Since deterministic systems have to adhere to the Principle of Least Action and humans consistently violate this principle, is this free will? — Agent Smith
I'm not sure if processes have to adhere to any principle. That's just our way to say that these processes have to obey to
our principle. If we mentally construct a variety of paths of a particle between two fixed points in space and time, and put forth the proposition that it's the path with least action that is the actual path taken, does that mean then that the particle has to adhere to The Principle? Isn't it the other way round?
If you throw yourself from the stairs at your mom's house, do you obey to The Principle? You start from a point at the top at noon, and end up down the steps somewhere, a fraction of a second later. Two defined points in space and time. The path taken is precisely the one for which the action is minimal. Your path is such that it ensures one can project this property on them.
The Principle seems to be teleological in nature, but it's just us giving the system initial and final conditions, like begin/end positions in space and time. If a particle finds itself, in the absence of force, at an origin, and 5 seconds later at a point 10 meters away from it, then you can let the particle move in 1001 possible ways between the two points. Let it go fast, then slow, and then fast again.
If you evaluate the integral of the kinetic energy over time you will note that the path (history) for which the kinetic energy is constant is the path with least action, or resistance.
As the particle moves in a force free region this is understandable. Any other path you let the particle follow in your mind involves a force. If you let it move fast in the first part and slow in the second you have to apply a force to decelerate it and change direction if the parts of the path have different ones.
This is even clearer if you let the particle in a circle first and then, kzjong! straight to the end point. The only path without using forces is a straight line with constant velocity. No resistance of forces is met.
Same for a free particle in a force field. The path of a particle in a gravitational field can be varied. Between the two given points in spacetime, the path on which you don't have to apply forces to the particle during its motion on the path is a parabolic trajectory on which the particle moves with varying velocity. The parabola with varying velocity in a gravity field is the equivalent of the straight line with constant velocity in the force free casus.
If the particle is constrained to a path, then it won't follow a path of least resistance. The particle experiences forces along the path (say a marble shot in a madly curved tube). You could have chosen the shape of the tube as a path to follow in the free particle case. The path would have given a higher value of the action then the straight line. But if you
force the particle, it obviously doesn't follow a path of least resistance, as that is a straight line (or a parabola). But given the constraint, the particle still moves with least resistance or least action. The forced path is always in disagreement with the free path.
The variational principle can't be applied if frictional forces are present. If you imagine a rough solid box on a rough table, and imagine it stands on on end at a given time and on the other side at a later time, what is the actual path taken?
The principle seems teleological insofar it seems that the particle chooses the right path. It seems to know that to arrive at a point in space at a given time, it has to start with a certain velocity, follow the right path, with varying (or not) velocities, and end up at the right time at the desired point. This is obvious nonsense. It's us who vary the paths and choose the right path.