Legendre Elliptic Integrals Research Articles

Subsurface stresses in an elliptical Hertzian contact

The traditional solution for the stresses below an elliptical Hertzian contact expresses the results in terms of incomplete Legendre elliptic integrals, so are necessarily based on the length of the semi-major axis a and the axis ratio k. The result is to produce completely different equations for the stresses in the x and y directions; and although these equations are now well-known, their derivation from the fundamental, symmetric, integrals is far from simple. When instead Carlson elliptic integrals are used, they immediately match the fundamental integrals, allowing the equations for the stresses to treat the two semi-axes equally, and so providing a single equation where two were needed before. The numerical evaluation of the Carlson integrals is simple and rapid, so the result is that more convenient answers are obtained more conveniently. A bonus is that the temptation to record the depth of the critical stresses as a fraction of the length of the semi-major axis is removed. Thomas and Hoersch’s method of finding all the stresses along the axis of symmetry has been extended to determine the full set of stresses in a principal plane. The stress patterns are displayed, and a comparison between the answers for the planes of the major and minor semiaxes is made. The results are unchanged from those found from equations given by Sackfield and Hills, but not previously evaluated. The present equations are simpler, not only in the simpler elliptic integrals, but also for the “tail” of elementary functions.

Defining the parameters for a brush with polypropylene bristle when uncovering the root system of maternal plants

The paper addresses the operation of a cylindrical brush with elastic rods of bristles at the disclosure of the root system of maternal plants. It has been established that the basic components of the resistance force in the operation of a given brush is the sum of resistances caused by the soil viscosity force, the static resistance of loose particles of soil and the resistance force at removing soil particles. Based on the minimization of costs of energy, the optimal kinematical parameters have been defined for the brush with a vertical rotation axis. The value for the total resistance force is largely affected by the angular speed of the brush. In addition to the resistance force, the rods of the brush bristles are exposed to the normal reaction from the soil base. By deriving the resultant of these two forces using a method of the Legendre elliptic integrals, we have established the optimal dimensional parameters for the rods of bristles made of polypropylene. A given method has made it possible to take into consideration their significant, compared with the rods' length, deformations as a result of their bending. The rods' length has been determined to be the maximally possible one to ensure the conditions for the removal of soil from a swath that covers the root system of maternal plants of clonal rootstock. In addition, the influence of the friction force during brush operation has been investigated. This implies friction among soil particles and against the surface of the bristle rods. It was determined that the forces of friction, as well as the normal reaction of soil, have little effect on the operation of a cylindrical brush while opening the root system of maternal plants. This is due to the lack of a solid foundation when a bristle rod is at work, which in turn makes it possible to arrange the bristle rods one by one at the surface of the brush. The absence of necessity in the considerable relative stiffness makes it possible for the bristle rods to remove soil from the swath while minimizing damage to the plants. By using the Legendre elliptic integrals of second kind, we have explored the deflection of bristle rods in a cylindrical brush while opening the root system of maternal plants. The magnitude of the rod deflection during operation affects the degree of it removing the soil particles. An increase in deflection changes the angle between a working facet of the bristle rod and the soil surface. This leads to a decrease in the removal of particles and to an increase in their compaction in a swath. Therefore, we have calculated the length of bristle rods that could ensure the required operating parameters under a predefined load.

Jacobian elliptic functions describe the properties of the diffuse charge region in narrow electrochemical cells

Diffuse double layers merge in narrow electrochemical cells. The distribution of ions, and the accompanying electrical properties, are predicted under static conditions that are mild enough not to engender electrochemical reactions. The predictions rest on assumptions that are not egregiously unreasonable. The potential and concentration profiles are respectively described by Legendre elliptic integrals and Jacobian elliptic functions. The special case of pure water is examined and the relevance of this study to water desalination is briefly addressed.

Analytical solution for axial flow of a Giesekus fluid in concentric annuli

An exact analytical solution for a Poiseuille flow of a Giesekus fluid in an annulus is obtained, taking into account the effects of the non-linearity of the constitutive equation. The fluid velocity is given in terms of Legendre elliptic integrals and Jacobi elliptic functions. The fluid behaviour is described using the Deborah number along the mobility factor and the radius ratio, while the tangential and the normal stresses are also evaluated.The shear-rate, the velocity and the stress tensor components as function of the involved parameters are graphically represented and analysed in detail.

On a Certain Sub-Riemannian Geodesic Flow on the Heisenberg Group

Under study is an integrable geodesic flow of a left-invariant sub-Riemannian metric for a right-invariant distribution on the Heisenberg group. We obtain the classification of the trajectories of this flow. There are a few examples of trajectories in the paper which correspond to various values of the first integrals. These trajectories are obtained by numerical integration of the Hamiltonian equations. It is shown that for some values of the first integrals we can obtain explicit formulae for geodesics by inverting the corresponding Legendre elliptic integrals.

Indefinite integrals of incomplete elliptic integrals from Jacobi elliptic functions

ABSTRACTIntegration formulas are derived for the three canonical Legendre elliptic integrals. These formulas are obtained from the differential equations satified by these elliptic integrals when the independent variable u is the argument of Jacobian elliptic function theory. This allows a limitless number of indefinite integrals with respect to the amplitude to be derived for these three elliptic integrals. Sample results are given, including the integrals derived from powers of the 12 Glaisher elliptic functions. New recurrence relations and integrals are also given for the 12 Glaisher elliptic functions.

Comparison of integral and discrete Ostrowski's inequalities in the plane

A comparison of integral and discrete Ostrowski's inequalities in the plane is con- sidered. An integral inequality is described by Legendre's elliptic integrals. A natural discrete analogue of the inequality is also given. The main point is to find a suitable decomposition of the radius in polar coordinates.

THE 2-D SEXTIC HAMILTONIAN OSCILLATOR

The 2-D sextic oscillator is studied as a family of axial symmetric parametric integrable Hamiltonian systems, presenting a bifurcation analysis of the different flows. It includes the "elliptic core" model in 1-D nonlinear oscillators, recently proposed in the literature. We make use of the energy-momentum mapping, which will give us the fundamental fibration of the four-dimensional phase space. Special attention is given to the singular values of the energy-momentum mapping connected with rectilinear and circular orbits. They are related to the saddle-center and pitchfork scenarios with the associated homoclinic and heteroclinic trajectories. We also study how the geometry of the phase space evolves during the transition from the one-dimensional to the two-dimensional model. Within an elliptic function approach, the solutions are given using Legendre elliptic integrals of the first and third kind and the corresponding Jacobi elliptic functions.

Fourier and other series containing associated Legendre functions for incomplete Epstein–Hubbell integrals and functions related to elliptic integrals

Fourier series containing associated Legendre functions are given for functions related to the Legendre elliptic integrals and for incomplete Epstein–Hubbell integrals. A number of definite and indefinite integrals related to these series are derived. Additional series of Legendre functions and integrals are derived from a binomial expansion of the reciprocal of the delta amplitude. A power series expansion for the elliptic integral of the third kind in terms of the parameter α is obtained. Series of associated Legendre functions are given for general monomials.

Fourier series for the Legendre elliptic integrals

Simple Fourier series in terms of the amplitude φ are derived for the common Legendre elliptic integrals. The coefficients in the various series contain Legendre functions of the second kind and half-integral degree, also known as toroidal functions. For the Legendre elliptic integrals of the first and second kinds, F(φ, k) and E(φ, k), repectively, the expansions given are simple sine series in the amplitude φ and an additional aperiodic term proportional to φ. These series are valid for φ∈ℝ. The complete elliptic integral of the third kind Π(α2, k) can be expressed in terms of Heuman’s lambda function Λ0(β, k) and the Jacobi zeta function Z(β, k), which in turn can be expressed in terms of the integrals of the first and second kinds. This enables simple sine series in terms of β to be derived for Λ0(β, k) and Z(β, k). The series for Λ0(β, k) has an aperiodic term in β but the series for Z(β, k) does not. The various series are obtained by first expanding the delta amplitude and Δ−1(φ, k) as cosine series in the amplitude φ and integrating term by term with repect to φ. The recurrence relation for the Legendre functions is frequently used to simplify or rearrange the various series.

On directional average of the local anisotropic damage

The derivation of thermodynamical damage state potentials and the formulation of damage constitutive laws always require averaging actual directional damage distribution. The analysis of anisotropic damage state is developed here by the hidden variable technique and presented in the two canonical variants – the energy and the entropy. Thermodynamic damage state potentials in their canonical forms are obtained. A variant of the canonical description of damage and the canonical definition of the directional damage variable consistent with the irreversible thermodynamics are discussed. These involve averages of the directional damage variable. A general algorithm of derivation of symmetric even rank damage tensors from the directional damage distribution is developed. Directional damage averaging procedures are considered. Various integral and series representations of the averaged damage are obtained. It is shown that damage averaging requires rapid computations of the canonical Legendre elliptic integrals. Numerical results are presented for two-, three- and five-parametric average damages in order to determine an availability for the averaged damage values range.

Age-redshift relation for standard cosmology

We present compact, analytic expressions for the age-redshift relation $\ensuremath{\tau}(z)$ for standard Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) cosmology. The new expressions are given in terms of incomplete Legendre elliptic integrals and evaluate much faster than by direct numerical integration.

Profile of a rotating string

The problem of a uniform flexible chain or string rotating about an axis with fixed ends is addressed in the context of both Newtonian mechanics and the variational principle. It is shown that, in the absence of gravity, the equations describing the profile of the string as well as the constraints of the system are all expressible in terms of Legendre's elliptic integrals of various kinds. The effectof gravity is discussed qualitatively, and it is shown that for high enough angularvelocities its effect is negligible. The profile of a rotating chain is obtained using stroboscopic photography and compared with the theoretical results. PACS No.: 46.00

Magnetic field evaluation for disk conductors

A method for calculating the magnetic field from a sector or an entire disk conductor in which a radial current flows is described. New analytic expressions for the components of the magnetic vector potential and field are given in closed form. These consist of suitably polynomial, inverse trigonometric and logarithmic functions of the coordinates of the point considered and of the dimensions of the conductor, and of Legendre's elliptic integrals of the first, second and third kind, having the same independent variables. The relations presented can be usefully adopted as efficient and time-saving algorithms in magnetic field and eddy current analysis.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Adiabatic pair potential of highly charged plates in an electrolyte

The effective interaction between highly charged plates immersed in an electrolyte solution is investigated using mean field theory. The Helmholtz free energy is formulated as a functional of the mean electric potential which satisfies the Poisson-Boltzmann equation. Exact solutions of the Poisson-Boltzmann equation under Dirichlet and Neumann boundary conditions enable us to express the free energy analytically in terms of Legendre's elliptic integrals of the first and second kinds. The adiabatic potentials of the charged plates, which are determined as the parts of the free energy that depend on the interplate distance, turn out to have long-range weak attractive parts as well as medium-range strong repulsive parts irrespective of the type of boundary conditions. While the repulsion originates mainly in the osmotic pressure of the excess ions trapped between the plates by the large surface charges, the attraction arises from an electric pull from the intermediate cloud of excess counterions between the ...

British Association Reports

Report of the Committee on Mathematical Tables. The objects for which the Committee were appointed at Edinburgh were twofold, viz., the preparation of a list of tables scattered about in books and mathematical journals and transactions, and the calculation of new tables. With regard to the first object, the tables were roughly divided into three classes, viz. (1) ordinary tables (such as trigonometrical and logarithmic) usually published in books; (2) tables of continuously varying quantities, generally definite integrals; and (3) theory-of-numberf tables. On the first class Mr. J. W. L. Glaisher had already written a report, to which it was intended, after the lapse of several years, to add a supplement; with the second some progress had been made; while Prof. Cayley proposed to undertake the third. The Committee had to acknowledge the assistance of several foreigners, and chiefly of Prof. Bierens de Haan, who had forwarded to them an account of 128 logarithmic and 105 non-logarithmic tables; to Dr. Carl Ohrtmann, of Berlin; and Profs. W. W. Johnson and J. M. Rice, of Annapolis, Maryland. The principal achievement, however, which the Committee had to report related to the second object, and was the completion of the tables of the Elliptic Functions, the commencement of which was noticed in NATURE nearly two years ago, and on which six or seven computers, under the superintendence of Mr. J. Glaisher, F.R.S., and Mr. J. W. L. Glaisher, have since been constantly engaged. These tables (which are of double entry) give the four theta functions, which form the numerators and denominators of the three elliptic functions, and their logarithms for 8,100 arguments; so that they contain nearly 65,000 tabular results. The calculation has been carried to ten figures, but only eight will be printed, the tabular portion of the work occupying 360 pages. Parts of the introduction will be written by Prof. Cayley, Sir William Thomson, and Prof. H. J. S. Smith, and. it is hoped that before the next meeting of the Association the whole work, which will form one of the largest tables that have appeared as the result of an original calculation, will be in print. It is perhaps desirable to state that the elliptic functions which have thus been tabulated are, as it were, generalised sines and cosines. Sines and cosines may be combined so as to represent any singly periodic function, as is well known; and in the same way elliptic functions represent every possible doubly periodic function; and no quantities can be of a higher degree of periodicity. The elliptic functions (which are in a sense inverse to Legendre's Elliptic Integrals) are thus quantities of the highest importance and generality in mathematics, and they are daily becoming of more importance in physics. They appear conspicuously in the investigation of the motion of a rigid body and in electrostatics, and have also numerous applications in the theory of numbers. The calculations were just completed before the meeting, and the printing will commence immediately: it is intended that the tables shall be stereotyped to ensure freedom from typographical errors.