There is several things I am confused about with gravitational potential energy. So, first of all, shouldn't U=mgh always be written ΔU=mgΔh, because isn't that equation only dealing with differences of potential energies when close to the surface of earth? Second, with the equation U=-GMm/r, there is a +K for the constant of integration at the end. What role does that play? Does that have something to do with reference frame, or is that just some random constant. I know that with differences of potential energies (ΔU=U(f)-U(i)) it will subtract itself and become irrelevant, but when you are just dealing with U(r) what significance does it have? Do you always just assume it's 0? Otherwise, wouldn't it make the single quantity U(r) useless if it could be anything? Mind you it is rather late and I wrote this in pure confusion so some of the things I wrote may not make sense.