Helmholtz decomposition theorem and Blumenthal's extension by regularization

Helmholtz decomposition theorem for vector fields is usually presented with too strong restrictions on the fields and only for time independent fields. Blumenthal showed in 1905 that decomposition is possible for any asymptotically weakly decreasing vector field. He used a regularization method in his proof which can be extended to prove the theorem even for vector fields asymptotically increasing sublinearly. Blumenthal's result is then applied to the time-dependent fields of the dipole radiation and an artificial sublinearly increasing field.Comment: 11 pages. arXiv admin note: text overlap with arXiv:1506.0023

On incorrectness of application of the Helmholtz decomposition to microscopic electrodynamics

The integral expressions served to decompose vector field into irrotational and divergence-free components represent modern version of the Helmholtz decomposition theorem. These expressions are also widely used to decompose the electromagnetic fields. However, an appropriate analysis of application of these expressions to electrodynamics shows that the improper integral arising in the procedure for calculating these components makes such a decomposition impossible.Comment: AMS-LaTeX, 6 page

Theoretical aspects of a multiscale analysis of the eigenoscillations of the earth

The elastic behaviour of the Earth, including its eigenoscillations, is usually described by the Cauchy-Navier equation. Using a standard approach in seismology we apply the Helmholtz decomposition theorem to transform the Fourier transformed Cauchy-Navier equation into two non-coupled Helmholtz equations and then derive sequences of fundamental solutions for this pair of equations using the Mie representation. Those solutions are denoted by the Hansen vectors Ln;j , Mn;j , and Nn;j in geophysics. Next we apply the inverse Fourier transform to obtain a function system depending on time and space. Using this basis for the space of eigenoscillations we construct scaling functions and wavelets to obtain a multiresolution for the solution space of the Cauchy-Navier equation.The elastic behaviour of the Earth, including its eigenoscillations, is usually described by the Cauchy­Navier equation. Using a standard approach in seis mology we apply the Helmholtz decomposition theorem to transform the Fourier transformed Cauchy­Navier equation into two non­coupled Helmholtz equations and then derive sequences of fundamental solutions for this pair of equations us ing the Mie representation. Those solutions are denoted by the Hansen vectors Ln,j , Mn,j , and Nn,j in geophysics. Next we apply the inverse Fourier trans form to obtain a function system depending on time and space. Using this basis for the space of eigenoscillations we construct scaling functions and wavelets to obtain a multiresolution for the solution space of the Cauchy­Navier equation. 2000 Mathematics Sub ject Classification: 35J05, 42C40

Direct derivation of Lienard Wiechert potentials, Maxwell's equations and Lorentz force from Coulomb's law

In 19th century Maxwell derived Maxwell equations from the knowledge of three experimental physical laws: the Coulomb's law, the Ampere's force law and Faraday's law of induction. However, theoretical basis for Ampere's force law and Faraday's law remains unknown to this day. Furthermore, the Lorentz force is considered as experimental phenomena, the theoretical foundation of this force is still unknown. To answer these fundamental theoretical questions, we derive Lienard Wiechert potentials, Maxwell's equations and Lorentz force from two simple postulates: (a) when all charges are at rest the Coulomb's force acts between the charges, and (b) that disturbances caused by charge in motion propagate away from the source with finite velocity. The special relativity was not used in our derivations nor the Lorentz transformation. In effect, it was shown all the electrodynamic laws, including the Lorentz force, can be derived from Coulomb's law and time retardation. This was accomplished by analysis of hypothetical experiment where test charge is at rest and where previously moving source charge stops at some time in the past. Then the generalized Helmholtz decomposition theorem, also derived in this paper, was applied to reformulate Coulomb's force acting at present time as the function of positions of source charge at previous time when the source charge was moving. From this reformulation of Coulomb's law the Lienard Wiechert potentials and Maxwell's equations were derived. In the second part of this paper, the energy conservation principle valid for moving charges is derived from the knowledge of electrostatic energy conservation principle valid for stationary charges. This again was accomplished by using generalized Helmholtz decomposition theorem. From this dynamic energy conservation principle the Lorentz force is derived