Showing papers on "Elliptic coordinate system published in 2019"

Caustics of Poncelet Polygons and Classical Extremal Polynomials

Abstract: A comprehensive analysis of periodic trajectories of billiards within ellipses in the Euclidean plane is presented. The novelty of the approach is based on a relationship recently established by the authors between periodic billiard trajectories and extremal polynomials on the systems of d intervals on the real line and ellipsoidal billiards in d-dimensional space. Even in the planar case systematically studied in the present paper, it leads to new results in characterizing n periodic trajectories vs. so-called n elliptic periodic trajectories, which are n-periodic in elliptical coordinates. The characterizations are done both in terms of the underlying elliptic curve and divisors on it and in terms of polynomial functional equations, like Pell’s equation. This new approach also sheds light on some classical results. In particular, we connect the search for caustics which generate periodic trajectories with three classical classes of extremal polynomials on two intervals, introduced by Zolotarev and Akhiezer. The main classifying tool are winding numbers, for which we provide several interpretations, including one in terms of numbers of points of alternance of extremal polynomials. The latter implies important inequality between the winding numbers, which, as a consequence, provides another proof of monotonicity of rotation numbers. A complete catalog of billiard trajectories with small periods is provided for n = 3, 4, 5, 6 along with an effective search for caustics. As a byproduct, an intriguing connection between Cayley-type conditions and discriminantly separable polynomials has been observed for all those small periods.

Analytical solution for diverging elliptic shear wave in bounded and unbounded transverse isotropic viscoelastic material with nonhomogeneous inner boundary.

Abstract: A theoretical approach was recently introduced by Guidetti and Royston [J. Acoust. Soc. Am. 144, 2312–2323 (2018)] for the radially converging elliptic shear wave pattern in transverse isotropic materials subjected to axisymmetric excitation normal to the fiber axis at the outer boundary of the material. This approach is enabled via a transformation to an elliptic coordinate system with isotropic properties. The approach is extended to the case of diverging shear waves radiating from a cylindrical rod that is axially oscillating perpendicular to the axis of isotropy and parallel to the plane of isotropy.A theoretical approach was recently introduced by Guidetti and Royston [J. Acoust. Soc. Am. 144, 2312–2323 (2018)] for the radially converging elliptic shear wave pattern in transverse isotropic materials subjected to axisymmetric excitation normal to the fiber axis at the outer boundary of the material. This approach is enabled via a transformation to an elliptic coordinate system with isotropic properties. The approach is extended to the case of diverging shear waves radiating from a cylindrical rod that is axially oscillating perpendicular to the axis of isotropy and parallel to the plane of isotropy.

Converging super-elliptic torsional shear waves in a bounded transverse isotropic viscoelastic material with nonhomogeneous outer boundary

Abstract: A theoretical approach was recently introduced [Guidetti and Royston, J. Acoust. Soc. Am. 144, 2312-2323 (2018)] for the radially converging slow shear wave pattern in transverse isotropic materials subjected to axisymmetric excitation normal to the axis of isotropy at the outer boundary of the material. This approach is enabled via transformation to an elliptic coordinate system with isotropic properties. The approach is extended to converging fast shear waves driven by axisymmetric torsional motion polarized in a plane containing the axis of isotropy. The approach involves transformation to a super-elliptic shape with isotropic properties and use of a numerically efficient boundary value approximation.

Analytical solutions of some internal boundary value problems of elasticity for domains with hyperbolic boundaries

Abstract: An analytical solution of two-dimensional problems of elasticity in the region bounded by a hyperbola in elliptic coordinates is constructed using the method of separation of variables. The stress–...

On the Equivalence Between Type I Liouville Dynamical Systems in the Plane and the Sphere

Abstract: Separable Hamiltonian systems either in sphero-conical coordinates on an S2 sphere or in elliptic coordinates on a \({\mathbb R}^2\) plane are described in a unified way. A back and forth route connecting these Liouville Type I separable systems is unveiled. It is shown how the gnomonic projection and its inverse map allow us to pass from a Liouville Type I separable system with a spherical configuration space to its Liouville Type I partners where the configuration space is a plane and back. Several selected spherical separable systems and their planar cousins are discussed in a classical context.

Study of stress–strain state of elastic body with hyperbolic notch

Abstract: The paper considers elastic stress distributions in infinite space with hyperbolic notch when normal or tangential stresses are given on the boundary of notch. The work considers plane deformation. So, exact (analytical) solution of two-dimensional boundary value problems of elasticity in the domain with hyperbolic boundary in the elliptic coordinate system is constructed using the method of separation of variables. The stress–strain state of a homogeneous isotropic infinite body with a hyperbolic cut is studied when there are non-homogeneous (nonzero) boundary conditions given on the hyperbolic cut. Finally, the numerical simulation is performed to the stress and displacement distributions over a finite size volume surrounding the notch and relevant graphs for the numerical results of some test problems are presented.

Scattering and Diffraction of a Uniform Complex-Source Beam by a Slit

Abstract: The two-dimensional problem of the scattering and diffraction of a TMz-polarized uniform complex-source beam by a slit is solved directly and analytically by using plane-polar coordinates. Suitable orthogonality relations finally yield a finite system of linear equations. Its numerical solution is successfully compared to the classical one obtained via the solution of the strip problem in elliptic coordinates using Mathieu functions and then applying Babinet's principle. The evaluation of the numerical results reveals that a suitably chosen uniform complex-source beam can replace an incident plane wave if the scattering features of a small part of the object are needed.

Recovering functions from the spherical mean transform with data on an ellipse using eigenfunction expansion in elliptical coordinates

Abstract: The aim of this paper is to introduce a new inversion procedure for recovering functions, defined on $$\mathbb R^{2}$$ , from the spherical mean transform, which integrates functions on a prescribed family $$\Lambda $$ of circles, where $$\Lambda $$ consists of circles whose centers belong to a given ellipse E on the plane. The method presented here follows the same procedure which was used by Norton (J Acoust Soc Am 67:1266–1273, 1980) for recovering functions in case where $$\Lambda $$ consists of circles with centers on a circle. However, at some point we will have to modify the method in [24] by using expansion in elliptical coordinates, rather than spherical coordinates, in order to solve the more generalized elliptical case. We will rely on a recent result obtained by Cohl and Volkmer (J Phys A Math Theor 45:355204, 2012) for the eigenfunction expansion of the Bessel function in elliptical coordinates.

Simulating Quantum Mechanics with Light: The Quantum Pendulum Via Mathieu Beams

Abstract: We simulate the nonlinear quantum pendulum using non-diffracting solutions of the 2-dimensional Helmholtz equation in elliptical coordinates. Stationary and wavepacket Mathieu spatial modes are in quantitative agreement with the quantum probabilities.

A symmetric integral identity for Bessel functions with applications to integral geometry

Abstract: In the article of Kunyansky (Inverse Probl 23(1):373–383, 2007) a symmetric integral identity for Bessel functions of the first and second kind was proved in order to obtain an explicit inversion formula for the spherical mean transform where our data is given on the unit sphere in \({\mathbb {R}}^{n}\). The aim of this paper is to prove an analogous symmetric integral identity in case where our data for the spherical mean transform is given on an ellipse E in \({\mathbb {R}}^{2}\). For this, we will use the recent results obtained by Cohl and Volkmer (J Phys A Math Theor 45:355204, 2012) for the expansions into eigenfunctions of Bessel functions of the first and second kind in elliptical coordinates.

Energy transfer of the tightly focused hybridly polarized vector optical fields with elliptic symmetry in free space.

Abstract: We theoretically and experimentally present hybridly polarized vector optical fields (HP-VOFs) with elliptic symmetry in an elliptic coordinate system. Compared with the traditional cylindrical HP-VOFs, there is an additional degree of freedom for this new kind of vector optical field, which is the interval between the two foci in the elliptic coordinate system. Except for discussing the singularities of the HP-VOFs, we concentrate on studying the energy transfer of the tightly focused HP-VOFs with elliptic symmetry in free space. We summarize the rules of the energy transfer and introduce a reference optical field to explain them. We hope these results can provide a new way to flexibly modulate tightly focused fields, which may be applied in realms such as optical machining, optical trapping, and information transmission.

Natural oscillation shapes for a heavy fluid in an elliptic vessel

Abstract: The problem of constructing of shapes of the natural oscillations in a vertical cylindrical basin (vessel) with an elliptic cross-section in linearized form of an ideal incompressible fluid is considered. Only the gravity force is acting on the fluid. It is required to define the frequencies and shapes of the natural oscillations of the fluid as a function of the system's parameters. The solution algorithm represented here is based on the method of separation of variables in elliptic coordinates, and the special developed method for the high-accuracy solution of the boundary problem. It used the variational principle and the procedure of continuation in the parameter. This approach shows its effectiveness and obtains the almost complete solution of the problem. For five lower oscillation modes, the natural shapes are determined with high degree of accuracy for a wide range of the eccentricity values.