Showing papers on "Elliptic coordinate system published in 2011"

The Kepler Problem: Group Theoretical Aspects, Regularization and Quantization, with Application to the Study of Perturbations

Abstract: Preface.- List of Figures.- 1 Introductory Survey.- 1.1 Part I - Elementary Theory.- 1.1.1 Basic Facts.- 1.1.2 Separation of Variables and Action-Angle Variables.- 1.1.3 Quantization of the Kepler Problem.- 1.1.4 Regularization and Symmetry.- 1.2 Part II - Group-Geometric Theory.- 1.2.1 Conformal Regularization.- 1.2.2 Spinorial Regularization.- 1.2.3 Return to Separation of Variables.- 1.2.4 Geometric Quantization.- 1.2.5 Kepler Problem with a Magnetic Monopole.- 1.3 Part III - Perturbation Theory.- 1.3.1 General Perturbation Theory.- 1.3.2 Perturbations of the Kepler Problem.- 1.3.3 Perturbations with Axial Symmetry.- 1.4 Part IV - Appendices.- 1.4.1 Differential Geometry.- 1.4.2 Lie Groups and Lie Algebras.- 1.4.3 Lagrangian Dynamics.- 1.4.4 Hamiltonian Dynamics.- I Elementary Theory 17.- 2 Basic Facts.- 2.1 Conics.- 2.2 Properties of the Keplerian Motion.- 2.2.1 Energy H 0.- 2.2.3 Energy H = 0.- 2.3 The Three Anomalies.- 2.3.1 Energy H 0.- 2.3.3 Energy H = 0.- 2.4 The Elements of the Orbit for H < 0.- 2.5 The Repulsive Potential.- Append.- 2.A The Kepler Equation.- 3 Separation of Variables and Action-Angle Coordinates.- 3.1 Separation of Variables.- 3.1.1 Spherical Coordinates.- 3.1.2 Parabolic Coordinates.- 3.1.3 Elliptic Coordinates.- 3.1.4 Spheroconical Coordinates.- 3.2 Action-Angle Variables.- 3.2.1 Delaunay and Poincare Variables.- 3.2.2 Pauli Variables.- 3.2.3 Monodromy.- 4 Quantization of the Kepler Problem.- 4.1 The Schrodinger Quantization.- 4.1.1 Spherical Coordinates.- 4.1.2 Parabolic Coordinates.- 4.1.3 Elliptic Coordinates.- 4.1.4 Spheroconical Coordinates.- 4.2 Pauli Quantization.- 4.2.1 Canonical Quantization.- 4.2.2 Pauli Quantization.- 4.3 Fock Quantization.- Append.- 4.A Mathematical Review.- 4.A.1 Second Order Linear Differential Equations.- 4.A.2 Laplacian on the Sphere and Homogeneous Harmonic Polynomials.- 4.A.3 Associated Legendre Functions.- 4.A.4 Generalized Laguerre Polynomials.- 4.A.5 Surface Measure on the Sphere and Gamma Function.- 4.A.6 Green Function of the Laplacian.- 5 Regularization and Symmetry.- 5.1 Moser Method.- 5.2 Souriau Method.- 5.2.1 Fock Parameters.- 5.2.2 Bacry-Gyorgyi Parameters.- 5.3 Kustaanheimo-Stiefel Transformation.- II Group-Geometric Theory 109.- 6 Conformal Regularization.- 6.1 The Conformal Group.- 6.2 The Compactified Minkowski Space.- 6.3 The Cotangent Bundle to Minkowski Space.- 6.4 Regularization of the Kepler Problem.- 7 Spinorial Regularization.- 7.1 The Homomorphism SU(2, 2) ? SO(2, 4).- 7.1.1 Two Bases for su(2, 2).- 7.1.2 SU(2, 2) and Compactified Minkowski Space.- 7.2 Return to the Kustaanheimo-Stiefel Map.- 7.3 Generalized Kustaanheimo-Stiefel Map.- 8 Return to Separation of Variables.- 8.1 Separable Orthogonal Systems.- 8.1.1 Stackel Theorem.- 8.1.2 Eisenhart Theorem.- 8.1.3 Robertson Theorem.- 8.2 Finding Coordinate Systems Separating Kepler Problem.- 8.2.1 Spherical Coordinates.- 8.2.2 Parabolic Coordinates.- 8.2.3 Elliptic Coordinates.- 8.2.4 Spheroconical Coordinates.- 8.3 Integrable Perturbations.- 8.3.1 Euler Problem.- 8.3.2 Stark Problem.- Append.- 8.A Jacobian Elliptic Functions.- 9 Geometric Quantization.- 9.1 Multiplier Representations.- 9.2 Quantization of Geodesics on the Sphere.- 9.3 Quantization of the Kepler Problem.- 10 Kepler Problem with Magnetic Monopole.- 10.1 Nonnull Twistors and Magnetic Monopoles.- 10.1.1 Bound Motions.- 10.1.2 Unbound Motions.- 10.1.3 Quantization.- 10.2 The MICZ System.- 10.3 The Taub-NUT System.- 10.4 The BPST Instanton.- III Perturbation Theory 235.- 11 General Perturbation Theory.- 11.1 Formal Expansions.- 11.1.1 Lie Series and Formal Canonical Transformations.- 11.1.2 Homological Equation and its Formal Solution.- 11.2 The Convergence Problem.- 11.2.1 Convergence of Lie Series.- 11.2.2 Homological Equation and its Solution.- 11.2.3 Kolmogorov Theorem.- 11.2.4 Nekhoroshev Theorem.- Appendices.- 11.AResults from Diophantine Theory.- 11.B Cauchy Inequality.- 12 Perturbations of the Kepler Problem.- 12.1 A More Convenient Hamiltonian.- 12.2 Normalization (or Averaging) Method.- 12.3 Numerical Integration.- 12.3.1 Symbolic Manipulation.- 12.3.2 Compiling Equations.- Appendices.- 12.AVariation of the Constants.- 12.B The Stabilization Method.- 13 Perturbations with Axial Symmetry.- 13.1 Reduction of Orbit Manifold.- 13.2 Lunar Problem.- 13.3 Stark and Quadratic Zeeman Effect.- 13.4 Satellite around Oblate Primary.- IV Appendices 321.- A Differential Geometry.- A.1 Rudiments of Topology.- A.2 Differentiable Manifolds.- A.2.1 Definition.- A.2.2 Tangent and Cotangent Spaces.- A.2.3 Push-forward and Pull-back.- A.3 Tensors and Forms.- A.3.1 Tensors.- A.3.2 Forms and Exterior Derivatives.- A.3.3 Lie Derivative.- A.3.4 Integration of Differential Forms.- A.4 Distributions and Frobenius Theorem.- A.5 Riemannian, Symplectic and Poisson Manifolds.- A.5.1 Riemannian Manifolds.- A.5.2 Symplectic Manifolds.- A.5.3 Poisson Manifolds.- A.6 Fibre Bundles.- A.6.1 Definition.- A.6.2 Principal and Associated Fibre Bundles.- B Lie Groups and Lie Algebras.- B.1 Definition and Properties.- B.2 Adjoint and Coadjoint Representation.- B.3 Action of a Lie Group on a Manifold.- B.4 Classification of Lie Groups and Lie Algebras.- B.5 Connection on a Principal Bundle.- C Lagrangian Dynamics.- C.1 Lagrange Equations.- C.2 Hamilton Principle.- C.3 Noether Theorem.- C.4 Reduced Lagrangian and Maupertuis Principle.- D Hamiltonian Dynamics.- D.1 From Lagrange to Hamilton.- D.2 The Hamilton-Jacobi Integration Method.- D.2.1 Canonical Transformations.- D.2.2 Hamilton-Jacobi Equation.- D.2.3 Geometric Description.- D.2.4 The Time-dependent Case.- D.3 Symmetries and Reduction.- D.3.1 The Moment Map.- D.3.2 Reduction of Symplectic Manifolds.- D.3.3 Reduction of Poisson Manifolds.- D.4 Action-Angle Variables.- D.4.1 Arnold Theorem.- D.4.2 Degenerate Systems.- D.4.3 Monodromy.

Scattering and Focusing of SH Waves by a Lower Semielliptic Convex Topography

Abstract: The SH -wave scattering induced by a lower semielliptic convex topography is studied here. A rigorous series solution is obtained via the region-matching technique. The method of separation of variables in elliptic coordinates is adopted to express the pertinent wave fields in terms of an infinite series containing products of radial and angular Mathieu functions with unknown coefficients. Steady-state responses for some parameters are calculated and discussed. Because the present surficial configuration may collect wave energy in much the same way as a concave mirror does with converging light waves, the potential focusing effects are further probed in the frequency domain. Numerical results demonstrate that the geometry under consideration is indeed capable of causing a localized high concentration of wave energy underground, which may have a great influence on the structures below the convex topography such as mountain tunnels.

A generalized Zakharov-Shabat equation with finite-band solutions and a soliton-equation hierarchy with an arbitrary parameter

Abstract: In this paper, a generalized Zakharov–Shabat equation (g-ZS equation), which is an isospectral problem, is introduced by using a loop algebra G ∼ . From the stationary zero curvature equation we define the Lenard gradients {gj} and the corresponding generalized AKNS (g-AKNS) vector fields {Xj} and Xk flows. Employing the nonlinearization method, we obtain the generalized Zhakharov–Shabat Bargmann (g-ZS-B) system and prove that it is Liouville integrable by introducing elliptic coordinates and evolution equations. The explicit relations of the Xk flows and the polynomial integrals {Hk} are established. Finally, we obtain the finite-band solutions of the g-ZS equation via the Abel–Jacobian coordinates. In addition, a soliton hierarchy and its Hamiltonian structure with an arbitrary parameter k are derived.

On orthogonal curvilinear coordinate systems in constant curvature spaces

Abstract: We describe a method for constructing an n-orthogonal coordinate system in constant curvature spaces. The construction proposed is actually a modification of the Krichever method for producing an orthogonal coordinate system in the n-dimensional Euclidean space. To demonstrate how this method works, we construct some examples of orthogonal coordinate systems on the two-dimensional sphere and the hyperbolic plane, in the case when the spectral curve is reducible and all irreducible components are isomorphic to a complex projective line.

A singular element of the graph model of an elastic medium in a cartesian coordinate system

Abstract: A numerical method for analyzing mechanical fields in a deformable body is presented. The graph model of an elastic medium in the form of a directed graph is employed in the proposed method. To describe a singularity that appears near a crack tip of an isotropic elastic material, a new type of a unit cell is introduced.

Mathieu Progressive Waves

Abstract: A new family of exact solutions to the wave equation representing relatively undistorted progressive waves is constructed using separation of variables in the elliptic cylindrical coordinates and one of the Bateman transforms. The general form of this Bateman transform in an orthogonal curvilinear cylindrical coordinate system is discussed and a specific problem of physical feasibility of the obtained solutions, connected with their dependence on the cyclic coordinate, is addressed. The limiting case of zero eccentricity, in which the elliptic cylindrical coordinates turn into their circular cylindrical counterparts, is shown to correspond to the focused wave modes of the Bessel-Gauss type.

Error estimates in coordinate measurements of the cross section parameters of cylindrical surfaces

Abstract: A simulation method and the results of simulating the errors in coordinate measurements of the cross section parameters of cylindrical surfaces are discussed. Data are presented from experimental studies that confirm the simulations.

Application of eigenfunctions method for solving time-fractional partial differential equations

Abstract: In this paper, the solution of a class of time fra ctional differential equations by using Green's function method is presented. Green's functions by using the Laplace trans- formation with respect to the time variable and the method of eigenfunction expansion with respect to space variables are derived. An analytic al form of the solution to the problem in rectangular, polar and elliptical coordinates has b een given.

The application of Fourier-Mellin moment in image recognize

Abstract: In order to test the ability of Fourier-Mellin moment in image recognize,research work are done that two ways of computation and image reconstructed in a polar coordinate and in a Cartesian coordinate system,noise robustness are implemented in this paper.The Fourier-Mellin moment can be computed directly in a Cartesian coordinate,and this method completely removes both approximation and geometrical errors produced by the conventional methods.The experiments show that the reconstructed image by the Fourier-Mellin moment in a Cartesian coordinate is more accurate than that in a polar coordinate and the Fourier-Mellin moment has strong robustness for noise images.

Covariant transformations of basis differential-difference schemes in a plane

Abstract: Consistent difference approximations to differential operators in vector and tensor analysis are constructed in curvilinear coordinates in a plane by applying the basis operator method. They are obtained as a transformation of basis approximations in a Cartesian coordinate system. For the continuum mechanics equations in Lagrangian variables, this approach yields theoretically justified differential-difference schemes whose conservation laws correspond to the continuous case.

Three-dimensional Coordinate Transformation with Consideration of Coordinate Errors in Two Coordinate Systems

Abstract: The paper presents a new coordinate transformation model with a consideration of the coordinate errors in both coordinate systems.The computation formulae of transformation parameters are derived based on the minimization of weighted sum of coordinate corrections in two systems.In addition,the coordinate corrections of transformed points are computed according to the covariance matrix of coordinates between public and transformed points with the least-squares collocation method.The experiments show that the new model can significantly improve the precision of the coordinate transformation especially in case that the strong correlation exists between public and transformed coordinates.

Elastodynamic Scalar Potential Functions in Cylindrical Coordinate Systems

Abstract: The aim of this paper is to present scalar potential functions for three-dimensional elastodynamic problems in transversely isotropic media defined by different cylindrical coordinate systems. To do so, the standard cylindrical coordinate system, elliptical cylindrical coordinate system and parabolic cylindrical coordinate system are considered. By virtue of Poisson’s representation, Lame solution and Chadwick-Trowbridge, the completeness of such representations is addressed. The representations given here are useful in the study of wave scattering from the circular, elliptical and parabolic edges.

An investigation into the expression of potential vorticity in the common meteorological coordinate systems

Abstract: The potential vorticity theory and diagnostic techniques are based on the potential vorticity equation and expression in the common meteorological coordinate systems. In this paper, the potential vorticity equation and expression in the isobaric and isoentropic coordinates are gotten via coordinate transformation with the two methods. First, starting from the three-dimensional vector motion equation, the potential vorticity equations and expressions are gotten by the combination of the three-dimensional vorticity equation, continuity equation, and thermodynamic equation. Second, the potential vorticity equations and expressions are directly gotten from the corresponding scalar motion equations in the isobaric and isoentropic coordinates. The results show that potential vorticity expression is different with one method from that with the other in the isobaric coordinate system, and it is the same as each other in the isoentropic coordinate system. It was found, based on further analysis of the physical nature of the coordinates, that the isobaric and isoentropic coordinates are essentially treated as a mathematical coordinate system with the first method despite the coordinate transformation made for the term of pressure gradient force in the vector motion equation. From the procedure for the second method it is clearly seen that the isobaric and isoentropic coordinate systems are the physical coordinate system under the assumption of static equilibrium, which are not simply used as a mathematical coordinate system. As far as the isobaric coordinate is concerned, only the potential vorticity equation obtained from the scalar motion equations is the strict potential vorticity equation. As for the isoentropic coordinate, owing to the potential temperature gradient perpendicular to the isoentropic plane, the potential vorticity equation and expression are the same regardless of the coordinate being viewed as the physical or the mathematical.

Scattering by a homogeneous anisotropic-coated conducting elliptic cylinder

Abstract: A solution to the two-dimensional scattering properties of a conducting elliptic cylinder coated with a homogeneous anisotropic elliptical shell is obtained. The conducting elliptic cylinder and the shell have the same eccentricity. The transmitted and scattered fields of the anisotropic shell are expressed as Mathieu functions in elliptic coordinates. The unknown coefficients of the scattered and transmitted fields are solved with the aid of the boundary conditions and the Galerkin's method. Only the transverse magnetic (TM) polarization is presented and the transverse electric (TE) polarization can be obtained in the same way. Some numerical results are presented and discussed. As expected the result is in agreement with that available when the coated elliptic cylinder degenerates to a coated circular one.

Criterion of transverse stability for ion traps with transverse motion integrable in elliptic coordinates

Abstract: Electrostatic traps with ideal spatiotemporal focusing of ions in one direction of their motion and the motion in transverse directions integrable in the elliptic coordinates are considered. Conditions for the confinement of ions in the trapping field are determined. The finite character of ion motion under the proposed confinement conditions is independently checked by numerical integration of the Lagrange equations.

Coordinate Systems and Systems of Equation

Abstract: A device simulator for nanoscale strained SiGe double gate PMOSFETs based on the self-consistent solution of the 1D SE, 2D PE, and multi subband 1D BTE is developed, which is applicable for arbitrary crystallographic orientations and arbitrary channel directions.