Page:PoincareDynamiqueJuillet.djvu/15

so that if we set

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

we get:

J' = J.

However, to justify this equality, the integration limits have to be the same; so far we have assumed that t varies from t₀ to t₁, and x, y, z from ∞ to + ∞. On this account the integration limits would be affected by the Lorentz transformation, but nothing prevents us from assuming t₀ =- ∞, t₁ = + ∞; with those conditions the limits are the same for J and J'.

We then compare the following two equations analogues to equation (10) of § 2:

(2)	$\left\{{\begin{aligned}\delta J&=-\int \sum X\delta U\ d\tau \ dt\\\delta J'&=-\int \sum X^{\prime }\delta U^{\prime }\ d\tau ^{\prime }\ dt^{\prime }.\end{aligned}}\right.$

For this, we must first compare δU with δU.

Consider an electron whose initial coordinates are x₀, y₀, z₀; its coordinates at the instant t are

$x=x_{0}+U,\quad y=y_{0}+V,\quad z=z_{0}+W.$

If one considers the electron after the corresponding Lorentz transformation, it will have as coordinates

$x^{\prime }=kl\left(x+\epsilon t\right),\quad y^{\prime }=ly,\quad z^{\prime }=lz$

where

$x^{\prime }=x_{0}+U^{\prime },\quad y^{\prime }=y_{0}+V^{\prime },\quad z^{\prime }=z_{0}+W^{\prime }$

but it will only attain these coordinates at the instant

$t^{\prime }=kl\left(t+\epsilon x\right).$

If we subject our variables to the variations δU, δV, δW, and when we give at the same time t an increasement δt, the coordinates x, y, z will experience a total increasement

$\delta x=\delta U+\xi \delta t,\quad \delta y=\delta V+\eta \delta t,\quad \delta z=\delta W+\zeta \delta t.$

We will also have:

$\delta x^{\prime }=\delta U^{\prime }+\xi ^{\prime }\delta t^{\prime },\quad \delta y^{\prime }=\delta V^{\prime }+\eta ^{\prime }\delta t^{\prime },\quad \delta z^{\prime }=\delta W^{\prime }+\zeta ^{\prime }\delta t^{\prime },$

and in virtue of the Lorentz transformation:

$\delta x^{\prime }=kl\left(\delta x+\epsilon \delta t\right),\quad \delta y^{\prime }=l\delta y,\quad \delta z^{\prime }=l\delta z,\quad \delta t^{\prime }=kl\left(\delta t+\epsilon \delta x\right).$

hence, assuming δt = 0, the relations:

$\left\{{\begin{aligned}\delta x^{\prime }&=\delta U^{\prime }+\xi ^{\prime }\delta t^{\prime }&=kl\delta U,\\\delta y^{\prime }&=\delta V^{\prime }+\eta ^{\prime }\delta t^{\prime }&=l\delta V,\\\delta t^{\prime }&=kl\epsilon \delta U.\end{aligned}}\right.$