In Feynman's lecture on the principle of least action, he show that you can describe electrostatics by saying that a certain integral is a minimum or maximum. He states the integral, and goes on to show how it can be used to approximate the capacitance of a coaxial line. I've been able to follow his argument and use a similar technique to solve the parallel plate capacitor. Later Feynman states that, "For any other shape, you can guess an approximate field with some unknown parameters like alpha and adjust them to get a minimum. You will get excellent numerical results for otherwise intractable problems." I can't seem to apply his method to anything but the simplest of cases. For example, coming up with an approximate field that meets complicated boundary conditions in 2D seems impossible....leaving conformal mapping as the only usable method of solving such problems. I also tried solving another problem that Feynman solves in a more traditional manner, "Colloidal particles in an electrolyte." using the least action principle...but what approximate field should I choose besides a polynomial?