Page:LorentzGravitation1915.djvu/2

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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

$\delta \int Ldt+\int \left(K_{1}\delta x+K_{2}\delta y+K_{3}\delta z\right)dt=0$

(1)

$L$ 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

$K_{4}=-\left(v_{1}K_{1}+v_{2}K_{2}+v_{3}K_{3}\right)$

(2)

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

$\delta \int Ldt+\Sigma \displaystyle (a)K_{a}\delta x_{a}\cdot dt=0$

(3)

and for (2) we may write^[1]
——————

1. In these formulae we have put between parentheses behind the sign of summation the index with respect to which the summation must be effected, which means that the values 1, 2, 3, 4 have to be given to it successively. In the same way two or more indices behind the sign of summation will indicate that in the expression under this sign these values have to be given to each of the indices. $\Sigma (ab)$ f. i. means that each of the four values of $a$ has to be combined with each of the four values of $b$.