Page:LorentzGravitation1915.djvu/4

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754

Motion of a system of incoherent material points.


§ 4. Let us now, following Einstein, 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 is the density and an element of volume we may write instead of (8)

(9)

If now we wish to extend equation (3) to the whole system we must multiply (9) by and integrate with respect to and .

In the last term of (3) we shall do so likewise after having replaced the components by , so that in what follows will represent the external force per unit of volume.

If further we replace by , an element of the four-dimensional extension , and put

(10)
(11)

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 , which are arbitrary continuous functions of the coordinates within an arbitrarily chosen finite space , at the limits of which they vanish. Then we have, if the integrals are taken over the space , and the quantities are left unchanged,

(12)

For the first term we may write

if denotes the change of at a fixed point of the space .

The quantity and therefore also the integral is invariant when we pass to another system of coordinates.[1]


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  1. This follows from the invariancy of , combined with the relations