Terms Of Elliptic Functions Research Articles

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.

New method for solving strong conservative odd parity nonlinear oscillators: Applications to plasma physics and rigid rotator

In the present work, a new method for solving a strong nonlinear oscillator equation of the form ẍ + F(x) = 0, where F(−x) = −F(x), is carried out. This method consists of approximating function F(x) by means of a suitable Chebyshev polynomial: F(x) ≈ P(x) = px + qx3 + rx5, and then, the original oscillator is replaced by the cubic–quintic Duffing equation ẍ + px + qx3 + rx5 = 0 with arbitrary initial conditions, which admits the exact solution in terms of elliptic functions. The efficacy of the present method is demonstrated through the fluid multi-ion plasma equations and a generalized pendulum problem. For the generalized pendulum problem, the governing motion is directly reduced to the cubic–quintic Duffing oscillator with the help of the Chebyshev polynomial, and the approximate analytical and exact solutions are obtained. In addition, the comparison between our solutions and the Runge–Kutta numerical solution is examined. Moreover, the periodic time formula of the oscillations for both the approximate analytical solution and the exact solution is deduced, and the comparison between them is implemented. With respect to the plasma application, the fluid plasma equations of its particles are reduced to the Extended Korteweg–de Vries (EKdV) equation utilizing a reductive perturbation method. Then, we proved for the first time that any undamped polynomial oscillator of the nth degree can be reduced to a (2n − 1)th odd parity Duffing. Accordingly and after applying the previous theory to the EKdV equation, it was converted to the cubic–quintic Duffing equation. Finally, we can deduce that our new solutions and theory help us to understand and investigate many nonlinear phenomena in various branches of science.

Numerical solution for time period of simple pendulum with large angle

In this study, the numerical solution of the ordinary kind of differential equation for a simple pendulum with large-angle of oscillation was introduced to obtain the time period. The analytical solution is obtained in terms of elliptic functions, and numerical solution of the problem was achieved by using two numerical quadrature methods, namely, Simpson?s 3/8 and Boole?s method. The period of a simple pendulum with large angle is presented. A comparison has been carried out between the analytical solution and the numerical integration results. In the case of error analysis, absolute and relative errors of the problem have been presented. A numerical algorithm has been developed by MATLAB software 2013R and used for analyzing the result. It is established that the results of the comparison guaranty the ability and the accuracy of the present method.

Explicit algebraic solution of Zolotarev's First Problem for low-degree polynomials

E.I. Zolotarev's classical so-called First Problem (ZFP), which was posed to him by P.L. Chebyshev, is to determine, for a given \(n\in{\mathbb N}\backslash\{1\}\) and for a given \(s\in{\mathbb R}\backslash\{0\}\), the monic polynomial solution \(Z^{*}_{n,s}\) to the following best approximation problem: Find\[\min_{a_k}\max_{x\in[-1,1]}|a_0+a_1 x+\dots+a_{n-2}x^{n-2}+(-n s)x^{n-1}+x^n|,\]where the \(a_k, 0\le k\le n-2\), vary in \(\mathbb R\). It suffices to consider the cases \(s>\tan^2\left(\pi/(2n)\right)\).
 In 1868 Zolotarev provided a transcendental solution for all \(n\geq2\) in terms of elliptic functions. An explicit algebraic solution in power form to ZFP, as is suggested by the problem statement, is available only for \(2\le n\le 5.^1\)
 We have now obtained an explicit algebraic solution to ZFP for \(6\le n\le 12\) in terms of roots of dedicated polynomials.
 In this paper, we provide our findings for \(6\le n\le 7\) in two alternative fashions, accompanied by concrete examples. The cases \(8\le n\le 12\) we treat, due to their bulkiness, in a separate web repository.
 \(^1\) Added in proof: But see our recent one-parameter power form solution for \(n=6\) in [38].

On Various Approaches to Asymptotics of Solutions to the Third Painlevé Equation in a Neighborhood of Infinity

We examine asymptotic expansions of the third Painleve transcendents for αδ ≠ = 0 and γ = 0 in the neighborhood of infinity in a sector of aperture <2π by the method of dominant balance). We compare intermediate results with results obtained by methods of three-dimensional power geometry. We find possible asymptotics in terms of elliptic functions, construct a power series, which represents an asymptotic expansion of the solution to the third Painleve equation in a certain sector, estimate the aperture of this sector, and obtain a recurrent relation for the coefficients of the series.

On Hamiltonian Systems Integrable in Elliptic Functions That Describe Waves over Underwater Banks and Ridges

We discuss the 4-dimensional Hamiltonian systems that describe waves over underwater banks and ridges. The systems are exactly integrable in terms of elliptic functions and of solutions to nontrivial transcendental equations involving the elliptic integrals (Weierstrass' $\zeta$-function).

НЕЛИНЕЙНАЯ ДИНАМИКА ДЛИННОВОЛНОВЫХ ВОЗМУЩЕНИЙ НЕВЯЗКОГО ТЕЧЕНИЯ КОЛМОГОРОВА

The nonlinear dynamics of long-wave perturbations of the inviscid Kolmogorov flow, which models periodically varying in the horizontal direction oceanic currents, is studied. To describe this dynamics, the Galerkin method with basis functions representing the first three terms in the expansion of spatially periodic perturbations in the trigonometric series is used. The orthogonality conditions for these functions formulate a nonlinear system of partial differential equations for the expansion coefficients (Kalashnik, Kurgansky, 2018). Based on the asymptotic solutions of this system, a linear, quasilinear and nonlinear stage of perturbation dynamics are identified. It is shown that the time-dependent growth of perturbations during the first two stages is succeeded by the stage of stable nonlinear oscillations. The corresponding oscillations are described by the oscillator equation containing a cubic nonlinearity, which is integrated in terms of elliptic functions. An analytical formula for the period of oscillations is obtained, which determines its dependence on the amplitude of the initial perturbation. Structural features of the field of the stream function of the perturbed flow are described, associated with the formation of closed vortex cells and meandering flow between them. The research was supported by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications» and by the Russian Foundation for Basic Research (Projects 18-05-00414, 18-05-00831).

On the Hamiltonian formulation, integrability and algebraic structures of the Rajeev-Ranken model

The integrable 1+1-dimensional SU(2) principal chiral model (PCM) serves as a toy-model for 3+1-dimensional Yang-Mills theory as it is asymptotically free and displays a mass gap. Interestingly, the PCM is ‘pseudodual’ to a scalar field theory introduced by Zakharov and Mikhailov and Nappi that is strongly coupled in the ultraviolet and could serve as a toy-model for non-perturbative properties of theories with a Landau pole. Unlike the ‘Euclidean’ current algebra of the PCM, its pseudodual is based on a nilpotent current algebra. Recently, Rajeev and Ranken obtained a mechanical reduction by restricting the nilpotent scalar field theory to a class of constant energy-density classical waves expressible in terms of elliptic functions, whose quantization survives the passage to the strong-coupling limit. We study the Hamiltonian and Lagrangian formulations of this model and its classical integrability from an algebraic perspective, identifying Darboux coordinates, Lax pairs, classical r-matrices and a degenerate Poisson pencil. We identify Casimirs as well as a complete set of conserved quantities in involution and the canonical transformations they generate. They are related to Noether charges of the field theory and are shown to be generically independent, implying Liouville integrability. The singular submanifolds where this independence fails are identified and shown to be related to the static and circular submanifolds of the phase space. We also find an interesting relation between this model and the Neumann model allowing us to discover a new Hamiltonian formulation of the latter.

Coordinate space calculation of two- and three-loop sunrise-type diagrams, elliptic functions and truncated Bessel integral identities

We integrate three-loop sunrise-type vacuum diagrams in D0=4 dimensions with four different masses using configuration space techniques. The finite parts of our results are in numerical agreement with corresponding three-loop calculations in momentum space. Using some of the closed form results of the momentum space calculation we arrive at new integral identities involving truncated integrals of products of Bessel functions. For the non-degenerate finite two-loop sunrise-type vacuum diagram in D0=2 dimensions we make use of the known closed form p-space result to express the moment of a product of three Bessel functions in terms of a sum of Clausen polylogarithms. Using results for the nondegenerate two-loop sunrise diagram from the literature in D0=2 dimensions we obtain a Bessel function integral identity in terms of elliptic functions.

Topologically embedded helicoidal pseudospherical cylinders**Authors partially supported by MIUR (Italy) under the PRIN project Varietà reali e complesse: geometria, topologia e analisi armonica; and by the GNSAGA of INDAM. The present research has also been partially supported by MIUR grant “Dipartimenti di Eccellenza” 2018-2022.

The class of traveling wave solutions of the sine-Gordon equation is known to be in 1–1 correspondence with the class of (necessarily singular) pseudospherical surfaces in Euclidean space with screw-motion symmetry: the pseudospherical helicoids. We explicitly describe all pseudospherical helicoids in terms of elliptic functions. This solves a problem posed by Popov (2014 Lobachevsky Geometry and Modern Nonlinear Problems (Berlin: Springer)). As an application, countably many continuous families of topologically embedded pseudospherical helicoids are constructed. A (singular) pseudospherical helicoid is proved to be either a dense subset of a region bounded by two coaxial cylinders, a topologically immersed cylinder with helical self-intersections, or a topologically embedded cylinder with helical singularities, called for short a pseudospherical twisted column. Pseudospherical twisted columns are characterized by four phenomenological invariants: the helicity , the parity , the wave number , and the aspect ratio , up to translations along the screw axis. A systematic procedure for explicitly determining all pseudospherical twisted columns from the invariants is provided.

Regimes of regular and chaotic motion of gyrostats in the central gravity field

In this article, the partial case of gyrostats angular motion in the central gravitational field is investigated at the condition of the collinearity of the angular momentum and the gravity field gradient. This case is closely connected with V.A. Stekloff's case of the rigid body motion and can be characterized as its partial generalization on the gyrostat motion. Considered dynamical circumstances can be realized in “conical precessions” regimes of rotating prolate gyrostat-satellites at the stabilization by the gravitational way along the local vertical direction. The corresponding analytical solution for the gyrostat/gyrostat-satellite angular motion parameters is obtained in terms of elliptic functions. In addition, the possibility of chaotization phenomena in the gyrostat-satellite angular motion are investigated.

Classical treatment of particle with position-dependent mass m(r) = 1/(1 + r4) in 1D and 2D subjected to harmonic potential

The purpose of this paper is to explore the motion of classical particle with position-dependent mass (PDM) m(r) = 1/(1 + r4) in the harmonic potential (1D and 2D). We establish a geometric model of the particle with mentioned PDM that involves constant-mass particles in curved space. With the help of Lagrangian formalism, we solve equations of motion and express the result in terms of elliptic functions. In general, we explore one of the possible developments of Mathews and Lakshmanan ideas and provide a new example of elliptic functions occurring in physics.

Nonlinear dynamics of long-wave perturbations of the Kolmogorov flow for large Reynolds numbers

The nonlinear dynamics of long-wave perturbations of the inviscid Kolmogorov flow, which models periodically varying in the horizontal direction oceanic currents, is studied. To describe this dynamics, the Galerkin method with basis functions representing the first three terms in the expansion of spatially periodic perturbations in the trigonometric series is used. The orthogonality conditions for these functions formulate a nonlinear system of partial differential equations for the expansion coefficients. Based on the asymptotic solutions of this system, a linear, quasilinear, and nonlinear stage of perturbation dynamics is identified. It is shown that the time-dependent growth of perturbations during the first two stages is succeeded by the stage of stable nonlinear oscillations. The corresponding oscillations are described by the oscillator equation containing a cubic nonlinearity, which is integrated in terms of elliptic functions. An analytical formula for the period of oscillations is obtained, which determines its dependence on the amplitude of the initial perturbation. Structural features of the field of the stream function of the perturbed flow are described, associated with the formation of closed vortex cells and meandering flow between them. As a supplement, an asymptotic analysis of nonlinear dynamics of long-wave perturbations superimposed on a damped by small viscosity Kolmogorov flow (very large, but finite Reynolds numbers) is made. It is strictly shown that all velocity components of the perturbed flow remain bounded in this case.

Exact solutions to $$(2+1)$$ ( 2 + 1 ) -dimensional Chaffee–Infante equation

In this paper, the canonical-like transformation method and trial equation method are applied to \((2+1)\)-dimensional Chaffee–Infante equation, and some exact solutions are obtained. In particular, a new solution in terms of elliptic functions is given.

Duality symmetries behind solutions of the classical simple pendulum

Describing the motion of the classical simple pendulum is one of the aims in every undergraduate classical mechanics course. Its analytical solutions are given in terms of elliptic functions, which are doubly periodic functions in the complex plane. The independent variable of the solutions is time and it can be considered either as a real variable or as a purely imaginary one, which introduces a rich symmetry structure in the space of solutions. When solutions are written in terms of the Jacobi elliptic functions this symmetry is codified in the functional form of its modulus, and is described mathematically by the six dimensional coset group Γ=Γ(2) where Γ is the modular group and Γ(2) is its congruence subgroup of second level. A discussion of the physical consequences that this symmetry has on the motions of the simple pendulum is presented in this contribution and it is argued they have similar properties to the ones termed as duality symmetries in other areas of physics, such as field theory and string theory. Thus by studying deeper a very familiar mechanical system, it is possible to get an insight to more abstract physical and mathematical concepts. In particular a single solution of pure imaginary time for all allowed values of the total mechanical energy is given and obtained as the S-dual of a single solution of real time, where S stands for the S generator of the modular group.

Modeling approaches for precise relativistic orbits: Analytical, Lie-series, and pN approximation

Accurate orbit modeling plays a key role in contemporary and future space missions such as GRACE and its successor GRACE-FO, GNSS, and altimetry missions. To fully exploit the technological capabilities and correctly interpret measurements, relativistic orbital effects need to be taken into account.Within the theory of General Relativity, equations of motion for freely falling test objects, such as satellites orbiting the Earth, are given by the geodesic equation. We analyze and compare different solution methods in a spherically symmetric background, i.e.for the Schwarzschild spacetime, as a test bed. We investigate satellite orbits and use direct numerical orbit integration as well as the semi-analytical Lie-series approach. The results are compared to the exact analytical reference solution in terms of elliptic functions. For a set of exemplary orbits, we determine the respective accuracy of the different methods.Within the post-Newtonian approximation of General Relativity, modified orbital equations are obtained by adding relativistic corrections to the Newtonian equations of motion. We analyze the accuracy of this approximation with respect to the general relativistic setting. Therefore, we solve the post-Newtonian equation of motion using the eXtended High Performance Satellite dynamics Simulator. For corresponding initial conditions, we compare orbits in the Schwarzschild spacetime to those in its post-Newtonian approximation. Moreover, we compare the magnitude of relativistic contributions to several typical perturbations of satellite orbits due to, e.g., solar radiation pressure, Earth’s albedo, and atmospheric drag. This comparison is done for our test scenarios and for a real GRACE orbit to highlight the importance of relativistic effects in geodetic space missions. For the considered orbits, first-order relativistic contributions give accelerations of about 20 nm/s2 and are dominant in the radial direction.

Pulsating strings with mixed three-form flux

Circular strings pulsating in AdS3 × S3 × T4 with mixed R-R and NS-NS three-form fluxes can be described by an integrable deformation of the one-dimensional Neumann-Rosochatius mechanical model. In this article we find a general class of pulsating solutions to this integrable system that can be expressed in terms of elliptic functions. In the limit of strings moving in AdS3 with pure NS-NS three-form flux, where the action reduces to the SL(2, ℝ) WZW model, we find agreement with the analysis of the classical solutions of the system performed using spectral flow by Maldacena and Ooguri. We use our elliptic solutions in AdS3 to extend the dispersion relation beyond the limit of pure NS-NS flux.

Satellite orbits in Levi-Civita space

In this paper we consider satellite orbits in central force field with quadratic drag using two formalisms. The first using polar coordinates in which the satellite angular momentum plays a dominant role. The second is in Levi-Civita coordinates in which the energy plays a central role. We then merge these two formalisms by introducing polar coordinates in Levi-Civita space and derive a new equation for satellite orbits which unifies these two paradigms. In this equation energy and angular momentum appear on equal footing and thus characterize the orbit by its two invariants. Using this formalism we show that equatorial orbits around oblate spheroids can be expressed analytically in terms of Elliptic functions. In the second part of the paper we derive in Levi-Civita coordinates a linearized equation for the relative motion of two spacecrafts whose trajectories are in the same plane. We carry out also a numerical verification of these equations.

Spinning strings in the η -deformed Neumann-Rosochatius system

The sigma-model of closed strings spinning in the $\eta$-deformation of $AdS_{5} \times S^{5}$ leads to an integrable deformation of the one-dimensional Neumann-Rosochatius mechanical system. In this article we construct general solutions to this system that can be written in terms of elliptic functions. The solutions correspond to closed strings with non-constant radii rotating with two different angular momenta in an $\eta$-deformed three-sphere. We analyse the reduction of the elliptic solutions for some limiting values of the deformation parameter. For the case of solutions with constant radii we find the dependence of the classical energy of the string on the angular momenta as an expansion in the 't Hooft coupling.

Multiplicative equations related to the affine Weyl group E8

We derive integrable equations starting from autonomous mappings with a general form inspired by the multiplicative systems associated with the affine Weyl group E8(1). Five such systems are obtained, three of which turn out to be linearisable and the remaining two are integrable in terms of elliptic functions. In the case of the linearisable mappings, we derive non-autonomous forms which contain a free function of the independent variable and we present the linearisation in each case. The two remaining systems are deautonomised to new discrete Painlevé equations. We show that these equations are in fact special forms of much richer systems associated with the affine Weyl groups E7(1) and E8(1), respectively.