Page:Prinzipien der Dynamik des Elektrons.djvu/2

$\int \int \int dv\ \varrho [{\mathfrak {r\ F}}].$

The "dynamic fundamental equations" now say:

The resultants of the inner and outer force and angular force are vanishing:

3)	${\begin{array}{rl}\int \int \int dv\ \varrho {\mathfrak {F}}+{\mathfrak {R}}&=0,\\\int \int \int dv\ \varrho [{\mathfrak {r\ F}}]+\Theta &=0.\end{array}}$

The kinematic fundamental equation (7), the field equations (2) and the dynamic fundamental equations (3), these are the foundations upon which the dynamics of the electron is built on. It may be emphasized, that the word "force" is at this place only an abbreviating denotation for certain vectors defined by the field strengths and by the velocity of translation and rotation of the electron; from ordinary mechanics, we only incorporate the purely geometric-kinematic concept into the foundations of the dynamics of the electron. But we choose the denotation of the derived quantities so that the analogy to ordinary mechanics clearly emerges.

Before passing to the treatment of special types of motion, two general theorems that follow from the field equations, may sent before. The first theorem formulates the energy principle, it reads:

4)	$\int \int dv\ \varrho ({\mathfrak {v\ F}})+\int \int do\ {\mathfrak {S}}_{v}=-{\frac {dW}{dt}}.$

There, $o$ denotes the boundary of the field, which can be determined by other bodies or by an (only imagined) surface; $({\mathfrak {v\ F}})$ is the inner (scalar) product of vectors ${\mathfrak {v}}$ and ${\mathfrak {F}}$, and the first member of the left-hand side is the "power of the inner forces". The second member contains the normal component of "Poyntings radiation vector".

${\mathfrak {S}}={\frac {c}{4\pi }}\cdot [{\mathfrak {E\ H}}]$

and gives the radiation sent through the boundary by the electron. Power of the inner forces and radiation appear at the cost of the scalar quantity $W$, which we will call "electromagnetic energy", and which occupies the field with the density

${\frac {1}{8\pi }}({\mathfrak {E}}^{2}+{\mathfrak {H}}^{2})$

Eq. (4) thus corresponds to the theorem of the living force. In a corresponding way – as it was first noticed by H. Poincaré – the theorems of momentum, or the "momentum theorems" can be obtained from Lorentz's theory. We have:

5)

${\begin{array}{cc}\int \int \int dv\ \varrho {\mathfrak {F}}+\int \int do\ {\mathfrak {P}}&=-\int \int \int dv{\frac {1}{c^{2}}}{\frac {\partial {\mathfrak {S}}}{\partial t}},\\\\\int \int \int dv\ \varrho [{\mathfrak {r\ F}}]+\int \int do\ [{\mathfrak {r\ P}}]&=-\int \int dv\left[{\mathfrak {r}},\ {\frac {1}{c^{2}}}{\frac {\partial {\mathfrak {S}}}{\partial t}}\right].\end{array}}$

Here, ${\mathfrak {P}}$ denotes the force, exerted by the so called "Maxwell tensions" of the field excited by the electron upon the unit surface of the boundary surface $o$, $[{\mathfrak {r\ P}}]$ is the static moment of this force, related to the center of the electron. According to equations (5), the forces exerted by the field upon the electron on one side, and on the boundary on the other side, are in general not canceling each other throughout; thus they contradict the third axiom of Newton. Eq. (4) would also contradict the energy principle in quite the same way, if we hadn't maintained it by the assumption of a new "electromagnetic energy". Also the third axiom can be saved by us, when we introduce a new "electromagnetic momentum", distributed over the field with the density ${\tfrac {1}{c^{2}}}\cdot {\mathfrak {S}}$. Eq. (5) is then to be interpreted as follows.

We imagine a framework rigidly connected with the electron. In all points of the field, at which the density of the electromagnetic momentum increases with time, a corresponding force acts at the framework. If all these individual forces are composed according to the rules of the statics of rigid bodies, one obtains the force and angular force, exerted by the field (limited by surface $o$) upon the framework. The consideration becomes simpler, when it is allowed to move surface $o$ into infinity, and to omit the surface integrals in Eq. (5) and (4); this is then the case – the proof would lead too far at this place –, when the influence of other bodies upon the electron, as far as it is not considered in vector ${\mathfrak {F}}_{h}$, becomes unnoticeable. If we furthermore assume this condition as satisfied, then we can fully replace the "inner forces" by means of Eq. (5) by the dynamic effects of the electromagnetic momentum.

We call the integral extended over the infinite space:

${\mathfrak {G}}={\frac {1}{c^{2}}}\int \int \int dv\ {\mathfrak {S}}$

the "momentum" of the electron, furthermore