Page:PoincareDynamiqueJuillet.djvu/17

 and in addition

$$x^{\prime\prime}=k^{\prime}l^{\prime}(x^{\prime}+\epsilon^{\prime}t^{\prime}),\quad y^{\prime\prime}=l^{\prime}y^{\prime},\quad z^{\prime\prime}=l^{\prime}z^{\prime},\quad t^{\prime\prime}=k^{\prime}l^{\prime}(t^{\prime}+\epsilon^{\prime}x^{\prime}),$$

with

$$k^{-2}=1-\epsilon^{2},\quad k^{\prime-2}=1-\epsilon^{\prime2}$$

it follows:

$$x^{\prime\prime}=k^{\prime\prime}l^{\prime\prime}(x+\epsilon^{\prime\prime}t),\quad y^{\prime\prime}=l^{\prime\prime}y,\quad z^{\prime\prime}=l^{\prime\prime}z,\quad t^{\prime\prime}=k^{\prime\prime}l^{\prime\prime}(t+\epsilon^{\prime\prime}x),$$

with

$$\epsilon^{\prime\prime}=\frac{\epsilon+\epsilon^{\prime}}{1+\epsilon\epsilon^{\prime}},\quad l^{\prime\prime}=ll^{\prime},\quad k^{\prime\prime}=kk^{\prime}(1+\epsilon\epsilon^{\prime})=\frac{1}{\sqrt{1-\epsilon^{\prime\prime2}}}.$$

If we take for l the value 1, and we suppose ε infinitely small,

$$x^{\prime}=x+\delta x,\quad y^{\prime}=y+\delta y,\quad z^{\prime}=z+\delta z,\quad t^{\prime}=t+\delta t$$

it follows:

$$\delta x=\epsilon t,\quad\delta y=\delta z=0,\quad\delta t=\epsilon x.$$

This is the infinitesimal generator of the transformation group, which I call the transformation T1, and which can be written in 's notation:

$$t\frac{d\varphi}{dx}+x\frac{d\varphi}{dt}=T_{1}.$$

If we assume ε = 0 and l = 1 + δl, we find instead

$$\delta x=x\delta l,\quad\delta y=y\delta l,\quad\delta z=z\delta l,\quad\delta t=t\delta l$$

and we would have another infinitesimal transformation t0 of the group (assuming that l and ε are regarded as independent variables) and we would have with 's notation:

$$T_{0}=x\frac{d\varphi}{dx}+y\frac{d\varphi}{dy}+z\frac{d\varphi}{dz}+t\frac{d\varphi}{dt}.$$

But we could give the y- or z-axes the special role, which we gave the x-axis; thus we have two further infinitesimal transformations:

which also would not alter the equations of.

We can form combinations devised by, such as

$$\left[T_{1},T_{2}\right]=x\frac{d\varphi}{dy}-y\frac{d\varphi}{dx},$$

but it is easy to see that this transformation is equivalent to a coordinate change, the axes are rotating a very small angle around the z-axis. We