Page:LorentzGravitation1915.djvu/6

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arbitrary variations that need not vanish at the limits of $S$, becomes

$\delta \mathrm {L} +\Sigma (a)K_{a}\delta x_{a}=-\Sigma (ab){\frac {\partial }{\partial x_{b}}}\left({\frac {u_{a}}{P}}\chi _{ab}\right)$

(20)

§ 6. We can derive from this the equations for the momenta and the energy.

Let us suppose that only one of the four variations $\delta x_{a}$ differs from 0 and let this one, say $\delta x_{c}$, have a constant value. Then (14) shows that for each value of $a$ that is not equal to $c$

$\chi _{ac}=-w_{a}\delta x_{c},\ \chi _{ca}=w_{a}\delta x_{c}$

(21)

while all $\chi$'s without an index $c$ vanish.

Putting first $b=c$ and then $a=c$, and replacing at the same time in the latter case $b$ by $a$, we find for the right hand side of (20)

$\Sigma (a){\frac {\partial }{\partial x_{c}}}\left({\frac {u_{a}w_{a}}{P}}\right)\delta x_{c}-\Sigma (a){\frac {\partial }{\partial x_{a}}}\left({\frac {u_{c}w_{a}}{P}}\right)\delta x_{c}.$^[1]

But, according to (15) and (16),

$\Sigma (a){\frac {u_{a}w_{a}}{P}}=P=-\mathrm {L}$

so that (20) becomes

$\delta \mathrm {L} =K_{c}\delta x_{c}=-{\frac {\partial \mathrm {L} }{\partial x_{c}}}\delta x_{c}-\Sigma (a){\frac {\partial }{\partial x_{a}}}\left({\frac {u_{c}w_{a}}{P}}\right)\delta x_{c}.$

(22)

It remains to find the value of $\delta \mathrm {L}$.

The material system together with its state of motion has been shifted in the direction of the coordinate $x_{c}$ over a distance $\delta x_{c}$. If the gravitation field had participated in this shift, $\partial \mathrm {L}$ would have been equal to $-{\dfrac {\partial \mathrm {L} }{\partial x_{c}}}\delta x_{c}$. As, however, the gravitation field has been left unchanged, ${\dfrac {\partial \mathrm {L} }{\partial x_{c}}}$ in this last expression must be diminished by $\left({\dfrac {\partial \mathrm {L} }{\partial x_{c}}}\right)_{w}$, the index $w$ meaning that we must keep constant the quantities $w_{a}$ and consider only the variability of the coefficients $g_{ab}$. Hence

$\delta \mathrm {L} =\left\{-{\frac {\partial \mathrm {L} }{\partial x_{c}}}+\left({\frac {\partial \mathrm {L} }{\partial x_{c}}}\right)_{w}\right\}\delta x_{c}$

Substituting this in (22) and putting
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1. The circumstance that (21) does not hold for $a=c$ might lead us to exclude this value from the two sums. We need not, however, introduce this restriction, as the two terms that are now written down too much, annul each other.