What does it mean to say that something moves on a straight wordline in terms of the principle of least action? I know it generally means that action is minimum or stationary but since I only really know some physics from a conceptual standpoint and not a mathematical one I don't really know what this means. Does it mean for geodesic motion that action is Zero? Another way of putting the question is, is there anyway to tell from the output of the Principle of Least Action that something is moving on a straight worldline as opposed to a curved one? Does one always reduce to 0 and the other is non-zero? Or is that wrong. I'm trying to figure out the privileged status of movement along a geodesic in terms of the principle of least action.