The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the action of a mechanical system, yields the equations of motion for that system. The principle states that the trajectories (i.e. the solutions of the equations of motion) are stationary points of the system's action functional. The term "least action" is a historical misnomer since the principle has no minimality requirement: the value of the action functional need not be minimal (even locally) on the trajectories.


https://en.wikipedia.org/wiki/Stationary-action_principle

I noticed something only today: I was walking around my mom's house. it has a few steps leading up to it. I could take the shortest route down these steps (a ball rolled down would follow a linear path, ceteris paribus). However, I chose to zig-zag my way down. If I were dead, my body rolled down these stairs would do so like the ball - straight down.

Interestintgly, when we feel the cold or have the chills, we roll up into a ball (fetal position). Makes sense physics-wise: reducing the surface area for a given volume means less heat lost from the skin.

The more chaotic a locality (town, city, etc.), the more inefficiently done the public works are, the more free the people are. Has anyone been to the orient (China is now an exception but not the whole country obviously)?

Human behavior, if you'll take the time to notice, breaks this easiest route rule - we do things in very inefficient ways, most of the times failing to take the shortest route between beginning (of a project) and its end. In essence we violate the Principle of Least Action. — Agent Smith

Yes, we behave inefficiently very frequently. BUT we have reasons for those behaviors. They may not be very good reasons, but they are reasons and thus, it's not "free" will in the traditional conception of the phrase.

Yes, we behave inefficiently very frequently. BUT we have reasons for those behaviors. They may not be very good reasons, but they are reasons and thus, it's not "free" will in the traditional conception of the phrase — Artemis

Methinks you're trying to eat the cake and have it to.

Perhaps a better way to express it is that everything takes the path of least resistance. What Artemis might be saying is that humans will find the easiest way to do the hardest part. The hardest part for one person may be totally irrational to someone else. And the easiest way may be willingly ignorant, inefficient and not the shortest route at all. But that was not the goal. The goal was for that person to find the easiest way.

Direct & Indirect means.

The shortest distance between two points is a straight line. — Math

The shortest distance between two points is a smooth curve. — Humans

Math says something, humans say something totally different. Free will?

Humans say the shortest distance between two points (of what they think they want), is a straight line. Math doesn't want anything.

Determinism is about the absence of choice (mathematical laws don't give you options).

Humans, it appears, can defy mathematical laws/principles. If I choose to, I needn't travel along a straight line from point A to point B.

If I choose to, I needn't travel along a straight line from point A to point B. — Agent Smith

No, but you will from point A to point A.1. A smooth curve is a series of infinite straight lines. You were never headed to point B, you were going to point A.1 first.

No, but you will from point A to point A.1. A smooth curve is a series of infinite straight lines. You were never headed to point B, you were going to point A.1 first. — Monitor

Yet, a curved line is longer than a straight line (irrational/free will). It doesn't make mathematical sense to meander, travel on a curve, when we can make a beeline to our destination.

The natural (nonliving) world follows the principle of easiest path (least resistance, like you said). You might point to a river, winding its way to the sea and say "there, that's not how things work!" However, if we bring math & physics to bear on the matter, you'll, I'm certain, discover that a river's course is mathematically sound.

For humans, this is most likely false. Our paths in life aren't mathematically sensible; one could even say that it's quite like a river flowing uphill!

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.

Humans, it appears, can defy mathematical laws/principles. If I choose to, I needn't travel along a straight line from point A to point B. — Agent Smith

Wouldn't that describe that math forced your choice? e.g I chose to not walk straight because of math.

Wouldn't that describe that math forced your choice? e.g I chose to not walk straight because of math. — john27

I would expect a mathematical system to exhibit behavior in line with mathematical principles one of which is always take the shortest distance between two points. Any deviation from linearity, in my humble opinion, can't be mathematical.

The shortest distance from my house to the convenience store is to cut across the park. But maybe I don't want to walk through a soccer game. Maybe the park is muddy and I'm wearing new shoes. There are all kinds of reasons that the math of it would not be my top priority. And those reasons and decisions are still determined.

Even when there are no compelling reasons (e.g. a soccer match, a man with a loaded gun) to take a nonlinear path, we can and do.

Any deviation from linearity, in my humble opinion, can't be mathematical. — Agent Smith

I would say that if you wanted to, I'm sure you could describe linear deviation in a mathematical fashion. Even more so, we are subject to linearity, just not all the time. (e.g I'm really hungry and want a sandwich, I naturally take the shortest way there, to the kitchen.)

All I can say is human beings seem to violate some mathematical principle reducible to linearity (straigth lines). Physics, chemistry don't (every physio-chemical process in our body takes the shortest route from start to finish). Yet we're able to take not just any route but even the longest one. It just doesn't add up? Nonphysicalidm? Free will?

The principle of least action, applying to inanimate, conservative stuff, determines the unconscious will and action of mindless, conservative being. Progressive creatures with a mindful or conscious will, can sidestep the principle of least resistance and move in a way it wouldn't if they were dead. I could move between two fixed points in total disagreement with and opposition to the Principle. So my will free and only a physical or mental leash can limit my free will.

All I can say is human beings seem to violate some mathematical principle reducible to linearity (straigth lines). — Agent Smith

Well you could as well turn it the other way around, and say that all of humanity is on a linear trajectory, from life to death. I think my problem is linear comparative to where? It seems subjective, where we place the two dots.

Human behavior, if you'll take the time to notice, breaks this easiest route rule - we do things in very inefficient ways, most of the times failing to take the shortest route between beginning (of a project) and its end. In essence we violate the Principle of Least Action. — Agent Smith

I'm pretty sure if you fire a human from a trebuchet, they will follow the path given by the principle of least action. So in an example that is equivalent to the physical behaviour you cite, people behave as expected.

What you seem to have in mind is that people do not always make the best possible decisions, and you're using action as a metaphor. But people are also ruled by other physical laws. One is the second law of thermodynamics, which forbids people from suddenly and opportunely knowing something they did not learn. As such, humans always make decisions in partial ignorance, leading (deterministically) to a high probability of suboptimal choices.

There are many other factors that influence decision-making (for instance, exercise in order to get fit), but none of them violate the principle of least action.

As such, humans always make decisions in partial ignorance, leading (deterministically) to a high probability of suboptimal choices — Kenosha Kid

But wouldn't a human's partial ignorance be funded by his free will?

I'm pretty sure if you fire a human from a trebuchet, they will follow the path given by the principle of least action — Kenosha Kid

But the question is: Is she subject to the principle, or is this so only in the mind uttering the principle? After the launch friction will influence the motion, so the principle is not applicable anymore, as only conservative forces are implied. Even on the Moon, friction is involved, gravitational friction. During flight you can wobble around your center of mass, causing friction, and this motion won't be determined by a minimum action principle.

I just can't see mechanistic domino effects producing symphonies or the works of Shakespeare.

I'm convinced!

But wouldn't a human's partial ignorance be funded by his free will? — john27

No, it's funded by the second law of thermodynamics. You can choose to be ignorant about something (or do your own research in contemporary parlance), but it doesn't follow that all ignorance is voluntary. We are born ignorant of almost everything.

Is she subject to the principle — Raymond

And the answer is what you quoted.

After the launch friction will influence the motion, so the principle is not applicable anymore, as only conservative forces are implied. — Raymond

Well you can look at that in three ways. One is that you'd have to perform the experiment under ideal conditions. Another is that you factor dissipation, which is still deterministic, into the calculation. Another is that you add "in principle" on the grounds that, on a molecular level, there are no dissipative forces but, on that level, the calculations are untenable.

None of which suggests that failure to obey the principle of least action is free will: indeed, you've just cited examples of its violation that are deterministic.

I just can't see mechanistic domino effects producing symphonies or the works of Shakespeare.

I'm convinced! — Yohan

I think you arrived already convinced, no?

Free will, as we all know, is central to ethics. Does ethics make scientific/mathematical sense? The 2^nd law of thermodynamics (entropy) implies that disorder (evil) is more likely than order (good).

What I meant, can one literally be under the influence of a principle we invented? I think it is the user of the principle projecting his will to power over the ones he projects it to.

Even if it were so, the principle doesn't hold for real processes, maybe by approximation only. Are we approximately subjected to it? The principle is even teleological, as it supposes a final point in spacetime that can't be known at the start, except for isolated systems.

None of which suggests that failure to obey the principle of least action is free will: indeed, you've just cited examples of its violation that are deterministic. — Kenosha Kid

If I act contrary to the principle, which I do, by every action I perform. My will is nor free, nor tied to determinism or any other abstract principle. The will simply is.

Does ethics make scientific/mathematical sense? — Agent Smith

Hm, well it could be said that ethical behaviour is cost effective in a social setting, which would make it more or less scientific.