Why this definition for "symmetry transformation"?

Question

This question concerns section 8.5.1 in these notes:

I don't understand why a symmetry transformation is defined as such. What implications is there if $\delta \mathcal L$ is a total derivative? Is there an intuitive way to see this?

Answer 1

That is a bit of a strange definition of "symmetry transformation", since there are other kinds of transformations which are symmetries (Gauge transformations, for instance). But at least here, if you change the Lagrangian by a surface term, the action will vanish if the fields vanish at infinity. This is a standard assumption of quantum field theory. Likewise you could work on a compact manifold and force the fields to vanish at the boundary.

Answer 2

The reason that is called a symmetry transformation is that the equations of motion which are derived from that action principle are invariant under such a symmetry. Notice a total derivative term integrates out under the conditions mentioned in levitopher's answer. That is my understanding of the motivation of the term. Physicists care about equations of motion (solutions of the Euler Lagrange equations derived from the action)