Maxwell Equations Of Electrodynamics Research Articles

Optimal control of the two-dimensional Vlasov-Maxwell system

The time evolution of a collisionless plasma is modeled by the Vlasov-Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We only consider a two-dimensional version of the problem since existence of global, classical solutions of the full three-dimensional problem is not known. We add external currents to the system, in applications generated by coils, to control the plasma properly. After considering global existence of solutions to this system, differentiability of the control-to-state operator is proved. In applications, on the one hand, we want the shape of the plasma to be close to some desired shape. On the other hand, a cost term penalizing the external currents shall be as small as possible. These two aims lead to minimizing some objective function. We restrict ourselves to only such control currents that are realizable in applications. After that, we prove existence of a minimizer and deduce first order optimality conditions and the adjoint equation.

Weak solutions of the relativistic Vlasov–Maxwell system with external currents

The time evolution of a collisionless plasma is modeled by the relativistic Vlasov–Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We consider the case that the plasma consists of N particle species, the particles are located in a bounded container , and are subject to boundary conditions on ∂Ω. Furthermore, there are external currents, typically in the exterior of the container, that may serve as a control of the plasma if adjusted suitably. We do not impose perfect conductor boundary conditions for the electromagnetic fields but consider the fields as functions on whole space and model objects, that are placed in space, via given matrix‐valued functions ε (the permittivity) and μ (the permeability). A weak solution concept is introduced and existence of global‐in‐time solutions is proved, as well as the redundancy of the divergence part of the Maxwell equations in this weak solution concept.

Confined steady states of the relativistic Vlasov–Maxwell system in an infinitely long cylinder

The time evolution of a collisionless plasma is modeled by the relativistic Vlasov-Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. In this work, the setting is two and one-half dimensional, that is, the distribution functions of the particles species are independent of the third space dimension. We consider the case that the plasma is located in an infinitely long cylinder and is influenced by an external magnetic field. We prove existence of stationary solutions and give conditions on the external magnetic field under which the plasma is confined inside the cylinder, i.e., it stays away from the boundary of the cylinder.

Hot Plasma in a Container---an Optimal Control Problem

The time evolution of a collisionless plasma is modeled by the relativistic Vlasov-Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We consider the case that the plasma is located in a bounded container $\Omega\subset\mathbb R^3$, for example a fusion reactor. Furthermore, there are external currents, typically in the exterior of the container, that may serve as a control of the plasma if adjusted suitably. We model objects, that are placed in space, via given matrix-valued functions $\varepsilon$ (the permittivity) and $\mu$ (the permeability). A typical aim in fusion plasma physics is to keep the amount of particles hitting $\partial\Omega$ as small as possible (since they damage the reactor wall), while the control costs should not be too exhaustive (to ensure efficiency). This leads to a minimizing problem with a PDE constraint. This problem is analyzed in detail. In particular, we prove existence of minimizers and establish an approach to derive first order optimality conditions.

Geometric Diversity of Living Organisms and Viruses

Recently, it has been celebrated a hundred year anniversary of Einstein's theory of general relativity. Since, underlying geometry of space and time became main source of inspiration of new theories. Though, much attention was paid to the geometric construction of the universe, geometric diversity of living organisms got largely omitted. Of course such reluctant approach has fundamental explanation. General relativity is formulated to explain the gravity, while the role of gravitation on the molecular level in the formation of living organisms is negligibly small. However, mathematical approach used in general relativity can be carried out for two dimensional surfaces. Therefore, scientists started to develop analogical differential geometric approach to predict static shapes of living organisms. As a result associated Euler-Lagrange equations, so called “shape equations” have been formulated. But, shape equations turned to be fourth order partial nonlinear differential equations, and finding a general analytical solution is a difficult problem, even though it has been numerically solved for some specific cases. The numerical solutions generated a large body of beautiful explanation of experimental results especially observed in diversity of lipid membranes shapes. Because of shape equations as well as numerical calculations are not exact equations of motion and analytical solutions, we have decided to employ different approach to the problem. Namely, we have used tensor calculus of moving surfaces to deduce exact equations of surface motion in electromagnetic field and then to analytically solve it for some simplified cases. Simplifications have produced such outstanding equations as Maxwell equations for electrodynamics and Navier-Stokes equations for dynamic fluid are (D. V. Svintradze, (2016), arXiv:1609.07765 [cond-mat.soft]). Also, analytic solutions to simplified equations for homogeneous surfaces, in equilibrium with environment, (D. V. Svintradze, (2016), arXiv:1608.01491 [physics.bio-ph]) have explained why constant mean curvature (CMC) surfaces are such abundant shapes among living organisms and viruses.

Magneto-thermo-elastic response of a rotating functionally graded cylinder

In this paper, an analytical solution of displacement, strain and stress field for rotating thickwalled cylinder made of functionally graded material subjected to the uniform external magnetic field and thermal field in plane strain state has been studied. Stress, strain and displacement field as a function of radial coordinates considering magneto-thermo-elasticity are derived analytically. According to the Maxwell electro-dynamic equations, Lorentz force in term of displacement is obtained in cylindrical coordinates. Also, symmetric temperature distribution along the thickness of hollow cylinder is obtained by solving Fourier heat transfer equation in cylindrical coordinates. Using equation of equilibrium and thermomechanical constitutive equations associated with Lorentz force, a second-order inhomogeneous differential equation in term of displacement is obtained and will be solved analytically. Except Poisson\'s ratio, other mechanical properties such as elasticity modulus, density, magnetic permeability coefficient, heat conduction coefficient and thermal expansion coefficient are assumed to vary through the thickness according to a power law. In results analysis, non-homogeneity parameter has been chosen arbitrary and inner and outer surface of cylinder are assumed to be rich metal and rich ceramic, respectively. The effect of rotation, thermal, magnetic field and non-homogeneity parameter of functionally graded material which indicates percentages of cylinder\'s constituents are studied on displacement, Von Mises equivalent stress and Von Mises equivalent strain fields.

On the “area” of an electromagnetic field

A general formulation of retained integral characteristics (“areas” of an electromagnetic field) has been obtained from the Maxwell equations for electrodynamics of continuous media. The relations found can be used to monitor the accuracy of computer simulation of the field dynamics.

Physical theories in hypercomplex geometric description

Compact description is given of algebras of poly-numbers: quaternions, bi-quaternions, double (split-complex) and dual numbers. All units of these (and exceptional) algebras are shown to be represented by direct products of 2D vectors of a local basis defined on a fundamental surface. In this math medium a series of equalities identical or similar to known formulas of physical laws is discovered. In particular, a condition of the algebras' stability with respect to transformations of the 2D-basis turns out equivalent to the spinor (Schrödinger–Pauli and Hamilton–Jacobi) equations of mechanics. It is also demonstrated that isomorphism of SO(3, 1) and SO(3, ℂ) groups leads to formulation of a quaternion relativity theory predicting all effects of special relativity but simplifying solutions of relativistic problems in non-inertial frames. Finely it is shown that the Cauchy–Riemann type equations written for functions of quaternion variable repeat vacuum Maxwell equations of electrodynamics, while a quaternion space with non-metricity comprises main relations of Yang–Mills field theory.

Theory of 4D-media with stationary dislocations

In earlier studies, the authors showed that an application of classical methods of mechanics of deformable media to the study of properties of 4D-space-time continuum permit stating consistent models of nonholonomic media mechanics consistent with the first and second laws of thermodynamics. In the present paper, we show that the classical methods of continuum mechanics are also promising when modeling physical processes. It is shown that, just as in the three-dimensional theory of stationary dislocations, there exist dislocations of three types for a generalized 4D-medium. They correspond to the decomposition of the free distortion tensor into a spherical tensor, a deviator tensor, and a pseudotensor of rotations. We interpret several particular models, thus showing that the proposed model describes the spectrum of known physical interactions: electromagnetic, strong, weak, and gravitational. We show that the resolving equations include the Maxwell equations of electrodynamics and the Yukawa equations for strong interactions as subsystems.

EQUIVALENCE GROUPS FOR FIRST-ORDER BALANCE EQUATIONS AND APPLICATIONS TO ELECTROMAGNETISM

We investigate the groups of equivalence transformations for first-order balance equations involving an arbitrary number of dependent and independent variables. We obtain the determining equations and find their explicit solutions. The approach to this problem is based on a geometric method that depends on Cartan's exterior differential forms. The general solutions of the determining equations for equivalence transformations for first-order systems are applied to a class of the Maxwell equations of electrodynamics.

Development of the method of characteristics for a system of maxwell electrodynamic equations

Expressions for the velocities of propagation of the discontinuity surfaces in a homogeneous anisotropic medium have been obtained and equations of bicharacteristics have been derived. Using them, the surfaces of inverse velocities and three‐dimensional wave fronts of electromagnetic waves have been constructed for different values of the permittivities and their features have been investigated.

Spectrum of normal oscillations of a dislocation ensemble in a viscoplastic medium

Dynamic equations of a dislocation ensemble are derived within the framework of the gauge model of an elastic body with defects. The governing relation between the defect field and the material continuum is obtained with allowance for the analogy between the equations obtained and Maxwell's electrodynamic equations. In the phenomenological theories of plasticity, this relation corresponds to the definition of a viscoplastic body. The dispersion relations and configurations of the normal oscillations of a dislocation ensemble in a viscoplastic medium are calculated.

Maxwell's Equations and the "Flux-Cutting" Principle

IN NATURE of February 5, Dr. C. V. Drysdale attempts a simplification of Maxwell's electrodynamic equations, on the grounds that these in their original form are difficult to comprehend and to apply, and not completely fundamental. As regards the latter criticism, it must be admitted that Maxwell rarely leaves much scope for improvement in his mathematical statements of physical laws, and so far as the complexity is concerned, many years' lecturing experience have convinced me that the average electrical engineering student of pass degree standard finds little difficulty in understanding and appreciating the concise and elegant expression of the laws of induction, which he has had ample opportunity of testing experimentally in his laboratory work, and usually finds the wider applications, to transmission, skin effect, and metallic reflexion, both easy and interesting.