I recently read about the construction of the Nambu-Goto action by looking at the proper area of the world-sheet. However, when one varies the action with an arbitrary space-time coordinate (here I treat it in Minkowski space-time X^(\mu)), there appears three terms and some of them would go to zero.

What I don't understand is why is that

(i) The momentum density running along the string P^(\mu)_(\sigma) at the open string endpoints \sigma (0, \pi) is zero?

(ii) Why is it that the momentum density transverse to the string, i.e. P^(\mu)_(\tau) \neq 0?

Has this got something to do with the conservation law, that energy is conserved?

Thanks