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 Lorentz 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 Lorentz 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 Lorentz 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

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