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The purpose of this paper is to demonstrate the existence of symmetric periodic solutions for a two-dimensional hydrogen atom subject to external electric and magnetic fields. We build upon the analysis of periodic orbits that was initially performed by De Bustos et al. (J Math Phys 53:082701, 2012). In this study, we utilize the symmetry of reflection and an appropriate set of variables to obtain results regarding the existence and stability of periodic solutions. Furthermore, we determine the presence of KAM 2-tori that enclose some of the linearly stable periodic solutions. [ABSTRACT FROM AUTHOR]

We investigate the existence of families of symmetric periodic solutions of second kind as continuation of the elliptical orbits of the two-dimensional Kepler problem for certain symmetric differentiable perturbations using Delaunay coordinates. More precisely, we characterize the sufficient conditions for its existence and its type of stability is studied. The estimate on the characteristic multipliers of the symmetric periodic solutions is the new contribution to the field of symmetric periodic solutions. In addition, we present some results about the relationship between our symmetric periodic solutions and those obtained by the averaging method for Hamiltonian systems. As applications of our main results, we get new families of periodic solutions for: the perturbed hydrogen atom with stark and quadratic Zeeman effect, for the anisotropic Seeligers two-body problem and to the planar generalized Størmer problem. [ABSTRACT FROM AUTHOR]

We study the nonholonomic motion of a point particle on the Heisenberg group around the fixed “sun” placed at the origin whose potential is given by the fundamental solution of the sub-Laplacian. In contrast with several recent papers that approach this problem as a variational one (hence a control problem) we study the equations of dynamical motion which are non-variational in nonholonomic mechanics. We find three independent first integrals of the system and show that its bounded trajectories are wound up around certain surfaces of the fourth order. We also describe some particular cases of trajectories. [ABSTRACT FROM AUTHOR]

In this paper, two models of interest for Celestial Mechanics are presented and analysed, using both analytic and numerical techniques, from the point of view of the possible presence of regular and/or chaotic motion, as well as the stability of the considered orbits. The first model, presented in a Hamiltonian formalism, can be used to describe the motion of a satellite around Earth, taking into account both the non-spherical shape of our planet and the third-body gravitational influence of Sun and Moon. Using semi-analytical techniques coming from Normal Form and Nekhoroshev theories it is possible to provide stability estimates for the orbital elements of its geocentric motion. The second dynamical system presented can be used as a simplified model to describe the motion of a particle in an elliptic galaxy having a central massive core; it is constructed as a refraction billiard where an inner dynamics, induced by a Keplerian potential, is coupled with an external one, where a harmonic oscillator-type potential is considered. The investigation of the dynamics is carried on by using results of ODEs' theory and is focused on studying the trajectories' properties in terms of periodicity, stability and, possibly, chaoticity. [ABSTRACT FROM AUTHOR]

This study presents generalized solutions to the Kepler problem involving the gravitational perturbation of a third body external to a two-body system. The spatial case of the non-coplanar motion of perturbing and perturbed bodies is considered. The expression for the relative position of the perturbing body relative to that of the perturbed body is expanded in Legendre polynomials. This expansion allows us to obtain the equation of motion and the energy of the system in terms of the orbital elements (I, Ω, ω, ν) for both the perturbed body and the perturbing body. By applying the Sundman transformation to regularize the dynamics of the problem, we eliminate a singularity and find two regularized equations of motion. Solutions to these equations are obtained by employing the families of the Battin and Stumpff functions. From these solutions, we derive formulas that are universally applicable and free from indeterminacy. This includes the expression for the radial distance, the generalized Kepler's equation, the closed-form expressions for the coefficients f and g, along with their derivative forms, and the expressions for the orbital state vectors. [ABSTRACT FROM AUTHOR]

The Landau problem and harmonic oscillator in the plane share a Hilbert space that carries the structure of Dirac's remarkable s o (2 , 3) representation. We show that the orthosymplectic algebra o s p (1 | 4) is the spectrum generating algebra for the Landau problem and, hence, for the 2D isotropic harmonic oscillator. The 2D harmonic oscillator is in duality with the 2D quantum Coulomb–Kepler systems, with the o s p (1 | 4) symmetry broken down to the conformal symmetry s o (2 , 3) . The even s o (2 , 3) submodule (coined Rac) generated from the ground state of zero angular momentum is identified with the Hilbert space of a 2D hydrogen atom. An odd element of the superalgebra o s p (1 | 4) creates a pseudo-vacuum with intrinsic angular momentum 1/2 from the vacuum. The odd s o (2 , 3) -submodule (coined Di) built upon the pseudo-vacuum is the Hilbert space of a magnetized 2D hydrogen atom: a quantum system of a dyon and an electron. Thus, the Hilbert space of the Landau problem is a direct sum of two massless unitary s o (2 , 3) representations, namely, the Di and Rac singletons introduced by Flato and Fronsdal. [ABSTRACT FROM AUTHOR]

We consider the Lorentz force equation d d t (m x ˙ 1 − | x ˙ | 2 / c 2 ) = q (E (t , x) + x ˙ × B (t , x)) , x ∈ R 3 , 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. [ABSTRACT FROM AUTHOR]

The recent detection of gravitational waves emanating from inspiralling black hole binaries has triggered a renewed interest in the dynamics of relativistic two-body systems. The conservative part of the latter are given by Hamiltonian systems obtained from so-called post-Newtonian expansions of the general relativistic description of black hole binaries. In this paper we study the general question of whether there exist relativistic binaries that display Kepler-like dynamics with elliptical orbits. We show that an orbital equivalence to the Kepler problem indeed exists for relativistic systems with a Hamiltonian of a Kepler-like form. This form is realised by extremal black holes with electric charge and scalar hair to at least first order in the post-Newtonian expansion for arbitrary mass ratios and to all orders in the post-Newtonian expansion in the test-mass limit of the binary. Moreover, to fifth post-Newtonian order, we show that Hamiltonians of the Kepler-like form can be related explicitly through a canonical transformation and time reparametrisation to the Kepler problem, and that all Hamiltonians conserving a Laplace – Runge – Lenz-like vector are related in this way to Kepler. [ABSTRACT FROM AUTHOR]

Abstract: We derive a new analytical solution for the first-order, short-periodic perturbations due to planetary oblateness and systematically compare our results to the classical Brouwer–Lyddane transformation. Our approach is based on the Milankovitch vectorial elements and is free of all the mathematical singularities. Being a non-canonical set, our derivation follows the scheme used by Kozai in his oblateness solution. We adopt the mean longitude as the fast variable and present a compact power-series solution in eccentricity for its short-periodic perturbations that relies on Hansen’s coefficients. We also use a numerical averaging algorithm based on the fast-Fourier transform to further validate our new mean-to-osculating and inverse transformations. This technique constitutes a new approach for deriving short-periodic corrections and exhibits performance that are comparable to other existing and well-established theories, with the advantage that it can be potentially extended to modeling non-conservative orbit perturbations.

We prove that the correspondence between Reeb and Beltrami vector fields presented in Etnyre & Ghrist (Etnyre, Ghrist 2000 Nonlinearity13, 441–458 (doi:10.1088/0951-7715/13/2/306)) can be made equivariant whenever additional symmetries of the underlying geometric structures are considered. As a corollary of this correspondence, we show that energy levels above the maximum of the potential energy of mechanical Hamiltonian systems can be viewed as stationary fluid flows, though the metric is not prescribed. In particular, we showcase the emblematic example of the n -body problem and focus on the Kepler problem. We explicitly construct a compatible Riemannian metric that makes the Kepler problem of celestial mechanics a stationary fluid flow (of Beltrami type) on a suitable manifold, the Kepler–Euler flow. [ABSTRACT FROM AUTHOR]

The phase space of the classical Kepler problem is incomplete, as certain Kepler motions do not continue indefinitely in time. To address this issue, a method called "regularization" was introduced in the 1960s and 1970s. While there are several versions of this scheme, the one developed by Ligon and Schaaf [Rep. Math. Phys. 9, 281–300 (1976)] is often preferred due to its global nature and lack of time reparameterization requirement. In this article, we present a new regularization method that is both global and requires no time reparameterization, while also exhibiting a higher degree of symmetry. Our method emphasizes symmetries over spaces and exhibits a duality similar to the compact-noncompact duality observed in symmetric spaces. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For the approximate solution of the Kepler equations and some related problems, a fourth-order convergent functionally-fitted block hybrid Falkner method which is based on the concepts of interpolation and collocation of the fitting function given as a linear combination of 1 , sin (ω x) , cos (ω x) , sinh (ω x) , cosh (ω x) is presented. The proposed method uses variable coefficients that are based on the product of the dominant frequency and the integration step length. This hybrid formula uses a block-wise implementation strategy to get over the difficulties of the predictor–corrector mode. In addition to being zero stable, the proposed method is applied to the Lambert–Watson linear stability test, which allows obtaining its stability region. Six numerical examples are provided to establish the performance of the proposed method. [ABSTRACT FROM AUTHOR]

The Lambert problem consists in connecting two given points in a given lapse of time under the gravitational influence of a fixed center. While this problem is very classical, we are concerned here with situations where friction forces act alongside the Newtonian attraction. Under some boundedness assumptions on the friction, there exists exactly one rectilinear solution if the two points lie on the same ray, and at least two solutions traveling in opposite directions otherwise. [ABSTRACT FROM AUTHOR]

In 1878, Hill found numerically, in his limiting case of the restricted three-body problem, the so-called Hill's lunar problem, a planar direct periodic orbit with a period of one synodic month. By virtue of the spatial system's invariance under a symplectic involution, whose fixed point set corresponds to the planar problem, we can assign to Hill's orbit planar and spatial Floquet multipliers and planar and spatial Conley–Zehnder indices. We show that these have deep astronomical significance because, on the one hand, we relate the anomalistic month to the planar Floquet multipliers and the planar Conley–Zehnder index. On the other hand, we relate the draconitic month to the spatial Floquet multipliers and the spatial Conley–Zehnder index. Knowledge of this lunar month dates back to the Babylonians, who lived until around 500 BCE. In order to determine the indices, we analyze analytically the bifurcation procedure of the fundamental families of planar direct and retrograde periodic orbits (traditionally known as families g and f) from the rotating Kepler problem for very low energies. [ABSTRACT FROM AUTHOR]

Through an appropriate change of reference frame and rescalings of the variables and the parameters introduced, the Hamiltonian of the three-body problem is written as a perturbed Kepler problem. In this system, new Delaunay variables are defined and a suitable configuration of the phase space and the mass parameters is chosen. In such a system, wide regions of librational and rotational motions where orbits are regular and stable are found. Close to the separatrix of these regions, the existence of chaotic motions presenting a double rotational and librational dynamics is proved, numerically, through Poincaré sections and the use of FLI. [ABSTRACT FROM AUTHOR]

Explicit methods to generate new canonical sets of elements are presented, together with an analysis on the structure of existing canonical elements that are used in the study of the perturbed two-body problem. New non-singular sets of canonical elements are introduced. Among these sets, the resonant canonical variables are non-singular for circular orbits and display an action-angle-like behaviour for the unperturbed Kepler problem, if their associated generalized frequency is tuned to match the conserved energy. They also prove to be an efficient tool for both qualitative and quantitative studies of perturbations. [ABSTRACT FROM AUTHOR]

The paper examines how the scarcely discussed problem of force obstruction contributed to shaping Kepler's concept of force, that is, the question whether corporeal forces are blocked when an intermediate body is interposed between the force source and the target body. We explore the role played by this problem in Kepler's works, focusing on his Optics, Foundations of Astrology, Astronomia Nova and Epitome. While the problem is instrumental in abandoning an initially physicalistic framework for astrology, Kepler attempts at different solutions when discussing the forces responsible for planetary motion. These forces are first modelled upon light, and then magnetism (and gravity). However, after considering several ways to get rid of the problem, he is eventually unable to decide why observable effects of obstruction are usually absent. So he accepts this absence at face value, similarly to Newton in his most relevant passages, allowing for a new ontological category that is physical on the one hand, but non-material on the other. [ABSTRACT FROM AUTHOR]

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 R 4 × R 4 phase space. [ABSTRACT FROM AUTHOR]

We explore the interaction between the Conley–Zehnder index and bifurcation points of symmetric planar as well as spatial periodic orbits in the spatial Hill three-body problem. We start with the fundamental families of planar periodic orbits which are those of direct (family g) and retrograde periodic orbits (family f). Since the spatial system is invariant under a symplectic involution, whose fixed point set corresponds to the planar problem, planar orbits have planar and spatial Floquet multipliers, and planar and spatial Conley–Zehnder indices. When the Floquet multipliers move through a root of unity, new families of periodic orbits bifurcate and the index jumps. For very low energies, the families g and f arise dynamically from the rotating Kepler problem, and in a recent work (Aydin From Babylonian lunar observations to Floquet multipliers and Conley-Zehnder Indices) we determined analytically their indices. By their numerical continuations for higher energies, we determine the index of various families of planar and spatial periodic orbits bifurcating from g and f. Since these families can bifurcate again and meet each other, this procedure can get complicated. This index leads to a grading on local Floer homology. Since the local Floer homology and its Euler characteristic stay invariant under bifurcation, the index provides important information about the interconnectedness of such families, which we illustrate in form of bifurcation graphs. Since the solutions of Hill's system may serve as orbits for space mission design or astronomical observations, our results promote the interaction between Symplectic Geometry and practical problems. [ABSTRACT FROM AUTHOR]

The anisotropic Kepler problem is a model of the motion of free electrons on an n-type semiconductor and is known to be a non-integrable Hamiltonian system. While many approximate periodic solutions have been found through numerical calculation (Sumiya et al. in Artif Life Robot 19:262–269, 2014), none have been rigorously proved to exist. In this paper, we first show that the action functional of the anisotropic Kepler problem has a minimizer under a fixed region condition with boundary conditions on a vertical half-line. Next, we identify the smallest collision trajectory that satisfies the same boundary conditions. By constructing an orbit with an action functional smaller than this collision orbit via local deformation, we show that the collision solution does not become the minimizer. This holds for any μ ∈ (0 , 1) . Reversibility allows the periodic orbit to be constructed from the minimizer obtained via the action functional. [ABSTRACT FROM AUTHOR]

Muñoz accused Kepler, moreover, of being unaware of the isosceles triangle proper to the heptagon, the pentagon, and the 15-sided polygon that was used in Cardano's and Candale's constructions. Muñoz, who accused Kepler of heresy, claimed to have discovered a method for tracing the heptagon, despite Kepler's contention that it could not be done. In this way, Muñoz argued that Kepler's theory of considering as geometrically unconstructible the heptagon, nonagon, and other figures with an odd number of sides did not comport with the metaphysical principle of God the omnipotent creator. Had it not been the case that one side had been given a fixed value of 4.000000, Muñoz could have stated, "In the circumcircle of the isosceles triangle of the heptagon (9, 4, 9), the side of the heptagon is equal to the base of this triangle (4).". [Extracted from the article]

In the study of systems' dynamics the presence of symmetry dramatically reduces the complexity, while in chemistry, symmetry plays a central role in the analysis of the structure, bonding, and spectroscopy of molecules. In a more general context, the principle of equivalence, a principle of local symmetry, dictated the dynamics of gravity, of space-time itself. In certain instances, especially in the presence of symmetry, we end up having to deal with an equation with multiple roots. A variety of optimal methods have been proposed in the literature for multiple roots with known multiplicity, all of which need derivative evaluations in the formulations. However, in the literature, optimal methods without derivatives are few. Motivated by this feature, here we present a novel optimal family of fourth-order methods for multiple roots with known multiplicity, which do not use any derivative. The scheme of the new iterative family consists of two steps, namely Traub-Steffensen and Traub-Steffensen-like iterations with weight factor. According to the Kung-Traub hypothesis, the new algorithms satisfy the optimality criterion. Taylor's series expansion is used to examine order of convergence. We also demonstrate the application of new algorithms to real-life problems, i.e., Van der Waals problem, Manning problem, Planck law radiation problem, and Kepler's problem. Furthermore, the performance comparisons have shown that the given derivative-free algorithms are competitive with existing optimal fourth-order algorithms that require derivative information. [ABSTRACT FROM AUTHOR]

We consider a (1 + N) -body problem in which one particle has mass m 0 ≫ 1 and the remaining N have unitary mass. We can assume that the body with larger mass (central body) is at rest at the origin, coinciding with the center of mass of the N bodies with smaller masses (satellites). The interaction force between two particles is defined through a potential of the form U ∼ 1 r α , where α ∈ [ 1 , 2) and r is the distance between the particles. Imposing symmetry and topological constraints, we search for periodic orbits of this system by variational methods. Moreover, we use Γ -convergence theory to study the asymptotic behaviour of these orbits, as the mass of the central body increases. It turns out that the Lagrangian action functional Γ -converges to the action functional of a Kepler problem, defined on a suitable set of loops. In some cases, minimizers of the Γ -limit problem can be easily found, and they are useful to understand the motion of the satellites for large values of m 0 . We discuss some examples, where the symmetry is defined by an action of the groups Z 4 , Z 2 × Z 2 and the rotation groups of Platonic polyhedra on the set of loops. [ABSTRACT FROM AUTHOR]

The Landau problem and harmonic oscillator in the plane share a Hilbert space that carries the structure of Dirac’s remarkable so(2,3) representation. We show that the orthosymplectic algebra osp(1|4) is the spectrum generating algebra for the Landau problem and, hence, for the 2D isotropic harmonic oscillator. The 2D harmonic oscillator is in duality with the 2D quantum Coulomb–Kepler systems, with the osp(1|4) symmetry broken down to the conformal symmetry so(2,3). The even so(2,3) submodule (coined Rac) generated from the ground state of zero angular momentum is identified with the Hilbert space of a 2D hydrogen atom. An odd element of the superalgebra osp(1|4) creates a pseudo-vacuum with intrinsic angular momentum 1/2 from the vacuum. The odd so(2,3)-submodule (coined Di) built upon the pseudo-vacuum is the Hilbert space of a magnetized 2D hydrogen atom: a quantum system of a dyon and an electron. Thus, the Hilbert space of the Landau problem is a direct sum of two massless unitary so(2,3) representations, namely, the Di and Rac singletons introduced by Flato and Fronsdal.

We consider the Kepler two-body problem in the presence of a cosmological constant Λ. Several dimensionless parameters characterizing the possible orbit typologies are used to identify open and closed trajectories. The qualitative picture of the two-body motion is described and critical parameters of the problem are found. [ABSTRACT FROM AUTHOR]

In this work, a perturbed system of the restricted three-body problem is derived when the perturbation forces are conservative alongside the corresponding mean motion of two primaries bodies. Thus, we have proved that the first and second types of periodic orbits of the rotating Kepler problem can persist for all perturbed two-body and circular restricted three-body problems when the perturbation forces are conservative or the perturbed motion has its own extended Jacobian integral. [ABSTRACT FROM AUTHOR]