Page:LorentzGravitation1915.djvu/4

 Motion of a system of incoherent material points.

§ 4. Let us now, following, consider a very large number of material points wholly free from each other, which are moving in a gravitation field in such a way that at a definite moment the velocity components of these points are continuous functions of the coordinates. By taking the number very large we may pass to the limiting case of a continuously distributed matter without internal forces.

Evidently the laws of motion for a system of this kind follow immediately from those for a single material point. If $$\varrho$$ is the density and $$dy\ dy\ dz$$ an element of volume we may write instead of (8)

If now we wish to extend equation (3) to the whole system we must multiply (9) by $$dt$$ and integrate with respect to $$x, y, z$$ and $$t$$.

In the last term of (3) we shall do so likewise after having replaced the components $$K_{a}$$ by $$K_{a}dx\ dy\ dz$$, so that in what follows $$K$$ will represent the external force per unit of volume.

If further we replace $$dx\ dy\ dz\ dt$$ by $$dS$$, an element of the four-dimensional extension $$x_{1},\dots x_{4}$$, and put

we find the following form of the fundamental theorem.

Let a variation of the motion of the system of material points be defined by the infinitely small quantities $$\delta x_{a}$$, which are arbitrary continuous functions of the coordinates within an arbitrarily chosen finite space $$S$$, at the limits of which they vanish. Then we have, if the integrals are taken over the space $$S$$, and the quantities $$g_{ab}$$ are left unchanged,

For the first term we may write

$\int\delta\mathrm{L}\cdot dS,$

if $$\delta\mathrm{L}$$ denotes the change of $$\mathrm{L}$$ at a fixed point of the space $$S$$.

The quantity $$\mathrm{L}dS$$ and therefore also the integral $$\int\mathrm{L}dS$$ is invariant when we pass to another system of coordinates.