Page:LorentzGravitation1915.djvu/5

 § 5. The equations of motion may be derived from (12) in the following way. When the variations $$\delta x_{a}$$ have been chosen, the varied motion of the matter is perfectly defined, so that the changes of the density and of the velocity components are also known. For the variations at a fixed point of the space $$S$$ we find

where

(Therefore: $$\chi_{ba}=-\chi_{ab},\ \chi_{aa}=0$$).

If for shortness we put

so that $$\mathrm{L}=-P$$, and

we have

$\begin{array}{c} \delta\mathrm{L}=-\Sigma(a)\dfrac{u_{a}}{P}\delta w_{a}=-\Sigma(ab)\dfrac{u_{a}}{P}\dfrac{\partial\chi_{ab}}{\partial x_{b}}=\\ \\ =-\Sigma(ab)\dfrac{\partial}{\partial x_{b}}\left(\dfrac{u_{a}}{P}\chi_{ab}\right)+\Sigma(ab)\chi_{ab}\dfrac{\partial}{\partial x_{b}}\left(\dfrac{u_{a}}{P}\right), \end{array}$|undefined

so that, with regard to (14),

{{MathForm2|(17)|$$\left.\begin{array}{c} \delta\mathrm{L}+\Sigma(a)K_{a}\delta x_{a}=-\Sigma(ab)\dfrac{\partial}{\partial x_{b}}\left(\dfrac{u_{a}}{P}\chi_{ab}\right)+\\ \\ +\Sigma(ab)\left(w_{b}\delta x_{a}-w_{a}\delta x_{b}\right)\dfrac{\partial}{\partial x_{b}}\left(\dfrac{u_{a}}{P}\right)+\Sigma(a)K_{a}\delta x_{a} \end{array}\right\} $$}}

If after multiplication by $$dS$$ this expression is integrated over the space $$S$$ the first term on the right hand side vanishes, $$\chi_{ab}$$ being 0 at the limits. In the last two terms only the variations $$\delta x_{a}$$ occur, but not their differential coefficients, so that according to our fundamental theorem, when these terms are taken together, the coefficient of each $$\delta x_{a}$$ must vanish. This gives the equations of motion

which evidently agree with (4), or what comes to the same, with

In virtue of (18) the general equation (17), which holds for