Hamilton form of Maxwell equations and its generalized solutions

TL;DR

This paper reformulates Maxwell's equations using Hamiltonian form, introduces generalized solutions for shock electromagnetic waves, and proves their uniqueness and conservation laws, advancing understanding of electromagnetic shock phenomena.

Contribution

It presents a Hamiltonian form of Maxwell equations and develops generalized solutions for shock waves, including conservation laws and uniqueness theorems.

Findings

Shock electromagnetic waves have tangential tension gaps.

Charges are absent on wave fronts.

Generalized solutions satisfy conservation laws and are unique.

Abstract

The complex form of Maxwell equations has been constructed as one equation for 3-dimensional complex A-vector. The real and imaginary parts of this vector are described with use of electric and magnetic tensions accordingly. With using a theory of generalized functions for new equations, the strong shock electro-magnetic waves with the gap of tensions on fronts are considered. The conditions on wave fronts have been received. It's shown that gap of the tensions is tangent to the front of a wave, i.e. shock electromagnetic waves are transverse and charges on the front of wave are absent. Generalized laws of conservation of energy and charges have been received including ones on the fronts of shock waves. The generalized solutions of this equations and solution of Caushy problem have been constructed and the theorems of their uniqueness have been proved including the shock waves.