Suppose I have a hill that goes up and down unevenly. It's frictionless, and I want to slide a point mass down the hill. I am interested in the path it takes. (By "path" I mean the trail it leaves behind, not the time-dependent trajectory.)
I can find the equations of motion of the point mass and, from there, find the path by finding the entire time-dependent motion. Is there a way to find the path directly, given the initial velocity of the particle? For example, does the correct path extremize something?