In all of the below remarks, I am considering an object (a "mass") moving around above the surface of the earth.
What are some physical examples of situations that demonstrate path independence in a gravitational field (by that I mean: the condition where the work done by gravity on the object, as the object moves from point A to point B is independent of path)? We have all seen the randomly curvy paths leading from point A to point B... but is this physics or is this an artifact of multivariable calculus books? It seems to me the only true demonstrations of this would be projectile motion and free fall. So why do physics books have the super curvy paths when they introduce path independence? Am I missing something? Is there something physical to these strange paths?
Perhaps I misread the figures from the physics books, I don't know.
But the question remains: in a perfect book should only parabolas and straight lines be used for demonstrating "the geometry of path independence" for work done by a gravitational field?
What about roller coasters with no friction for making the demonstration of "the geometry of path independence" for work done by a gravitational field? For that matter, what about me holding a mass that I move around in the air so long as I promise to only provide the normal force like the track did for the roller coaster? Is the final word that, strictly speaking, constraints such as tracks are not allowed if you want to purely probe the interaction force between our mass of interest and the earth which is nothing other than the conservative force of gravity? So are roller coasters out [not because they exert a force normal to the movement which does no work, but, rather, because they introduce any force at all other that the one creating the conservative field... which necessarily takes us beyond the conditions we are considering]?
And finally, what if I want to move the mass from point A to point B with all different kinds of pushes and pulls with my hand... introducing tangential accelerations at will. Sometimes I move the mass from A to B very quickly and in a straight line. Sometimes I take a long path but with varying speeds. Once in a while I go slow at first and then accelerate to the fastest I can possibly go over a particularly long path. Do all of these paths have a place in the physics book when we do a picture of path independence? If yes, then obviously we are not constricting our focus to a situation where the only relevant force is the interaction force making the conservative field. So fine, we can still do the bookkeeping on the height-changes all the same... however this feels like a strange scenario for discussing a conservative field.
What is your take on the physicist's view on how one demonstrates, using a specific physical system, path independence in a conservative field.