Page:LorentzGravitation1915.djvu/6

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

§ 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 $$c$$

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)

$\Sigma(a)\frac{\partial}{\partial x_{c}}\left(\frac{u_{a}w_{a}}{P}\right)\delta x_{c}-\Sigma(a)\frac{\partial}{\partial x_{a}}\left(\frac{u_{c}w_{a}}{P}\right)\delta x_{c}.$ |undefined

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

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

so that (20) becomes

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

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

Substituting this in (22) and putting