Chemistry - Maxwell's nonlinear equations

Maxwell's nonlinear equations in physics are a nonlinear version of Maxwell's equations of electromagnetic theory (EM theory).

Write the electron's electric gauge as e, and use CGS units. The following implicitly uses Clifford algebra formalism.

Take the four-vector of the electromagnetic potential

A(g,x)

in the dimensionless form

A(g,x) = gA(z),

with the Lorentz gauge condition

$D\cdot A=0$ ,

where D is the four-gradient, (1/c ∂/∂t, -∇).

Then

e = ag,

where a and g are scales for lengths and EM potential,

x = za.

The simplest nonlinear action for the EM field is the following:

$8\pi cS=e^2\int [(DA)^2+k^2(Du)^2+A\cdot J+\gamma u\cdot J]\,d^4z$

where all the quantities after the integral sign are non-dimensional. And the jets fourvector of EM jets

J = FuF

is not variable by EM potential. Then the nonlinear Maxwell equations for non-dimensional potentials are the following:

$\nabla \cdot \mathbf{H}=0$

$\partial_0\mathbf{H}+ \nabla \times \mathbf{E}=0$

$\nabla \cdot \mathbf{E}=u_0\left(E^2+H^2\right)+2\mathbf{u}\cdot\left(\mathbf{H}\times\mathbf{E}\right)$

$\partial_0\mathbf{E}-\nabla \times \mathbf{H}=\mathbf{u}(E^2+H^2)+2u_0\mathbf{H}\times\mathbf{E}-2\mathbf{E}(\mathbf{u}\cdot \mathbf{E})-2\mathbf{H}(\mathbf{u}\cdot \mathbf{H})$

Here u(x) is the local four-velocity vector of the EM field. Itself it is the potential of the w-field.

There exists a system of many equations for the four-velocity vector. For the vacuum state

D 2 u = 0

with restriction on the unknown integration constant, which is the coherence condition, in the form

$\int\delta u\cdot\left[FAF+2\gamma FuF\right]\,d^4z=0 \int (A+\gamma u)\cdot \delta _A J=d^4z=0.$

For Glauber's state

k 2 D 2 u = F A F

with the coherence condition

$\int[(A+\gamma u)\cdot \delta_A J+2\gamma u\cdot \delta _uJ]\,d^4z.$

Energy and jet constraints determine the forces as boundary conditions. Coherence conditions determine the values of the unknown integration constants.

Nonlinear effects for the EM field are essential at the scale

A > 10⁸ V.

But because there elementary particles are needless, any field is a nonlinear field. As examples see nonlinear Coulomb field, nonlinear magnetic field.

Categories: Electromagnetism | Equations