The function for the action

is given. are the coordinates and the velocities, respectively, for a system of particles.

Then the change in

When this difference is expanded in powers of and in the integrand, the leading terms are of the first order

or, affecting the variation (2),

The conditions show that the integrated term is zero. There remains an integral which must vanish for all values of . This can be so only if the integrand is zero identically. Thus we have

What is meant by the sentence at (1)? Which calculations aren't shown here?

How is this translated to affect the variation at (2)?

How did we get to (3)?

Edit: Having studied this further the last couple of days, I've realized what a stupid question it was. I'm gonna read up on calculus of variations and understand this fully!

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