# Page:AbrahamMinkowski1.djvu/9

§ 5. Determination of momentum density and energy density.

The various theories of electrodynamics of moving bodies, are differing by the relations assumed between the four vectors $\mathfrak{E'H'DB}$ arising in the main equations. However, before we pass to the discussion of special theories, we want to pursue the general developments; there, only a quite general presupposition shall be made about the form of these relations: The vectors $\mathfrak{E'H'DB}$ shall be connected by equations, which namely contain the velocity vector $\mathfrak{w}$ itself, though not any derivatives of it with respect to time or coordinates.

Main equation (IV) gives:

$\mathrm{div}\mathfrak{S}'=c\{\mathfrak{H}'\mathrm{curl}\mathfrak{E}'-\mathfrak{E}'\mathrm{curl}\mathfrak{H}'\}$

this becomes with respect to the two first main-equations:

$\mathfrak{JE}'+\mathrm{div}\mathfrak{S}'=-\mathfrak{E}'\frac{\partial'\mathfrak{D}}{\partial t}-\mathfrak{H}'\frac{\partial'\mathfrak{B}}{\partial t}$

From main equation (III) and relation (14) it follows:

 (14a) $\frac{\delta\psi}{\delta t}-\mathfrak{w}\frac{\delta\mathfrak{g}}{\delta t}-P'=\mathfrak{E}'\frac{\partial'\mathfrak{D}}{\partial t}+\mathfrak{H}'\frac{\partial'\mathfrak{B}}{\partial t}$

a condition, which one can also be written in accordance with (4):

 (14b) $\frac{\delta\psi}{\delta t}-\mathfrak{w}\frac{\delta\mathfrak{g}}{\delta t}-P'=\mathfrak{E}'\frac{\delta\mathfrak{D}}{\delta t}+\mathfrak{H}'\frac{\delta\mathfrak{B}}{\delta t}-\mathfrak{E}'(\mathfrak{D}\nabla)\mathfrak{w}-\mathfrak{H}'(\mathfrak{B}\nabla)\mathfrak{w}$

and which finally, by using main equation (V), passes into:

 (15) $\frac{\delta\psi}{\delta t}-\mathfrak{w}\frac{\delta\mathfrak{g}}{\delta t}=\mathfrak{E}'\frac{\delta\mathfrak{D}}{\delta t}+\mathfrak{H}'\frac{\delta\mathfrak{B}}{\delta t}-\frac{1}{2}[\mathfrak{E'D+H'B}\}\mathrm{div}\mathfrak{w}$

This relation serves to determine the densities of energy and momentum in their dependence from the electromagnetic vectors.

They read with respect to (2a):

 (15a) $\dot{\psi}=\mathfrak{w\dot{g}}+(\psi-\mathfrak{wg})\mathrm{div}\mathfrak{w}=\mathfrak{E'\dot{D}+H'\dot{B}}+\frac{1}{2}[\mathfrak{E'D+H'B}\}\mathrm{div}\mathfrak{w}$

Since the way of time differentiation now employed, is satisfying the calculation rules, it follows when it is set for brevity's sake:

 (16) $\psi-\mathfrak{wg}=\varphi$
 (17) $\dot{\varphi}+\mathfrak{g\dot{w}}-\mathfrak{E'\dot{D}-H'\dot{B}}+\left\{ \varphi-\frac{1}{2}\mathfrak{E'D}-\frac{1}{2}\mathfrak{H'B}\right\} \mathrm{div}\mathfrak{w}=0$

As mentioned in the beginning of the paragraph, the relations which connect $\mathfrak{D,B}$ with $\mathfrak{E'H'}$ shall contain the velocity vector $\mathfrak{w}$, though not its derivative with respect to time and space. The same is to be demanded from the expressions,