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,

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

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

(3)	$u={\frac {d\gamma }{dy}}-{\frac {d\beta }{dz}}.$

This relationship gives us (by partial integration):

$\int \sum Fud\tau =\int \sum F\left({\frac {d\gamma }{dy}}-{\frac {d\beta }{dz}}\right)d\tau =\int \sum \left(\beta {\frac {dF}{dz}}-\gamma {\frac {dF}{dy}}\right)d\tau =$ $=\int \sum \alpha \left({\frac {dH}{dy}}-{\frac {dG}{dz}}\right)d\tau ,$

or

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

hence finally:

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

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

(5)	$\sum {\frac {d\delta f}{dx}}=\delta \rho$

and for convenience we write:

(6)	$\delta J=\int dt\ d\tau \left[\sum fdf-\sum F\delta u-\psi \left(\sum {\frac {d\delta f}{dx}}-\delta \rho \right)\right].$

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,

(7)	$\delta J=\int dt\ d\tau \sum \delta f\left(f+{\frac {dF}{dt}}+{\frac {d\psi }{dx}}\right)+\int dt\ d\tau \left(\psi \delta \rho -\sum F\delta \rho \xi \right).$

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