Page:PoincareDynamiqueJuillet.djvu/10

 We should have

$$\delta J=\int dt\ d\tau\left[\sum\alpha\left(\frac{d\delta H}{dy}-\frac{d\delta G}{dz}\right)-\sum u\delta F\right]=0,$$

or, integrating by parts,

whence, by setting the arbitrary coefficient δF equal to zero,

This relationship gives us (by partial integration):

or

$$\int\sum Fud\tau=\int\sum\alpha^{2}d\tau,$$

hence finally:

Now, thanks to equation (3), δJ is independent from δF and thus δα; let us vary now the other variables

It follows, by returning to expression (1) of J,

$$\delta J=\int dt\ d\tau\left(\sum f\delta f-\sum F\delta u\right).$$

But f, g, h are first subject to conditions (2), so that

and for convenience we write:

The principles of variation calculus tells us that we must do the calculation as if ψ is an arbitrary function, as if δJ is represented by (6), and as if the changes were no longer subject to the condition (5).

We have in addition:

$$\delta u=\frac{d\delta f}{dt}+\delta\rho\xi,$$

whence, after partial integration,

If we assume at first that the electrons do not undergo a variation,