Page:PoincareDynamiqueJuillet.djvu/36



§ 8. — Arbitrary motion
The above results apply only to quasi-stationary motion, but it is easy to extend them to the general case; it suffices to apply the principles of § 3, that is to say, the principle of least action.

For the expression of the action

$$J=\int dt\ d\tau\left(\frac{\sum f^{2}}{2}-\frac{\sum\alpha^{2}}{2}\right),$$

it is convenient to add a term representing the additional potential F of § 6; this term will obviously have the form:

$$J_{1}=\int\sum(F)dt$$

where Σ(F) represents the sum of the additional potential due to the different electrons, each of which is proportional to the volume of the corresponding electron.

I write (F) in brackets to avoid confusion with the vector F, G, H.

The total action is then J + J1. We saw in § 3 that J is not altered by the transformation, we must show now that it is the same for J1.

We have for one electron,

$$(F)=\omega_{0}\tau\,$$

ω0 being a special coefficient of the electron and τ its volume; so I can write:

$$\sum(F)=\int\omega_{0}d\tau,$$

the integral has to be extended to the entire space, but so that the coefficient ω0 is zero outside the electrons, and that within each electron it is equal to the special coefficient of that electron. Then we have:

$$J_{1}=\int\omega_{0}d\tau\ dt$$

and after the transformation:

$$J_{1}^{\prime}=\int\omega_{0}^{\prime}d\tau^{\prime}\ dt^{\prime}.$$

Now we have ω0 = ω'0; for if a point belong to an electron, the corresponding point after the transformation still belongs to the same electron. On the other hand, we found in § 3;

$$d\tau'dt'=l^{4}d\tau\ dt$$

and since we now assume l = 1

$$d\tau'dt'=d\tau\ dt$$