Here is a picture from Wikipedia, a free body diagram used to derive the maximum load in Euler's buckling theory. The moment is calculated around point A. It is claimed that the load $P$ causes a moment $Pw$ around point A, but why is this? In the left picture above, we can see that the load $P$ is applied directly towards point A without offset. As the vertical distance from the point is zero, shouldn't the moment be zero?
I understand that the same force $P$ is also present on the top of the buckle. So let's consider a similar situation:
We could imagine the rectangular arch to be any semi-circular object. If we push the ends (marked with the arrows) with equal forces, we have equilibrium in the x-direction. But is there a net moment around either end here? I think not. I tried this in real life with several objects and the ends simply bend towards inside, no rotation. So what is happening in the buckling case? Why is there a moment from the force $P$, if its distance is actually zero from the horizontal line?