whence, by setting the arbitrary coefficient δF equal to zero,
(3)
This relationship gives us (by partial integration):
or
hence finally:
(4)
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,
But f, g, h are first subject to conditions (2), so that
(5)
and for convenience we write:
(6)
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:
whence, after partial integration,
(7)
If we assume at first that the electrons do not undergo a variation,