Kepler Problem Research Articles

Classical mechanics in noncommutative spaces: confinement and more

We consider a semi-classical approximation to the dynamics of a point particle in a noncommutative space. In this approximation, the noncommutativity of space coordinates is described by a Poisson bracket. For linear Poisson brackets, the corresponding phase space is given by the cotangent bundle of a Lie group, with the Lie group playing the role of a curved momentum space. We show that the curvature of the momentum space may lead to rather unexpected physical phenomena such as an upper bound on the velocity of a free nonrelativistic particle, bounded motion for repulsive central force, and no-fall-into-the-centre for attractive Coulomb potential. We also consider a superintegrable Hamiltonian for the Kepler problem in 3-space with su(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {su}(2)$$\\end{document} noncommutativity. The leading correction to the equations of motion due to noncommutativity is shown to be described by an effective monopole potential.

Periodic solutions to relativistic Kepler problems: a variational approach

Periodic solutions to relativistic Kepler problems: a variational approach

Bertrand’s theorem and the double copy of relativistic field theories

Which relativistic field theories give rise to Kepler dynamics in the two-body problem? We consider a class of Hamiltonians that is the unique relativistic extension of the Kepler problem preserving its so(4) algebra, and have orbits related through time reparametrisation to orbits of the original Kepler problem. For three explicit examples, we give a natural interpretation in terms of spin-0,-1 and -2 interacting field theories in 5D. These are organically connected via the classical double copy, which therefore preserves maximal superintegrability.

Circularization in the Damped Kepler Problem

Circularization in the Damped Kepler Problem

Dynamics in the Kepler problem on the Heisenberg group

Dynamics in the Kepler problem on the Heisenberg group

On the Generalized Kepler Problem under the Effect of Outer Third-Body Perturbation

On the Generalized Kepler Problem under the Effect of Outer Third-Body Perturbation

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.

Non-integrability of the restricted three-body problem

Abstract The problem of non-integrability of the circular restricted three-body problem is very classical and important in the theory of dynamical systems. It was partially solved by Poincaré in the nineteenth century: he showed that there exists no real-analytic first integral which depends analytically on the mass ratio of the second body to the total and is functionally independent of the Hamiltonian. When the mass of the second body becomes zero, the restricted three-body problem reduces to the two-body Kepler problem. We prove the non-integrability of the restricted three-body problem both in the planar and spatial cases for any non-zero mass of the second body. Our basic tool of the proofs is a technique developed here for determining whether perturbations of integrable systems which may be non-Hamiltonian are not meromorphically integrable near resonant periodic orbits such that the first integrals and commutative vector fields also depend meromorphically on the perturbation parameter. The technique is based on generalized versions due to Ayoul and Zung of the Morales–Ramis and Morales–Ramis–Simó theories. We emphasize that our results are not just applications of the theories.

Nonintegrability of dissipative planar systems

We consider dissipative autonomous perturbations of planar Hamiltonian systems and give a sufficient condition for them not to be complex-meromorphically Bogoyavlenskij-integrable such that the first integral or commutative vector field also depends complex-meromorphically on the small parameter. We illustrate the theoretical result for three examples including systems with the Morse potential and effective potential of the Kepler problem.

Revisiting the Kepler problem with linear drag using the blowup method and normal form theory

In this paper, we revisit the Kepler problem with linear drag. With dissipation, the energy and the angular momentum are both decreasing, but in Margheri et al (2017 Celest. Mech. Dyn. Astron. 127 35–48) it was shown that the eccentricity vector has a well-defined limit in the case of linear drag. This limiting eccentricity vector defines a conserved quantity, and in the present paper, we prove that the corresponding invariant sets are smooth manifolds. These results rely on normal form theory and a blowup transformation, which reveals that the invariant manifolds are (nonhyperbolic) stable sets of (limiting) periodic orbits. Moreover, we identify a separate invariant manifold which corresponds to a zero limiting eccentricity vector. This manifold is obtained as a generalized center manifold over the zero eigenspace of a zero-Hopf point. Finally, we present a detailed blowup analysis, which provides a geometric picture of the dynamics. We believe that our approach and results will have general interest in problems with blowup dynamics, including the Kepler problem with generalized nonlinear drag.

Transverse foliations in the rotating Kepler problem

We construct finite energy foliations and transverse foliations of neighbourhoods of the circular orbits in the rotating Kepler problem for all negative energies. This paper would be a first step towards our ultimate goal that is to recover and refine McGehee’s results on homoclinics [23] and to establish a theoretical foundation to the numerical demonstration of the existence of a homoclinic–heteroclinic chain in the planar circular restricted three-body problem [20], using pseudoholomorphic curves.

Projective integrable mechanical billiards

In this paper, we use the projective dynamical approach to integrable mechanical billiards as in (Zhao 2021 Commun. Contemp. Math. 24 2150085) to establish the integrability of natural mechanical billiards with the Lagrange problem, which is the superposition of two Kepler problems and a Hooke problem, with the Hooke center at the middle of the Kepler centers, as the underlying mechanical systems, and with any finite combinations of confocal conic sections with foci at the Kepler centers as the reflection wall, in the plane, on the sphere, and in the hyperbolic plane. This covers many previously known integrable mechanical billiards, especially the integrable Hooke, Kepler and two-center billiards in the plane, as has been investigated in (Takeuchi and Zhao 2021 arXiv:2110.03376), as subcases. The approach of (Takeuchi and Zhao 2021 arXiv:2110.03376) based on conformal correspondence has been also applied to integrable Kepler billiards in the hyperbolic plane to illustrate their equivalence with the corresponding integrable Hooke billiards on the hemisphere and in the hyperbolic plane as well.

Orbital Regularity of Exoplanets and the Symmetries of the Kepler Problem

We investigate the question whether or not the orbitals of exoplanets follow the symmetry-governed sequence found by Barut from the dynamical group of the Kepler problem. In particular, we consider their star distances, periods, and velocities. Previous studies have shown the validity of this regularity for our solar system, and for some selected exoplanet systems. Here, we study all the systems which are known with four or more planets. A remarkable result is found: 63 out of 100 systems show a better agreement between the theory and observation than our solar system. We discuss the relation between the symmetry-inspired transformations and the generalized Titius–Bode (gTB) rule. It turns out that the gTB rule, which has been considered purely empirical, can be obtained from the transformations corresponding to the dynamical group of the Kepler problem.

Infinitely many periodic solutions to a Lorentz force equation with singular electromagnetic potential

We consider the Lorentz force equationddt(mx˙1−|x˙|2/c2)=q(E(t,x)+x˙×B(t,x)),x∈R3, in the physically relevant case of a singular electric field E. Assuming that E and B are T-periodic in time and satisfy suitable further conditions, we prove the existence of infinitely many T-periodic solutions. The proof is based on a min-max principle of Lusternik-Schnirelmann type, in the framework of non-smooth critical point theory. Applications are given to the problem of the motion of a charged particle under the action of a Liénard-Wiechert potential and to the relativistic forced Kepler problem.

Kepler Problem under Repulsive Inverse-square Central Forces

In this article, I present important results of the Kepler problem under repulsive inverse-square central forces. While motions of this type have been discussed thoroughly in the classical scattering theory, I do not find available literature that treats the problem as an analog of the Kepler problem under an attractive central force. My results are applicable to motions of dust particles subject to solar radiation pressure accelerations exceeding the solar gravitational acceleration in the two-body problem.

Bases and interbasis expansions in the generalized MIC–Kepler problem in the continuous spectrum and the scattering problem

Bases and interbasis expansions in the generalized MIC–Kepler problem in the continuous spectrum and the scattering problem

A computational approach to solving some applied rigid second-order problems

When a differential equation exhibits chaos, stiffness, damping and/or oscillation in its solution component, such a differential equation is termed rigid. Over the years, solving such problems has posed a serious challenge to researchers. A new computational approach was employed in this research by proposing a Two-Point Hybrid Block Method (TPHBM) for the solutions of some applied rigid second order problems. The method was constructed via the interpolation and collocation techniques within a two-step integration interval [xn,xn+2]. The procedure for the construction of the method involved the interpolation of the basis function at two selected points, collocation at all points and the incorporation of four off-grid points. This is aimed at improving the accuracy of the method, circumventing the Dahlquist-barrier as well as optimizing the order of the method while maintaining a low step-number. The paper further analyzed some basic properties of the method. Rigid second order problems like Bessel, Duffing and Kepler problems were solved using the TPHBM and the results generated showed that it outperformed other existing methods.

An efficient method of finding new symplectic schemes for Hamiltonian mechanics problems with the aid of parametric Gröbner bases

The explicit symplectic difference schemes are considered for the numerical solution of molecular dynamics problems described by systems with separable Hamiltonians. A general method for finding symplectic schemes of high order of accuracy using parametric Gröbner bases, resultants, permutations of variables, and optimization techniques is proposed. The implementation of the method is described by the example of four-stage partitioned Runge–Kutta (PRK) schemes of the Forest–Ruth and Yoshida families. 97 new PRK schemes of Forest–Ruth family have been obtained. The error functionals for five new schemes FR47, FR50, FR51, FR52, and FR59 are smaller by more than two orders of magnitude than in the case of the fourth-order schemes, which were obtained previously in the explicit form in this family. In the family of Yoshida schemes, 115 new schemes have been found. Verification of the schemes was carried out on the Kepler's problem, which has the exact solution. Calculations of this task performed at large time steps showed that the FR50 scheme significantly exceeds in terms of accuracy the four-stage Forest–Ruth schemes known in the literature, the Verlet scheme, schemes from the Yoshida family Y7, Y95, Y110, and a six-stage fourth-order scheme.

NASA DART mission: A preliminary mathematical dynamical model and its nonlinear circuit emulation

Abstract On September 22, 2022, a spacecraft, designed by the Double Asteroid Redirection Test (DART) team, successfully attempted to deflect the orbit of the asteroid Dimorphos, which together with Didymos, constitutes a binary system of near-Earth asteroids orbiting around the Sun. The effect of the impact of the spacecraft was to shorten the orbit of Dimorphos of about 33 min with respect to the original one. In this communication, a simple nonlinear circuit emulator based on a mathematical model allowing the emulation of the DART mission behavior is presented. The modeling is approached referring to the Kepler problem that leads to a highly nonlinear dynamical model. The problem is approached numerically, by using appropriate integration algorithms for both the two-body and three-body formulations of the problem, and experimentally, by means of an analog/digital electronic circuit emulator of the system that allows us to realize faster and qualitative more efficient experiments.

Periodic, Transition and Escape Trajectories for 3D-Kepler 2-Body Problem of Classical Electrodynamics

In a previous paper we studied the Kepler problem for the extended Synge’s 2-body problem of classical electrodynamics. We have used the radiation terms introduced in our previous papers and prove an existence–uniqueness of a periodic orbit in polar coordinates which confirmed the Bohr's hypothesis of the existence of the stationary states in the frame of classical electrodynamics. Our main aim here is to show the existence of trajectories of transition оf the particle orbiting the nucleus from one stationary state to another excited state. We also prove the existence of escape trajectories. This is made by a choice of suitable function space and applying fixed point method.