Page:Zur Elektrodynamik bewegter Systeme II.djvu/3

where ${\tfrac {d}{dt}}\left({\tfrac {\partial }{\partial t}}\right)$ denotes the differentiation with respect to a fixed material point (space point). Furthermore P is rotation, $\Gamma$ is divergence, $\nabla$ is gradient,

$(A\cdot \nabla )=A_{x}\cdot {\frac {\partial }{\partial x}}+A_{y}\cdot {\frac {\partial }{\partial y}}+A_{z}\cdot {\frac {\partial }{\partial z}}$.

§ 2. Transformation to a moving coordinate system and local time.

We decompose the velocities $u$ into a common translational velocity $p$ (which is constant with respect to time) of the whole system, and the "relative" velocity $v$:

$u=p+v,\ p=const.,$

(3)

and we denote a differentiation with respect to time, in relation to a relatively stationary point, by ${\tfrac {\delta }{\delta t}}$:

${\frac {\delta }{\delta t}}={\frac {\partial }{\partial t}}+(p\cdot \nabla )={\frac {d}{dt}}-(v\cdot \nabla )$

(4)

Then it is given

${\frac {\overline {dA}}{dt}}={\frac {\delta A}{\delta t}}+\Gamma (v)A-(A\cdot \nabla )v+(v\cdot \nabla )A$

$={\frac {\delta A}{\delta t}}+\Gamma (A)v-{\mathsf {P}}[vA)$.

(5)

Simultaneously, instead of the "general time" $t$ we introduce the "local time" $t'$. It is defined at a point whose radius vector is $r$, by:

$r'=r-(p\cdot r)\,$

(6)

Differentiations with respect to relative coordinates, in which local time is assumed as the fourth independent variable, shall be denoted by an upper index prime. Then it is

${\frac {\delta }{\delta t}}={\frac {\delta }{\delta t'}}$ ${\mathsf {P}}={\mathsf {P}}'-\left[p\cdot {\frac {\delta }{\delta t'}}\right]$ $\Gamma =\Gamma '-\left[p\cdot {\frac {\delta }{\delta t'}}\right]$

(7)

Eventually, we decompose ${\mathfrak {E}}$ and ${\mathfrak {M}}$:

${\begin{matrix}{\mathfrak {E}}={\mathfrak {E}}_{0}-[p{\mathsf {M}}]\\\\{\mathfrak {M}}={\mathfrak {M}}_{0}+[p{\mathsf {E}}]\end{matrix}}$

(8)