Terms Of Elliptic Integrals Research Articles

Geodesics on an arbitrary ellipsoid of revolution

The algorithms given in Karney (J. Geodesy 87:43–55, 2013), to compute geodesics on terrestrial ellipsoids, are extended to apply to ellipsoids of revolution with arbitrary eccentricity. For the direct and inverse geodesic problems, this entails implementing the formulation in terms of elliptic integrals given by Legendre and Cayley. The integral for the area of geodesic polygons is computed in terms of the discrete sine transform of the integrand. In all cases, accuracy close to full machine precision is achieved. An open-source implementation of these algorithms is available.

Frequency response of a mass grounded by linear and nonlinear springs in series: An exact analysis

An exact analytical solution for the frequency response of a system consisting of a mass grounded by linear and nonlinear springs in series was derived. The system is applicable in the study of beams carrying intermediate rigid mass. The exact solution was derived by naturally transforming the integral of the governing nonlinear ODE into a form that can be expressed in terms of elliptic integrals. Hence, the exact frequency–amplitude solution was derived in terms of the complete elliptic integral of the first and third kinds. Periodic solutions were found to exist for all real values of [Formula: see text] and for [Formula: see text]. On the other hand, linear or weakly nonlinear frequency–amplitude responses were found to occur during small-amplitude vibrations ([Formula: see text]), very large-amplitude vibrations with strong hardening nonlinearity ([Formula: see text] and [Formula: see text]), and for all amplitudes when [Formula: see text]. Simulations showed that the system’s periodic response is significantly influenced by the nonlinearity parameter ([Formula: see text]), linearity parameter ([Formula: see text]), and amplitude of vibration ([Formula: see text]). Besides, the existence of bifurcation points at [Formula: see text] and different values of [Formula: see text] was confirmed. Lastly, an approximate frequency solution obtained using the He’s frequency–amplitude formulation was found to produce errors less than 1.0% for a wide range of input parameters. In conclusion, the present study provides a benchmark solution for verification of other approximate solutions, and the He’s frequency–amplitude formulation can be used to obtain fast and accurate solutions for complex nonlinear systems.

Analytical solutions of the generalized Kaup–Newell equation

The generalized Kaup–Newell equation for describing the pulse propagation in a nonlinear optical medium is considered. The traveling wave reduction of this equation is taken into account. The method of direct transformations is used to construct a general solution of a nonlinear ordinary differential equation. Analytical solutions are found, expressed in terms of elliptic integrals in canonical form. The implicit function method is applied to find some degenerate solutions under constraints on the model parameters. Among the solutions obtained there are doubly periodic solutions on the complex plane, as well as optical solitons.

The geodesic flow of the BGPP metric is Liouville integrable

We prove that the geodesics equations corresponding to the BGPP metric are integrable in the Liouville sense. The symmetry of the model allows to reduce the system from four to two degrees of freedom. Moreover, solutions of the reduced system and its degenerations can be given explicitly or reduced to a certain quadrature. In degenerated cases BGPP metric coincides with the Eguchi–Hanson metric and for this case the mentioned quadrature can be calculated explicitly in terms of elliptic integrals.

Time Averages and Periodic Attractors at High Rayleigh Number for Lorenz-like Models

Revisiting the Lorenz ’63 equations in the regime of large of Rayleigh number, we study the occurrence of periodic solutions and quantify corresponding time averages of selected quantities. Perturbing from the integrable limit of infinite ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document}, we provide a full proof of existence and stability of symmetric periodic orbits, which confirms previous partial results. Based on this, we expand time averages in terms of elliptic integrals with focus on the much studied average ‘transport,’ which is the mode reduced excess heat transport of the convection problem that gave rise to the Lorenz equations. We find a hysteresis loop between the periodic attractors and the nonzero equilibria of the Lorenz equations. These have been proven to maximize transport, and we show that the transport takes arbitrarily small values in the family of periodic attractors. In particular, when the nonzero equilibria are unstable, we quantify the difference between maximal and typically realized values of transport. We illustrate these results by numerical simulations and show how they transfer to various extended Lorenz models.

Generalizing Fowler–Nordheim Tunneling Theory for an Arbitrary Power Law Barrier

Herein, the canonical Fowler–Nordheim theory is extended by computing the zero‐temperature transmission probability for the more general case of a barrier described by a fractional power law. An exact analytical formula is derived, written in terms of Gauss hypergeometric functions, that fully capture the transmission probability for this generalized problem, including screened interaction with the image potential. First, the quality of approximation against the so far most advanced formulation of Fowler–Nordheim, where the transmission is given in terms of elliptic integrals, is benchmarked. In the following, as the barrier is given by a power law, in detail, the dependence of the transmission probability on the exponent of the power law is analyzed. The formalism is compared with results of numerical calculations and its possible experimental relevance is discussed. Finally, it is discussed how the presented solution can be linked in some specific cases with an exact quantum‐mechanical solution of the quantum well problem.

Tidal properties of D-dimensional Tangherlini black holes

This paper investigates tidal forces in multidimensional spherically symmetric spacetimes. We consider geodesic deviation equation in Schwarzschild-Tangherlini metric and its electrically charged analog. It was shown that for radial geodesics these equations can be solved explicitly as quadratures in spaces of any dimension. In the case of five, six and seven dimensional spaces, these solutions can be represented in terms of elliptic integrals. For spacetimes of higher dimension, we find the asymptotics of the solution. It was found that in the physical singularity vicinity tidal stretch along the radial direction is the stronger the greater the dimension of space. Whereas the tidal compression in transverse to radial directions, starting from a certain dimension, does not change in the main order. Also in the case of non-radial geodesics, the presence of black hole electric charge does not affect the force of transverse compression in the leading order. For non-radial geodesics with non-zero angular momentum, the local properties of solutions of geodesic deviation equations in the vicinity of a singularity are studied.

Off-axis electric field due to cylindrical geometries of charge distribution

Off-axis electric field due to cylindrical distribution of charge is studied in various geometries, including solid cylinder, cylindrical shell, disk, and ring. The electric fields of the charged solid cylinder and cylindrical shell can be found by numerical calculations, but closed forms expressed in terms of elliptic integrals are derived for the charged disk and ring. By investigating the radial and axial components of the electric fields systematically we reveal many interesting and new features that are implicit or unseen in the ordinary approach based on the cylindrical symmetry.

Analytical study of gravitational lensing in Kerr-Newman black-bounce spacetime

We investigate the equatorial deflection angle of light rays propagating in Kerr-Newman black-bounce spacetime. Furthermore, we analyze the light ray trajectories and derive a closed-form formula for deflection angle in terms of elliptic integrals. The deflection angle increases with the decrease of charge and regularisation parameter for a particular impact parameter. We also study the strong field limit of the deflection angle. Using this strong deflection angle formula and lens equation, we find the radius of the first Einstein ring and study its dependence on the charge and the regularisation parameter. We demonstrate that the charge has a robust effect on the size of the Einstein rings, but the effect of the regularization parameter on the ring size is negligible. We also investigate the non-equatorial lensing and the caustic structures for small polar inclination, and the same observations appear to hold. These results directly affect the observational appearance of the Kerr-Newman black-bounce.

Analytic solution for perturbed Keplerian motion under small acceleration using averaging theory

A novel approach is developed for analytic orbit propagation based on asymptotic expansion with respect to a small perturbative acceleration. The method improves upon existing first order asymptotic expansions by leveraging on linear systems and averaging theories. The solution starts with the linearization of Gauss planetary equations with respect to both the small perturbation and the six orbital elements. Then, an approximate solution is obtained in terms of secular and short period components. The method is tested on a low-thrust maneuver scenario consisting of a Keplerian orbit perturbed by a constant tangential acceleration, for which a solution can be obtained in terms of elliptic integrals. Results show that the positional propagation error is about one order of magnitude smaller with respect to state-of-the-art methods. The position accuracy for a LEO orbit, apart from pathological cases, is typically in the range of tens of meters for a tangential acceleration of 10−7 km/s2 after 5 orbital periods propagation.

Analytic periods via twisted symmetric squares

We study the symmetric square of Picard-Fuchs operators of genus one curves and the thereby induced generalized Clausen identities. This allows the computation of analytic expressions for the periods of all one-parameter K3 manifolds in terms of elliptic integrals. The resulting expressions are globally valid throughout the moduli space and allow the explicit inversion of the mirror map and the exact computation of distances, useful for checks of the Swampland Distance Conjecture. We comment on the generalization to multi-parameter models and provide a two-parameter example.

Baroclinic instability of spatially-periodic flows in a discrete surface quasi-geostrophic model with two levels

Within the framework of a discrete surface quasi-geostrophic (SQG) model with two levels, the hydrodynamic stability of a spatially periodic flow with sinusoidal velocity profile (Kolmogorov flow) is investigated. Such spatially quasi-periodic flows are often observed in the atmospheres of the giant planets and in the Southern Ocean. To describe the dynamics of perturbations, the Galerkin method with three basis trigonometric functions was used. It provides a tool to study both the linear and nonlinear perturbation dynamics. Analytical expressions are obtained for the growth rates of perturbations in the linear theory of stability, and it is shown that the fastest growing perturbations have a wavelength on the order of the spatial period of the main flow. Using the Galerkin method, it is also shown that the exponential growth of perturbations at the linear stage of development is replaced by the stage of stable nonlinear oscillations. Such oscillations are analogous to the nonlinear fluctuations, or vacillations, found in R. Hide's laboratory experiments. In terms of elliptic integrals, an analytical expression for the period of nonlinear oscillations is obtained, and it is shown that for typical values of the parameters relevant to the Earth’s atmosphere, the period of oscillations is about one month. The paper also proposes a long-wavelength version of the discrete SQG model, which describes the dynamics of perturbations with a horizontal scale much larger than the baroclinic Rossby radius of deformation. It is shown that the use of this version simplifies the study of stability and leads to asymptotically correct results.

Quantum Rajeev–Ranken model as an anharmonic oscillator

The Rajeev–Ranken (RR) model is a Hamiltonian system describing screw-type nonlinear waves of wavenumber k in a scalar field theory pseudodual to the 1 + 1D SU(2) principal chiral model. Classically, the RR model is Liouville integrable. Here, we interpret the model as a novel 3D cylindrically symmetric quartic oscillator with an additional rotational energy. The quantum theory has two dimensionless parameters. Upon separating variables in the Schrödinger equation, we find that the radial equation has a four-term recurrence relation. It is of type [0, 1, 16] and lies beyond the ellipsoidal Lamé and Heun equations in Ince’s classification. At strong coupling λ, the energies of highly excited states are shown to depend on the scaling variable λk. The energy spectrum at weak coupling and its dependence on k in a double-scaling strong coupling limit are obtained. The semi-classical Wentzel-Kramers-Brillouin (WKB) quantization condition is expressed in terms of elliptic integrals. Numerical inversion enables us to establish a (λk)2/3 dispersion relation for highly energetic quantized “screwons” at moderate and strong coupling. We also suggest a mapping between our radial equation and the one of Zinn-Justin and Jentschura that could facilitate a resurgent WKB expansion for energy levels. In another direction, we show that the equations of motion of the RR model can also be viewed as Euler equations for a step-3 nilpotent Lie algebra. We use our canonical quantization to uncover an infinite dimensional reducible unitary representation of this nilpotent algebra, which is then decomposed using its Casimir operators.

A review on buckling and postbuckling of thin elastic beams

This paper provides a review of models and solutions for the buckling and postbuckling of beams available from the 50s of the last century to date. Beams with axially unrestrained (movable) ends and restrained (immovable) ends are covered. In each class, the formulation of the nonlinear buckling problem for the buckling loads and the postbuckling states is discussed and the underlying assumptions are highlighted. For relatively large-amplitude buckling of beams with movable ends, approximate analytical solutions up to the third order are presented and compared with the exact solutions expressed in terms of elliptic integrals. For beams with immovable ends, buckling involves midplane stretching that makes the nonlinear buckling problem takes the same form of the standard eigenvalue problem and, hence, exact solutions are affordable. This review combines the research outcomes on buckled beams from two scientific viewpoints: the structural dynamics and the nonlinear vibration viewpoints, respectively. Moreover, it presents in one place the formulation and the exact solutions for the buckling of beams to serve as an on-demand resource for researchers concerned with the buckling and postbuckling of beams.

Near lightcone thermal conformal correlators and holography

Heavy–heavy–light–light (HHLL) correlators of pairwise identical scalars in CFTs with a large central charge in any number of dimensions admit a double scaling limit where the ratio of the heavy conformal dimension to the central charge becomes large as the separation between the light operators becomes null. In this limit the stress tensor sector of a generic HHLL correlator receives contributions from the multi stress tensor operators with any number of stress tensors, as long as their twist is not increased by index contractions. We show how one can compute this leading twist stress tensor sector when the conformal dimension of the light operators is large and the stress tensor sector approximates the thermal CFT correlator. In this regime the value of the correlator is related to the length of the spacelike geodesic which approaches the boundary of the dual asymptotically AdS spacetime at the points of light operator insertions. We provide a detailed description of the infinite volume limit. In two spacetime dimensions the HHLL Virasoro vacuum block is reproduced, while in four spacetime dimensions the result is written in terms of elliptic integrals.

Roots of generalised Hermite polynomials when both parameters are large

We study the roots of the generalised Hermite polynomials H m,n when both m and n are large. We prove that the roots, when appropriately rescaled, densely fill a bounded quadrilateral region, called the elliptic region, and organise themselves on a deformed rectangular grid, as was numerically observed by Clarkson. We describe the elliptic region and the deformed grid in terms of elliptic integrals and their degenerations.

Surface gravity of rotating dumbbell shapes

We investigate the problem of determining the shape of a rotating celestial object – e.g., a comet or an asteroid – under its own gravitational field. More specifically, we consider an object symmetric with respect to one axis – such as a dumbbell – that rotates around a second axis perpendicular to the symmetry axis. We assume that the object can be modeled as an incompressible fluid of constant mass density, which is regarded as a first approximation of an aggregate of particles.In the literature, the gravitational field of a body is often described as a multipolar expansion involving spherical coordinates (Kaula, 1966). In this work we describe the shape in terms of cylindrical coordinates, which are most naturally adapted to the symmetry of the body, and we express the gravitational potential generated by the rotating body as a simple formula in terms of elliptic integrals. An equilibrium shape occurs when the gravitational potential energy and the rotational kinetic energy at the surface of the body balance each other out. Such an equilibrium shape can be derived as a solution of an optimization problem, which can be found via the variational method. We give an example where we apply this method to a two-parameter family of dumbbell shapes, and find approximate numerical solutions to the corresponding optimization problem.

Плотность энергетического спектра электрона в поле изображения и запирающем электрическом поле

In the semiclassical approximation, the density of the electron energy spectrum near the metal surface is described, when electron is bound by the image field and the blocking electrostatic field. In the system under consideration, the confinement mechanism is realized, and the energy spectrum for the motion of an electron in the direction perpendicular to the metal surface is completely discrete. The density of states of the energy spectrum is expressed in terms of elliptic integrals, the argument of which is a sigmoidal function. When the field is turned off, it becomes the Heaviside step function. A dimensionless energy parameter is introduced, which determines the intervals with qualitatively different changes in the width of the classically accessible region of motion. For large positive values of the energy parameter, the spectrum density asymptotically tends to the density in the triangular potential with the addition of the Coulomb logarithmic correction, and for negative values of the energy parameter, the spectrum density tends to dependence for a one-dimensional Coulomb potential. Approximate expressions are obtained for the spectrum density in terms of elementary functions in a wide range of electron energies and electric field strength.

Drop squeezing between arbitrary smooth obstacles

Abstract

A Method for Calculating the Mutual Inductance of a Coil System Using a Model of an Axially Magnetized Cylinder

A method for calculating the mutual inductance of a system of tightly wound coils through the energy of axially magnetized cylinders is proposed. Formulas for calculating 3D fields of a cylinder and coils expressed in terms of elliptical integrals are obtained. Тhe volume-average demagnetization factor of the cylinder is shown to be equal to the Nagaoka coefficient for the self-inductance of short coils.