Page:PoincareDynamiqueJuillet.djvu/16

 Note that

It follows, by replacing δt' by its value

If we recall the definition of k, we draw from this:

and also

hence

Now, in virtue of equations (2) we must have:

By replacing ΣXδU by its value (3) and by identifying, it follows:

These are the equations (11) of § 1. The principle of least action leads us to the same result as the analysis of § 1.

If we turn to formulas (1), we see that Σf² - Σα² is not affected by the transformation, except one constant factor; it is not the case with expression Σf² + Σα² which represents the energy. If we confine ourselves to the case where ε is sufficiently small, so that the square can be neglected so that k = 1, and if we also assume l = 1, we find:

or by addition

§ 4. — The group
It is important to note that the transformations form a group.

Indeed, if we set: