On a class of vector fields

G. G. Islamov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 19 (2015) no. 4, pp. 680-696

Résumé

It is shown that a simple postulate “The displacement field of the vacuum is a normalized electric field”, is equivalent to three parametric representation of the displacement field of the vacuum: $$ u(x;t) = P(x) \cos k(x)t + Q(x) \sin k(x)t. $$ Here $t$ — time; $k(x)$ — frequency vibrations at the point of three-dimensional Euclidean space; $P(x), Q(x)$ — a pair of stationary orthonormal vector fields; $(k,P, Q)$ — parameter list of the displacement field. In this case, the normalization factor has dimension $T^{-2}$. The speed of the displacement field $$ v(x;t) = \frac{\partial u(x;t)}{\partial t} = k(x)(Q(x) \cos k(x)t - P(x) \sin k(x)t). $$ The electric field corresponding to this distribution of the displacement field of vacuum, is given by the formula $$ E(x;t) = -\frac{\partial v(x;t)}{\partial t} = k^2(x)u(x;t). $$ Moreover, the magnetic induction $$ B(x;t) = \mathop{\mathrm{rot }} v(x; t). $$ These constructions are used in the determination of local and global solutions of Maxwell's equations describing the dynamics of electromagnetic fields.

Détail

Keywords: local and global solutions of Maxwell's equations, spectral problem for rotor operator, the small flow of the displacement field.