Ah - but lift doesn't act at the "aerodynamic centre" (or "center"). If it did there would be no pitching moment at the AC.
The AC is simply the point where d(PM)/d(alpha) is zero, but as the diagram shows, PM need not be zero.
There are two competing ideas stated above, and I'll try to disentangle them if I can.
Originally Posted by JohnFTEng
The idea of a fixed centre is to make the maths simpler. We fix the centre of lift and have a variable moment - twisitng force - about the same point. So all the forces act at this one point simplifying the equations.
What you're talking about there is a fixed moment reference centre. In principle you could choose any given point for an aircraft type, and as long as you're CONSISTENT and stick to that reference point then you can have relative simplicity in how you apply your equations of motion. That reference point is often, for historical reasons now as much as anything, defined to be at the 25% position on the mean aerodynamic chord, but it doesn't have to be, and there's no particular reason why that position should be special for a "real" aircraft.
The Centre of Lift is the point at which the summed lift on the aircraft (or some portion of it: you can have a "centre of lift" for a tail or a fin alone, too) actually may be assumed to act. It's the integration of the pressures over the surface, and as stated moves about all over the place with changes in Mach, alpha, camber, etc. It's a pretty useless thing to use a reference, but the fact that it does move about is of course rather important.
The Aerodynamic Centre is indeed the point where d(Pitching Moment)/d(alpha) is zero. It's important because the position of the cg relative to the AC is important in determining the longitudinal stability of the aircraft. The idea of the Neutral Point is closely related to the concept of the AC.