Page:Grundgleichungen (Minkowski).djvu/54

For a virtual displacement in a space-time sickle (with the previously applied designation) the value of the integral

(17)	$W+\delta W=\int \int \int \int \left({\underset {h,k}{\sum }}s_{hk}{\frac {\partial (x_{k}+\delta x_{k})}{\partial x_{h}}}\right)dx\ dy\ dz\ dt$

extended over the whole range of the sickle, may be called the tensional work of the virtual displacement.

The sum which comes forth here, written in real magnitudes, is

$X_{x}+Y_{y}+Z_{z}+T_{t}$

$+X_{x}{\frac {\partial \delta x}{\partial x}}+X_{y}{\frac {\partial \delta x}{\partial y}}+\dots +Z_{z}{\frac {\partial \delta z}{\partial z}}$ $-X_{t}{\frac {\partial \delta x}{\partial t}}-\dots +T_{x}{\frac {\partial \delta t}{\partial x}}+\dots +T_{t}{\frac {\partial \delta t}{\partial t}}$.

we can now postulate the following minimum principle in mechanics.

If any space-time sickle be bounded, then for each virtual displacement in the sickle, the sum of the mass-works, and tension works shall always he an extremum for that process of the space-time line in the sickle which actually occurs.

The meaning is, that for each virtual displacement,

(18)	$\left({\frac {d(\delta {\mathsf {N}}+\delta W)}{d\vartheta }}\right)_{\vartheta =0}=0$

By applying the methods of the Calculus of Variations, the following four differential equations at once follow from this minimal principle by means of the transformation (14), and the condition (16).

(19)	$\nu {\frac {dw_{h}}{d\tau }}=K_{h}+\varkappa w_{h}\qquad (h=1,2,3,4)$,

whence

(20)	$K_{h}={\frac {\partial S_{1h}}{\partial x_{1}}}+{\frac {\partial S_{2h}}{\partial x_{2}}}+{\frac {\partial S_{3h}}{\partial x_{3}}}+{\frac {\partial S_{4h}}{\partial x_{4}}}$

are components of the space-time vector 1st kind K = lor S, and $\varkappa$ is a factor, which is to be determined from the relation $w{\overline {w}}=-1$. By multiplying (19) by $w_{h}$, and