Maxwell theory

Historical prelude Kepler s laws Historical prelude Maxwell Theory Axiomatic teaching of Quantum Mechanics Problem lack of reference points Problem imprecise boundaries Problem inaccurate formulation Solution reference points from a journey Solution precise boundaries Solution accurate formulation Intuitive teaching of Quantum Mechanics Conclusion... 

The Maxwell theory of X-ray scattering by stable systems, both solids and liquids, is described in many textbooks. A simple and compact presentation is given in Chapter 15 of Electrodynamics of Continuous Media [20]. The incident electric and magnetic X-ray helds are plane waves Ex(r, f) = Exo exp[i(q r — fixO] H(r, t) = H o exp[/(q r — fixO] with a spatially and temporally constant amplitude. The electric field Ex(r, t) induces a forced oscillation of the electrons in the body. They then act as elementary antennas emitting the scattered X-ray radiation. For many purposes, the electrons may be considered to be free. One then finds that the intensity /x(q) of the X-ray radiation scattered along the wavevector q is... 

This result was experimentally discovered in the nineteenth century, but it could not be explained by Maxwells theory of electromagnetism. (James Clerk Maxwell was a Scottish physicist whose formulation of the laws of electricity and magnetism were... 

The Maxwell theory of X-ray scattering by stable systems, both solids and liquids, is described in many textbooks. A simple and compact presentation is given in Chapter 15 of the Landau-Lifshitz volume. Electrodynamics of Continuous Media [20]. The incident electric and magnetic X-ray fields are plane waves... 

Similarly, the 4-current J depends directly on the curvature tensor / [1], and there can exist no 4-current in the Heaviside-Maxwell theory, so the... 

The Maxwell theory is well known to be a material fluid flow theory [6],4 since the equations are hydrodynamic equations. In principle, anything that can be done with fluid theory can be done with electrodynamics, since the fundamental equations are the same mathematics and must describe consistent analogous functional behavior and phenomena [5]. This means that EM systems with electromagnetic energy winds from their active external atmosphere ... 

V. Local Equivalence and Global Difference with the Standard Maxwell Theory... 

The electric and magnetic fields are dual to one another and have the same properties in Maxwell theory in empty space. Given the divergenceless vector field B, we have defined the magnetic helicity as... 

We end this section with a comment referring to the Cauchy data for the scalars. In standard Maxwell theory, the Cauchy data are the eight functions A(i,6o<4M, and there is gauge invariance. In this topological model, they are the four complex functions (r, 0), 0 (r. 0), that is, eight real functions, constrained by the two conditions x V< >k) (V0 x V0 ) =0, k = 1,2, to ensure that the level curves of k will be orthogonal to those of 0. It is not necessary to prescribe the time derivatives 9o4>, 000 since they are determined by the duality conditions (138), as explained above. 

V. LOCAL EQUIVALENCE AND GLOBAL DIFFERENCE WITH THE STANDARD MAXWELL THEORY... 

U(l) Symmetry Form (Traditional Maxwell Theory) SU(2) Symmetry Form... 

A soliton is a solitary wave that preserves its shape and speed in a collision with another solitary wave [12,13]. Soliton solutions to differential equations require complete integrability and integrable systems conserve geometric features related to symmetry. Unlike the equations of motion for conventional Maxwell theory, which are solutions of U(l) symmetry systems, solitons are solutions of SU(2) symmetry systems. These notions of group symmetry are more fundamental than differential equation descriptions. Therefore, although a complete exposition is beyond the scope of the present review, we develop some basic concepts in order to place differential equation descriptions within the context of group theory. 

Maxwell theory, soliton flows are Hamiltonian flows. Such Hamiltonian functions define symplectic structures6 for which there is an absence of local invariants but an infinite-dimensional group of diffeomorphisms which preserve global properties. In the case of solitons, the global properties are those permitting the matching of the nonlinear and dispersive characteristics of the medium through which the wave moves. 

In order to clarify the difference between conventional Maxwell theory which is of U(l) symmetry, and Maxwell theory extended to SU(2) symmetry, we can describe both in terms of mappings of a field /(x). In the case of U(l) Maxwell theory, a mapping v / > v / is... 

A spectacular advance in the understanding of the distribution of charges around an ion in solution was achieved in 1923 by Debye and Hiickel. It is as significant in the understanding of ionic solutions as the Maxwell theory of the distribution of velocities is in the understanding of gases. 

Maxwell, Theory of Heat, 1897, 113, 121 (emphasising the different p v diagrams for steam and carbon dioxide), 134. 

In these relations, A is the conductivity of the suspension, and the subscripts m, o, and w refer to the microemulsion, oil and emulsifier combined, and water. The Hanai expression can be considered to be an extension of the Maxwell theory that more consistently accounts for the presence of neighboring particles (8) for the 0/W microemulsions considered here, the predictions of the Maxwell and Hanai formulas (as well as various other mixture theories) are not greatly different. Moreover, while these theories were developed for suspensions of spherical particles, the predictions of the mixture theories are not expected to vary greatly with the geometry of the dispersed particles, provided that the droplets are prolate or oblate ellipsoids whose axial ratios are not greatly... 

If m - n, as included in the standard Maxwell theory, the extra four conservation equations above reduce to 0=0, which is an ambiguity. However, with the restriction from Faraday s interpretation that requires that m / n, the ambiguity is removed and the extra conservation equations remain. 

The covariance groups underlying the tensor forms of the respective Einstein and the Maxwell held equations are reducible. This is because they entail reflection symmetry, not required by relativity theory, as well as the required continuous symmetry of the Einstein group E. When the Einstein held equations are factorized, they yield the irreducible form, which are then in terms of the quaternion and spinor variables, rather than the tensor variables. Such a generalization must then extend the physical predictions of the usual tensor forms of general relativity of gravitation and the standard vector representation of the Maxwell theory (both in terms of second-rank tensor helds, one symmetric and the other antisymmetric) because the new factorized variables have more degrees of freedom than did the earlier version variables. 

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