Page:LorentzGravitation1915.djvu/5

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§ 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

$\delta w_{a}=\Sigma (b){\frac {\partial \chi _{ab}}{\partial x_{b}}}$

(13)

where

$\chi _{ab}=w_{b}\delta x_{a}-w_{a}\delta x_{b}$

(14)

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

If for shortness we put

$P={\sqrt {\Sigma (ab)g_{ab}w_{a}w_{b}}}$

(15)

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

$\Sigma (b)g_{ab}w_{b}=u_{a}$

(16)

we have

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

so that, with regard to (14),

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

(17)

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^[1]

$K_{a}=\Sigma (b)w_{b}\left[{\frac {\partial }{\partial x_{a}}}\left({\frac {u_{b}}{P}}\right)-{\frac {\partial }{\partial x_{b}}}\left({\frac {u_{a}}{P}}\right)\right]$

(18)

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

$\Sigma (a)w_{a}K_{a}=0$

(19)

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

1. In the term

   $-\Sigma (ab)w_{a}\delta x_{b}{\frac {\partial }{\partial x_{b}}}\left({\frac {u_{a}}{P}}\right)$

   the indices $a$ and $b$ must first be interchanged.