# 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 ${\displaystyle {\mathfrak {E'H'DB}}}$ 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 ${\displaystyle {\mathfrak {E'H'DB}}}$ shall be connected by equations, which namely contain the velocity vector ${\displaystyle {\mathfrak {w}}}$ itself, though not any derivatives of it with respect to time or coordinates.

Main equation (IV) gives:

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

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

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

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

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

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

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

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

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

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

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

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

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