I am referring to "Newton's bucket" -- essentially a hollow cylinder filled with water inside which rotates about the cylinder's axis. The result is that the surface of the water is a parabola. The argument for this is that the surface normal of the water must be directly opposing the other forces at the surface. This is fine. However, I am wondering if such a problem can be approached with calculus of variations to solve for a function describing the surface. The problem I ran into, when naively trying to do it this way, is that the energy will of course be minimum when the height of the water is minimum, leading to obtaining no stationary solution, and the useless conclusion that the energy is reduced when the total amount of water is reduced. So, I want only want to consider variations of the height function that satisfy and additional constraint that the total amount of water is the same.