Helmholtz Theorem Research Articles

Propagation of plane wave in non-homogeneously saturated soils

Based on Biot’s model for fluid-saturated media, which takes the inertial, fluid viscous, mechanical couplings, compressibility of grains and fluid into account, the dispersion equations of plane waves in non-homogeneously saturated soil are established by using reverberation ray matrix method (RRMM) with the aid of Helmholtz theorem. The non-homogeneity considered is a gradient variation in material properties with depth. The propagation characteristic of elastic waves in non-homogeneously saturated soil is analyzed by numerical example in this paper. The results show that the wave number and dissipation change little for two kinds of compression along the variation direction of the material properties, however, the non-homogeneity has significant effect on the wave number and dissipation of shear wave.

NEW APPLICATION OF DECOMPOSITION OF U(1) GAUGE POTENTIAL: AHARONOV–BOHM EFFECT AND ANDERSON–HIGGS MECHANISM

In this paper, we study the Aharonov–Bohm (A–B) effect and Anderson–Higgs mechanism in Ginzburg–Landau model of superconductors from the perspective of the decomposition of U (1) gauge potential. By the Helmholtz theorem, we derive exactly the expression of the transverse gauge potential A⊥ in A–B experiment, which is gauge-invariant and physical. For the case of a bulk superconductor, we find that the gradient of the total phase field θ provides the longitudinal component A‖, which reflects the Anderson–Higgs mechanism. For the case of a superconductor ring, the gradient of the longitudinal phase field θ1 provides the longitudinal component A‖, while the transverse phase field θ2 produces new physical effects such as the flux quantization inside a superconducting ring.

루프-스타(Loop-Star) 기저 함수와 전제 조건(Preconditioner)을 이용한 모멘트법의 계산 효율 향상에 대한 연구

본 논문에서는 전계 적분 방정식 (Electric Field Integral Equation: EFIE)을 사용하는 모멘트 법의 저주파 오차(low frequency breakdown) 문제를 해결하기 위한 방법으로 루프-스타(loop-star) 기저 함수를 사용하였다. 또한, 모멘트 법의 해를 계산하기 위하여 conjugate gradient method(CGM)과 같은 반복법을 적용할 경우 반복 횟수를 줄이기 위한 기법으로 p-Type Multiplicative Schwarz preconditioner(pMUS)를 이용하였다. 헬름홀쯔 정리(Helmholtz theorem)에 기반한 루프-스타(loop-star) 기저 함수와 주파수 정규화 기법을 이용하여 전계 적분 방정식에서 Rao-Wilton-Glisson(RWG) 기저 함수를 사용하였을 때 발생하는 저주파 오차(low frequency instability) 문제를 해결할 수 있다. 하지만, RWG 기저 함수를 비발산(solenoidal) 성분과 비회전성(irroatational) 성분으로 분해함으로써 발생하는 행렬 방정식의 높은 조건 수(condition number)로 인하여 CGM과 같은 반복법을 사용할 경우 해를 계산하기 위하여 많은 반복 횟수가 요구된다. 본 논문에서는 이러한 문제점을 해결하기 위한 방안으로 pMUS 전제 조건 기법을 이용하여 CGM의 반복 횟수를 줄였다. 수치 해석 결과, pMUS와 같은 희소성(sparsity)을 가진 블럭 대각 전제 조건(Block Diagonal Precondtioner: BDP)과 비교하였을 때 pMUS는 BDP보다 빠르게 해를 계산할 수 있다. This paper uses loop-star basis functions to overcome the low frequency breakdown problem in method of moments (MoM) based on electric field integral equation(EFIE). In addition, p-Type Multiplicative Schwarz preconditioner (p-MUS) technique is employed to reduce the number of iterations required for the conjugate gradient method(CGM). Low frequency instability with Rao Wilton Glisson(RWG) basis functions in EFIE can be resolved using loop-start basis functions and frequency normalized techniques. However, loop-star basis functions, consisting of irrotational and solenoidal components of RWG basis functions, require a large number of iterations to calculate a solution through iterative methods, such as conjugate gradient method(CGM), due to high condition number. To circumvent this problem, in this paper, the pMUS preconditioner technique is proposed to reduce the number of iterations in CGM. Simulation results show that pMUS preconditioner is much faster than block diagonal preconditioner(BDP) when the sparsity of pMUS is the same as that of BDP.

Longitudinal and transverse components of a vector field

A unified account, from a pedagogical perspective, is given of the longitudinal and transverse projective delta functions proposed by Belinfante and of their relation to the Helmholtz theorem for the decomposition of a three-vector field into its longitudinal and transverse components. It is argued that the results are applicable to fields that are time-dependent as well as fields that are time-independent.

Electromagnetic potentials without gauge transformations

In this paper, we show that the use of the Helmholtz theorem enables the derivation of uniquely determined electromagnetic potentials without the necessity of using gauge transformation. We show that the electromagnetic field comprises two components, one of which is characterized by instantaneous action at a distance, whereas the other propagates in retarded form with the velocity of light. In our attempt to show the superiority of the new proposed method to the standard one, we argue that the action-at-a-distance components cannot be considered as a drawback of our method, because the recommended procedure for eliminating the action at a distance in the Coulomb gauge leads to theoretical subtleties that allow us to say that the needed gauge transformation is not guaranteed. One of the theoretical consequences of this new definition is that, in addition to the electric E and magnetic B fields, the electromagnetic potentials are real physical quantities. We show that this property of the electromagnetic potentials in quantum mechanics is also a property of the electromagnetic potentials in classical electrodynamics.

Gauss's law, infinite homogenous charge distributions and Helmholtz theorem

We rediscuss the validity of Gauss's law in the case of homogenous discrete and continuous charge distributions fulfilling all space.

3-D Modeling of High-$T_{c}$ Superconductor for Magnetic Levitation/Suspension Application—Part I: Introduction to the Method

A magnetic levitation technique with high- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Tc</i> superconductor (HTS) has received significant interest for a wide range of applications after its discovery due to its unique inherent stability, which gives a fundamental significance to evaluate the HTS magnetic levitation in both experiment and calculation. To numerically investigate the HTS magnetic levitation, a 3-D model describing the electromagnetic property of the HTS, including its anisotropic behavior, was established by incorporating the current vector potential and Helmholtz's theorem. In addition to the commonly considered nonlinear <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> - <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> characteristic in the reported calculation, we introduce an elliptical model to formulate the angular dependence of the critical current density <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Jc</i> resulting from the anisotropic behavior of the HTS. To numerically resolve the governing equations of the 3-D model, Galerkin's finite-element method and the Crank-Nicolson-θ method were employed to discretize the governing equations in space and time domains, respectively. The obtained algebraic equations were firstly linearized by the Newton-Raphson method, and then an extended format of the incomplete Cholesky-conjugate gradient method was applied to solve the linear algebraic equations. The 3-D model was implemented by a self-written numerical program based on a VC++ platform to calculate the magnetic force of a bulk HTS exposed to applied field generated by a permanent magnet guideway (PMG) assembled by the Nd-Fe-B magnets. In this paper, we present the numerical results of the levitation force of a moving bulk HTS above the PMG with different mesh densities and number of time steps. This presents a preliminary validation of the 3-D model proposed in this paper.

Mapping between dissipative and Hamiltonian systems

Theoretical studies of nonequilibrium systems are complicated by the lack of a general framework. In this work we first show that a transformation recently introduced by Ao (2004 J. Phys. A: Math. Gen. 37 L25) is related to previous works of Graham (1977 Z. Phys. B 26 397) and Eyink et al (1996 J. Stat. Phys. 83 385), which can also be viewed as the generalized application of the Helmholtz theorem in vector calculus. We then show that systems described by ordinary stochastic differential equations with white noise can be mapped to thermostated Hamiltonian systems. A steady-state of a dissipative system corresponds to the equilibrium state of the corresponding Hamiltonian system. These results provide a solid theoretical ground for corresponding studies on nonequilibrium dynamics, especially on nonequilibrium steady state. Mapping permits the application of established techniques and results for Hamiltonian systems to dissipative non-Hamiltonian systems, those for thermodynamic equilibrium states to nonequilibrium steady states. We discuss several implications of this work.

Neuroelectric source imaging using 3SCO: A space coding algorithm based on particle swarm optimization and l0 norm constraint

The electroencephalogram (EEG) neuroelectric sources inverse problem is usually underdetermined and lacks a unique solution, which is due to both the electromagnetism Helmholtz theorem and the fact that there are fewer observations than the unknown variables. One potential choice to tackle this issue is to solve the underdetermined system for a sparse solution. Aiming to the sparse solution, a novel algorithm termed 3SCO (Solution Space Sparse Coding Optimization) is presented in this paper. In 3SCO, after the solution space is coded with some particles, the particle-coded space is compressed by the evolution of particle swarm optimization algorithm, where an l0 constrained fitness function is introduced to guarantee the selection of a suitable sparse solution for the underdetermined system. 3SCO was first tested by localizing simulated EEG sources with different configurations on a realistic head model, and the comparisons with minimum norm (MN), LORETA (low resolution electromagnetic tomography), l1 norm solution and FOCUSS (focal underdetermined system solver) confirmed that a good sparse solution for EEG source imaging could be achieved with 3SCO. Finally, 3SCO was applied to localize the neuroelectric sources in a visual stimuli related experiment and the localized areas were basically consistent with those reported in previous studies.

Maxwell equations and the redundant gauge degree of freedom

On transformation to the Fourier space (k, ω), the partial differential Maxwell equations simplify to algebraic equations, and the Helmholtz theorem of vector calculus reduces to vector algebraic projections. Maxwell equations and their solutions can then be separated readily into longitudinal and transverse components relative to the direction of the wave vector k. The concepts of wave motion, causality, scalar and vector potentials and their gauge transformations in vacuum and in materials can also be discussed from an elementary perspective. In particular, the excessive freedom of choice associated with the gauge dependence of the scalar and the longitudinal vector potentials stands out with clarity in Fourier spaces. Since these potentials are introduced to represent the instantaneous longitudinal electric field, the actual cancellation in the latter of causal contributions arising from these potentials separately in most velocity gauges becomes an important issue. This cancellation is explicitly demonstrated both in the Fourier space, and for pedagogical reasons again in spacetime. The physical origin of the gauge degree of freedom in the masslessness of the photon, the quantum of electromagnetic wave, is elucidated with the help of special relativity and quantum mechanics.

Statistical mechanical proof of the second law of thermodynamics based on volume entropy

In a previous work [Campisi, M. (2005). On the mechanical foundations of thermodynamics: The generalized Helmholtz theorem. Studies in History and Philosophy of Modern Physics, 36, 275–290] we have addressed the mechanical foundations of equilibrium thermodynamics on the basis of the generalized Helmholtz theorem. It was found that the volume entropy provides a good mechanical analogue of thermodynamic entropy because it satisfies the heat theorem and it is an adiabatic invariant. This property explains the “equal” sign in Clausius principle (Sf⩾Si) in a purely mechanical way and suggests that the volume entropy might explain the “larger than” sign (i.e. the law of entropy increase) if non-adiabatic transformations are considered. Based on the principles of microscopic (quantum or classical) mechanics we prove here that, provided the initial equilibrium satisfies the natural condition of decreasing ordering of probabilities, the expectation value of the volume entropy cannot decrease for arbitrary transformations performed by some external source of work on an insulated system. This can be regarded as a rigorous quantum-mechanical proof of the second law. We discuss how this result relates to the minimal work principle and how it improves on previous attempts. The natural evolution of entropy is towards larger values because the natural state of matter is at positive temperature. Actually the law of entropy decrease holds in artificially prepared negative temperature systems.

Reply to ‘Comment on “Helmholtz Theorem and the V-Gauge in the Problem of Superluminal and Instantaneous Signals in Classical Electrodynamics” by A. Chubykalo et al.’ by J.A. Heras [Found. Phys. Lett. vol. 19(6) p. 579 (2006)

This is the reply to `COMMENT ON ``HELMHOLTZ THEOREM AND THE V-GAUGE IN THE PROBLEM OF SUPERLUMINAL AND INSTANTANEOUS SIGNALS IN CLASSICAL ELECTRODYNAMICS" BY A. CHUBYKALO ET AL' BY J. A. HERAS [FOUND. PHYS. LETT. vol. 19(6) p. 579 (2006)]

Practical electromagnetics: from biomedical sciences to wireless communication

Preface. 1. INTRODUCTION. 1.1 Electrical sources and fundamental quantities. 1.2 Static and dynamic fields. 1.3 Working with complex numbers and functions. 2. VECTORS AND FIELDS. 2.1 Working with vectors. 2.2 Coordinate systems. 2.3 Differentiation and integration of vectors. 2.4 Gradient of the scalar field and its applications. 2.5 Divergence of the vector field and its applications. 2.6 Curl of the vector field and its applications. 2.7 The divergence theorem. 2.8 Stokes' theorem. DELTA. 2.9 Other operations involving 2.10 Helmholtz theorem. 3. BASIC LAWS OF ELECTROMAGNETICS. 3.1 Maxwell's equations in large scale/integral form. 3.2 Maxwell's equations in point/differential form. 3.3 Constitutive relations. 3.4 Boundary conditions. 3.5 Lorentz force equation. 3.6 Poynting vector and power flow. 4. UNIFORM PLANE WAVES. 4.1 The wave equation and uniform plane wave solutions. 4.2 Plane electromagnetic waves in Lossy media. 4.3 Uniform plane wave incident normally on an interface. 4.4 Uniform plane wave incident obliquely on an interface. 5. TRANSMISSION LINES. 5.1 Transmission line equations. 5.2 Finite length transmission line. 5.3 Smith chart. 5.4 Transients on transmission lines. 6. MODIFIED MAXWELL'S EQUATIONS AND POTENTIAL FUNCTIONS. 6.1 Magnetic charge and current. 6.2 Magnetic vector and electric scalar potentials. 6.3 Electric vector and magnetic scalar potentials. 6.4 Construction of solution in rectangular coordinates. 6.5 Construction of solution in cylindrical coordinates. 6.6 Construction of solution in spherical coordinates. 7. SOURCE IN INFINITE SPACE. 7.1 Fields of an infinitesimal source. 7.2 Antenna parameters. 7.3 Linear antennas. 7.4 Antenna arrays. 7.5 Friis transmission formula and the radar range equation. 8. ELECTROSTATIC FIELDS. 8.1 Laws of electrostatic fields. 8.2 Gauss' law. 8.3 Poisson's and Laplace's equations. 8.4 Capacitors and energy storage. 8.5 Further applications of Poisson's and Laplace's equations. 9. MAGNETOSTATIC FIELDS. 9.1 Laws of magnetostatic fields. 9.2 Inductors and energy storage. 9.3 Magnetic materials. 9.4 Magnetic Circuits. 10. WAVEGUIDES AND CAVITY RESONATORS. 10. 1 Metallic rectangular waveguide. 10. 2 Metallic circular cylindrical waveguide. 10.3 Rectangular cavity resonators. 10.4 Circular cylindrical cavity resonators. 11. NUMERICAL TECHNIQUES. 11.1 Finite difference methods. 11.2 The method of moments. 11.3 Scattering of plane EM waves from an infinitely long cylinder. Appendix A. Mathematical formulas. Appendix B. Delta function and evaluation of fields in unbounded media. Appendix C. Bessel functions. Appendix D. Legendre functions. Appendix E. Characteristics of selected materials. Appendix F. Physical constants. Appendix G. Decibels and Neper. Appendix H. Nomenclature and characteristics of standard rectangular waveguides. SELECTED REFERENCE BOOKS . Index.

On Helmholtz's Theorem and its Interpretations

This paper presents a direct, natural proof of a generalized Helmholtz's theorem for piecewise continuously differentiable vector functions in vector analysis and mathematical physics and its precise statement. Based on the generalized Helmholtz's identity, it is pointed out that Helmholtz's theorem is an operator-based decomposition theorem of a vector function. As a mathematical identity, although it is compatible with some uniqueness theorems (especially those in electromagnetics), it does not indicate directly any uniqueness theorems for boundary value problems. Most existing versions of Helmholtz's theorem are commented. As an important application of the generalized Helmholtz's identity, the definitions of irrotational and solenoidal vector functions are revisited and complete definitions are proposed as a result. The generalized Helmholtz's theorem and the present conclusions should have important indications in vector analysis related disciplines such as electromagnetics.

A RIGOROUS AND COMPLETED STATEMENT ON HELMHOLTZ THEOREM

There are some limitations on the statement of classic Helmholtz theorem although it has broad applications. Actually, it only applies to simply connected domain with single boundary surface and does not provide any conclusion about the domain where discontinuities of field function exist. However, discontinuity is often encountered in practice, for example, the location of surface sources or interface of two kinds of medium. Meanwhile, most existing versions of Helmholtz theorem are imprecise and imperfect to some extent. This paper not only tries to present a precise statement and rigorous proof on classic Helmholtz theorem with the accuracy of mathematical language and logical strictness, but also generalizes it to the case of multiply connected domain and obtains a generalized Helmholtz theorem in the sense of Lebesgue measure and Lebesgue integral defined on three-dimensional Euclidean space. Meanwhile, our proof and reasoning are more sufficient and perfect. As an important application of the generalized Helmholtz theorem, the concepts of irrotational and solenoidal vector function are emphasized. The generalized Helmholtz theorem and the present conclusion should have important reference value in disciplines including vector analysis such as electromagnetics.

Comment on “Helmholtz Theorem and the V-Gauge in the Problem of Superluminal and Instantaneous Signals in Classical Electrodynamics,” by A. Chubykalo et al.

Fundamental errors in a paper by Chubykalo et al. [2] are highlighted. Contrary to their claim that “… the irrotational component of the electric field has a physical meaning and can propagate exclusively instantaneously,” it is shown that this instantaneous component is physically irrelevant because it is always canceled by a term contained in the solenoidal component. This result follows directly from the solution of the wave equation that satisfies the solenoidal component. Therefore the subsequent inference of these authors that there are two mechanisms of transmission of energy and momentum in classical electrodynamics, one retarded and the other one instantaneous, has no basis. The example given by these authors in which the full electric field of an oscillating charge equals its instantaneous irrotational component on the axis of oscillations is proven to be false.

Comment on “A generalized Helmholtz theorem for time-varying vector fields,” by Artice M. Davis [Am. J. Phys. 74, 72–76 (2006)

In a recent paper Davis formulated a generalized Helmholtz theorem for a time-varying vector field in terms of the Lorenz gauge retarded potentials. The purposes of this comment are to point out that Davis's generalization of the theorem is a version of the extension of the Helmholtz theorem formulated some years ago by McQuistan and also by Jefimenko and more recently by the present author and to show that Davis's expression for the time-dependent vector field is also valid for potentials in gauges other than the Lorenz gaug

Helmholtz Theorem and the V-Gauge in the Problem of Superluminal and Instantaneous Signals in Classical Electrodynamics

In this work we substantiate the applying of the Helmholtz vector decomposition theorem (H-theorem) to vector fields in classical electrodynamics. Using the H-theorem, within the framework of the two-parameter Lorentz-like gauge (so called v-gauge), we show that two kinds of magnetic vector potentials exist: one of them (solenoidal) can act exclusively with the velocity of light c and the other one (irrotational) with an arbitrary finite velocity $v$ (including a velocity more than c . We show also that the irrotational component of the electric field has a physical meaning and can propagate exclusively instantaneously.

On the chaotic rotation of a liquid-filled gyrostat via the Melnikov–Holmes–Marsden integral

The non-linear motions of a gyrostat with an axisymmetrical, fluid-filled cavity are investigated. The cavity is considered to be completely filled with an ideal incompressible liquid performing uniform rotational motion. Helmholtz theorem, Euler's angular momentum theorem and Poisson equations are used to develop the disturbed Hamiltonian equations of the motions of the liquid-filled gyrostat subjected to small perturbing moments. The equations are established in terms of a set of canonical variables comprised of Euler angles and the conjugate angular momenta in order to facilitate the application of the Melnikov–Holmes–Marsden (MHM) method to investigate homoclinic/heteroclinic transversal intersections. In such a way, a criterion for the onset of chaotic oscillations is formulated for liquid-filled gyrostats with ellipsoidal and torus-shaped cavities and the results are confirmed via numerical simulations.

A generalized Helmholtz theorem for time-varying vector fields

This paper develops a generalized Helmholtz theorem for an arbitrary localized time-varying vector function F⃗(r⃗,t) and shows that the mathematical relations commonly referred to as Maxwell’s equations can be derived from that theorem alone.