefunction of $$\Psi (r,\theta ,\varphi ) =\Upsilon (r,\theta ,\varphi ) J(r,\theta ,\varphi )^{-1/2}$$, where $$J(r,\theta ,\varphi )$$ is the Jacobian of the spherical coordinates. The first derivative terms of these SSAEs were absent. We considered the Kepler problem as applied to the hydrogen atom as an example. These SSAEs can easily be interpreted in terms of (semi)classical physics, and besides, these equations have exhibited an IHAM even for the ground state, i.e., without any excitation. This IHAM can be considered as a zero-point angular momentum.

Minimizers for the Kepler Problem

We characterize the minimizing geodesics for the Kepler problem endowed with the Jacobi-Maupertuis metric. We focus on the positive energy case, but do all energies. The more complicated negative energy case was solved in Jacobi (Crelles J 17:68–82, 1837. https://doi.org/10.1515/crll.1837.17.68), with his work translated and completed by Todhunter (Researches in the Calculus of Variations, Principally on the Theory of Discontinuous Solutions. Macmillan and Co., Cambridge, 1871), and later summarized in Wintner’s book. Our discussion of these old results includes a new proof for the positive energy case and perspectives coming from metric and differential geometry. For the negative energy result we need Lambert’s theorem which we discuss.