Least action on Birkhouse Moor

I've been reading a lot of Classical Mechanics recently. I've come across the Lagrangian for the first time and the Principle of Least Action. Lagrange based his mechanics on the arithmetic difference between kinetic and potential energies summed over different paths. The actual path was the one with least action. The book by Leonard Susskind that I'm reading is adamant that it should really be stationary action. He says that dA should be zero. I was looking at the book and then thinking about it on Birkhouse Moor yesterday.

This is a saddle point. From where I was looking the ground goes down and then up to the summit. From the left it comes up to the saddle and then down to the right. If A were to be the height about sea level, then going from where I was stood to the top of the fell, A would go down and up again. If I stood in the middle of the saddle and moved a tiny distance in any direction, A would not change. It would be stationary. This seems to be connected to going from the Lagrangian to the Euler-Lagrange equations.