Page:LorentzGravitation1915.djvu/2

 occurring in the real motion in the way defined by the infinitely small quantities $$\delta x,\delta y,\delta z$$. If, in the varied motion, the position $$x+\delta x,\ y+\delta y,\ z+\delta z$$ is reached at the same time $$t$$ as the position $$x,y,z$$ in the real motion, we shall have the equation

$$L$$ being the Lagrangian function and the integrals being taken over an arbitrary interval of time, at the beginning and the end of which the variations of the coordinates are zero. $$K$$ is supposed to be a force acting on the material point beside the forces that are included in the Lagrangian function.

§ 2. We may also suppose the time $$t$$ to be varied, so that in the varied motion the position $$x+\delta x,\ y+\delta y,\ z+\delta z$$ is reached at the time $$t+\delta t$$. In the first term of (1) this does not make any difference if we suppose that for the extreme positions also $$\delta t=0$$. As to the second term we remark that the coordinates in the varied motion at the time $$t$$ may now be taken to be $$x+\delta x-v_{1}\delta t$$, $$y+\delta y-v_{2}\delta t$$, $$z+\delta z-v_{3}\delta t$$, if $$v_{1},v_{2},v_{3}$$ are the velocities in the real motion. In the second term we must therefore replace $$\delta x,\delta y,\delta z$$ by $$\delta x-v_{1}\delta t$$, $$\delta y-v_{2}\delta t$$, $$\delta z-v_{3}\delta t$$. In the equation thus found we shall write $$x_{1},x_{2},x_{3},x_{4}$$ for $$x,y,z,t$$. For the sake of uniformity we shall add to the three velocity components a fourth, which, however, necessarily must have the value 1 as we take for it $$\dfrac{dx_{4}}{dx_{4}}$$. We shall also add to the three components of the force $$K$$ a fourth component, which we define as

and which therefore represents the work of the force per unit of time with the negative sign. Then we have instead of (1)

and for (2) we may write