Page:Zur Elektrodynamik bewegter Systeme II.djvu/9

terrestrial time, or the stationary celestial fixed stars and celestial time.

That our equations, interpreted in the one or the other form, correctly represent the influence of relative motions, was demonstrated partly by me l.c., and partly by others. A summary and a comparison with other theories, is planned by me to be given soon.

§ 5. Energy conservation and mechanical forces.

To obtain the energy equation, we decompose the magnitude ${\tfrac {\overline {dA}}{dt}}$ of equation (I) in two parts:

${\frac {\overline {dA}}{dt}}={\frac {d'A}{dt}}+A_{def}$.

(13)

Here, ${\tfrac {d'A}{dt}}$ shall denote the change of vector $A$ relative to moving matter, or with other words, the change that $A$ suffers by the change in a fixed space-point caused by translation and by rotation. This would be the total value of ${\tfrac {\overline {dA}}{dt}}$, when matter is not deformed. Therefore, $A$ is the contribution that stems from the deformation. In the notation:

${\frac {d'A}{dt}}={\frac {dA}{dt}}+{\frac {1}{2}}[A\cdot {\mathsf {P}}(u)]$

(14)

$(A_{def})_{x}=A_{x}\left({\frac {\partial u_{y}}{\partial y}}+{\frac {\partial u_{z}}{\partial z}}\right)-A_{y}{\frac {1}{2}}\left({\frac {\partial u_{y}}{\partial x}}+{\frac {\partial u_{x}}{\partial y}}\right)-A_{z}{\frac {1}{2}}\left({\frac {\partial u_{z}}{\partial x}}+{\frac {\partial u_{x}}{\partial z}}\right)$; etc.

(15)

or

$A_{def}=A\cdot \Gamma (u)+A_{\delta }\,$

(16)

where

$(A_{\delta })_{x}=-A_{x}{\frac {\partial u_{x}}{\partial x}}-A_{y}{\frac {1}{2}}\left({\frac {\partial u_{y}}{\partial x}}+{\frac {\partial u_{x}}{\partial y}}\right)-A_{z}{\frac {1}{2}}\left({\frac {\partial u_{z}}{\partial x}}+{\frac {\partial u_{x}}{\partial z}}\right)$; etc.

(17)

One can easily convince oneself, that ${\tfrac {\overline {dA}}{dt}}$, calculated by (13), (14), (15), gives the value in (I). From the defining equations (14) it follows:

$\left({\frac {d'A}{dt}}\cdot B\right)+\left(A\cdot {\frac {d'B}{dt}}\right)={\frac {d}{dt}}(A\cdot B)$

(18)

$\left[{\frac {d'A}{dt}}\cdot B\right]+\left[A\cdot {\frac {d'B}{dt}}\right]={\frac {d'}{dt}}[A\cdot B]$

(19)