Does the Principle of Least Action Have a Physical Meaning?

[crastinus]

I have found that some people say “yes, definitely”, and other days “no, definitely not”.

Those who say “no” seem to regard PLA as merely a neat way of packaging the equations. Those who say “yes” seem to regard PLA as somehow fundamental. (There have actually been two recent books on this, Coopersmith’s The Lazy Universe and Albert Rojo’s book, the name of which slips my mind. Both seem to take PLA as fundamental.)

Is there a consensus among physicists that I am unaware of? If not, why not?

What could the physical meaning of PLA even be?

 

[FactChecker]

The PLA expresses itself in innumerable physical ways. It can be used to derive the equations of motion in many settings (see https://en.wikipedia.org/wiki/Principle_of_least_action ). So the way things move are a direct physical consequence of the PLA. If that is not enough for a consensus, I don't know what would be.

 

[crastinus]

Someone recently argued to me that PLA is not fundamental. This is the gist of what he argued:

If we consider a case like refraction, for example, the photon moves forward “locally”, and there doesn’t seem to be an sign of least action or optimization. But if we fix it’s initial and final points and study the path between them, we will find that the path is least in some variational sense. But that appears to be just an after-the-fact, purely mathematical, “global” result of our having summed the local actions. Since it is purely mathematical, there’s no reason to think of it as fundamental. It is a result of how we organize our mathematical expressions.

 

[Dale]

Is there a consensus among physicists that I am unaware of? If not, why not?

What is the criteria used to judge if something has “physical meaning”? The lack of consensus on the answer is probably more due to a lack of consensus about “physical meaning” than about the principle of least action.

 

[crastinus]

Dale, that’s certainly a good point. What I find is some physicists saying it’s just mathematical and some saying it has something that isn’t just mathematical about it. I think of Taylor et al.’s efforts in the 90s to use PLA to reform the physics undergrad introductory mechanics courses; their idea was that PLA is more meaningful even for students than the concepts of force, etc. Not to get into that, but that just suggests some idea of at least how the phrase gets used in these contexts.

 

[ZapperZ]

What exactly does "physical meaning" mean? What are you looking for?

For example, what is the "physical meaning" of Newton's laws?

Zz.

 

[FactChecker]

In a standard case like refraction, for example, the wave or ray moves forward locally, and there's no sign of least action or optimization.

Saying that it moves forward, rather than drifting sideways, seems like a simple, limited, local expression of the PLA. There are simple cases where the PLA is so obvious that it is not envoked by name. But that does not make it less fundamental -- just anonymous.

But if you then fix beginning and end points of a ray and look at the path taken between them, it's least in some variational sense. But that's just an ex-post-facto, purely mathematical, global result of the sum of the local actions. Being purely mathematical, there’s no reason to think of it as fundamental. It is a result of how we organize our mathematical expressions.

The mathematical process of the calculus of variation is free to come up with any number of solutions in other contexts. The physics application of the PLA in larger, complicated, examples will just lead to results that reflect the PLA applied anonymously in the simple, obvious, physics case.

 

[Dale]

What I find is some physicists saying it’s just mathematical and some saying it has something that isn’t just mathematical about it.

Well, if that is the criteria then it clearly has physical meaning. The people claiming it is just mathematical are obviously wrong given the mountain of experimental evidence supporting it.

 

[crastinus]

What exactly does "physical meaning" mean? What are you looking for?

For example, what is the "physical meaning" of Newton's laws?

Zz.

Let me try to give an example from some of this literature. (I am not fully clear on how the term is used either.)

In one piece, the authors are trying to set up PLA just from reasoning about energy. They ask whether an object might minimize the sum of its KE and PE. They work out that, if this were so, then certain physical consequences would follow (things would accelerate away from each other, etc.). Since this is not what actually happens, they say that our guess that objects minimize the sum of KE and PE is “unphysical”. (Article: http://eftaylor.com/pub/ForceEnergyPredictMotion.pdf)

That example is a little removed from the more general question of “physical meaning”, but I hope that including it can help clarify something anyway.

 

[ZapperZ]

Let me try to give an example from some of this literature. (I am not fully clear on how the term is used either.)

In one piece, the authors are trying to set up PLA just from reasoning about energy. They ask whether an object might minimize the sum of its KE and PE. They work out that, if this were so, then certain physical consequences would follow (things would accelerate away from each other, etc.). Since this is not what actually happens, they say that our guess that objects minimize the sum of KE and PE is “unphysical”. (Article: http://eftaylor.com/pub/ForceEnergyPredictMotion.pdf)

That example is a little removed from the more general question of “physical meaning”, but I hope that including it can help clarify something anyway.

It doesn't.

If you ask for the "physical meaning" of something, then you must know of something in which you DO have a physical meaning that you are OK with. Clearly, since you didn't ask about the physical meaning of Newton's laws, you are fine with it. So I want to know, what is the "physical meaning" of Newton's laws that you are comfortable with that you do not have with the Least Action principle?

Zz.

 

[Dale]

(I am not fully clear on how the term is used either

I think no one is, hence the disagreement. In other words, the discussion is not about physics but about semantics, specifically the semantics of “physical meaning”.

 

[crastinus]

To answer that, ZapperZ, let me say what I think the importance of this question is.

In his Scholarpedia article on PLA, Chris Gray makes the point that action principles have been and are some of the most productive areas for research on unifying principles in physics (here: http://www.scholarpedia.org/article/Principle_of_least_action). If that is so, then it would seem to be worthwhile to examine them more carefully, so as to get a richer sense of what they do mean. Gray, for example, talks about “true” trajectories; by which he seems to mean those that a particle takes, as opposed to ones it doesn’t.

My view is that PLA is fundamental in a physical way (i.e. more than mathematical). So, I agree with what basically everyone here has said so far.

BUT I am interested in two things: 1. Why is it sometimes treated as if it does not have physical meaning? What are the arguments that it does not? 2. Why is it the case (as in my experience) that those who think it has physical meaning and those who think it is purely mathematical each regard their view as more or less obvious without giving the other view much consideration?

If action principles are being studied for their potential to unify theories, then it would seem important to deal with problems in how to interpret them.

That does not answer your question about what I think Newton’s laws mean, of course. But I’m trying to do something bigger than my own view here. (And, as I said, I do think PLA has some kind of physical meaning, although I don’t claim to know what that may be (beyond what one standardly finds in reference sources about minimizing action).)

B. Hartmann, from Perimeter Institute, argues that PLA has the physical meaning of “minimal steering effort”. (Here: https://arxiv.org/abs/1307.0499) Hartmann’s goal in outlying this, as he says, is to find a solid ground for research on unifying principles of physics. But he doesn’t deal with claims that PLA is purely mathematical. Hence, my interest in those kinds of claims.

 

[ZapperZ]

You still have not given me ANY example of something that, to you, has a "physical meaning", and why.

Zz.

 

[Dale]

let me say what I think the importance of this question is

Before you can establish the importance of a question I think you need to establish the meaning of the question. Surely a meaningless question cannot be important.

 

[crastinus]

This all reminds me of Dale’s Insight piece on why discussions of what energy is go astray. I think the trouble here is that there is no textbook definition of physical meaning. Nevertheless. physicists, chemists, and engineers use the phrase “physical meaning” frequently enough and without seeming incoherence. On Physics Forums itself, there are a number of discussions about some equation or operation or other and its “physical meaning” —i.e. of the Lorentz transformation, etc. (Check it out: https://www.physicsforums.com/search/87328339/?q=“physical+meaning”&o=relevance .) On Google Scholar too, there are lots of books and articles from scientific journals and publishers (Springer, Taylor Francis, etc) on “the physical meaning of X”, where X is entropy, gauge-invariant variables, scratch hardness, etc. Of course, what each author means by it may well be different somewhat, but it can’t be the case that everyone means something completely different, since there’s so much written that tries to address it in some ways. (One could argue that it’s all rubbish, but I tend to side with the ordinary practice of scientists myself.)

Really, all I’m looking for is to find out what the arguments are and what the evidence is that PLA has no physical meaning (whatever one might mean by that) or that it is purely mathematical. So far, everything we’ve talked about here has suggested that it does have physical meaning of some kind. I just want to be sure that that’s right by looking at all the evidence and arguments for the opposing view.

ZapperZ, since my goal is to understand what the other views are on this, I don’t see that coming up with examples of what I think physical meaning is will do anything but distract from what this thread is about. I am willing to consider anything a reputable physicist or mathematician has called “physical meaning (or lack thereof) of PLA”. I agree with you that, if I were only willing to consider some limited types of account of the physical meaning of PLA, then, yes, I think I would have to give my own view in order to focus the discussion. But at this stage there is no reason to focus it that way. Since whether I should give examples of my view or not is not really a question about physics or the practice of physics, perhaps we should not go into it anymore here. (Feel free to message me to continue the discussion, though, if you’re interested.)

 

[Dale]

Nevertheless. physicists, chemists, and engineers use the phrase “physical meaning” frequently enough and without seeming incoherence.

I disagree. In your OP you specifically mentioned the divergence of opinions. That is a clear indication of incoherence.

Of course, what each author means by it may well be different somewhat, but it can’t be the case that everyone means something completely different, since there’s so much written that tries to address it in some ways.

It certainly can be the case, and to my view clearly it must be the case since the conclusions differ. Assuming that the scientists involved did not make an error then there must be a substantive difference in meaning.

I am willing to consider anything a reputable physicist or mathematician has called “physical meaning (or lack thereof) of PLA”. I agree with you that, if I were only willing to consider some limited types of account of the physical meaning of PLA, then, yes, I think I would have to give my own view in order to focus the discussion.

Then let’s do this. Pick two references you like, one pro and one con, and let’s see if we can identify either a substantive difference in their usage of “physical meaning” or an error in their analysis.

 

[fresh_42]

Does the Principle of Least Action Have a Physical Meaning?

I have always considered it as the most fundamental physical principles of all together with the principle of least energy, which I think is correlated. You can basically develop the entire classical physics from there and even more. If not these have physical meaning, then nothing has.

 

[crastinus]

I found two interesting discussions of this over at Stack Exchange.

https://physics.stackexchange.com/q...meaning-of-the-action-in-lagrangian-mechanics

In this one, Neumaier (a contributor here too) says this (the voted best answer):

“The action has no immediate physical interpretation, but may be understood as the generating function for a canonical transformation; see e.g., http://en.wikipedia.org/wiki/Hamilton-Jacobi_equation”

It’s interesting that he either doesn’t try to argue against the other views or he regards them as consistent with his. Possibly the latter?

The answer right after his claims that PLA has a fundamental meaning in relativistic quantum mechanics, namely “least phase change”. Fascinating point! I’d never encountered it before.

https://physics.stackexchange.com/questions/9686/the-meaning-of-action

This discussion has different points to add, but I can’t see that anyone maintains here that PLA is just mathematical. (Maybe Lubos Motl’s point about definitions of concepts being used in tautologous ways, but it’s not clear to me.)

 

[fresh_42]

I got the impression that you confuse physical meaning by mathematical meaning. To me, and I think most of us, comes physical first and mathematical is merely a description of it in order to make certain situations quantifiable, and not the other way around. Even if you define physical meaning by the existence of a mathematical description, that description won't carry physical meaning in itself, and even less generates one.

 

[crastinus]

I got the impression that you confuse physical meaning by mathematical meaning. To me, and I think most of us, comes physical first and mathematical is merely a description of it in order to make certain situations quantifiable, and not the other way around. Even if you define physical meaning by the existence of a mathematical description, that description won't carry physical meaning in itself, and even less generates one.

My mistake, then!

I definitely think physical meaning “comes first”. The reason I’m interested in this question with regard to PLA is because I was disturbed by how it is sometimes said that PLA is “purely mathematical” (for one example of which see above). I started the thread to see what the arguments for that view are.

But what you said made me remember something: in every instance I know of in which someone claims that PLA is “purely mathematical”—with the possible exception of Neumaier—the person who made the claim was a mathematician. So, maybe you’re right that this really is the root of why they say PLA doesn’t have a physical meaning.

Thanks for the insight!

 

[crastinus]

In this article (https://arxiv.org/pdf/1203.2736.pdf), the authors begin by quoting Poincare to the effect that PLA cannot possibly have physical meaning because it attributes directedness to physical systems, which they (to him) cannot have. (Poincare is concerned that the particle seems to “know” what path to choose.)

Then, more interestingly, they make the point that there are really two distinct action quantities: the Euler-Lagrange action and the Hamilton-Jacobi action. With the Euler-Lagrange action, we must know the initial and fina states (I’m ignoring details here); with the Hamilton-Jacobi action, they say we only need an initial state which is itself the action of the system at its initial position.

They propose that “nature” actually follows the Hamilton-Jacobi action but we as observers must begin and reason (mostly) from the Euler-Lagrange action.

I’m not sure one needs to go to such lengths. To me, PLA is clearly fundamental in some sense. But the authors do seem motivated by a concern to show why objections like Poincare’s don’t hold up.

 

[Dale]

I found two interesting discussions of this over at Stack Exchange.

I was actually hoping for scientific references which are held to a higher standard than Internet forums. People argue on Internet forums that the Earth is flat.

 

[Vanadium 50]

(Poincare is concerned that the particle seems to “know” what path to choose.

How does a thermos know to keep cold liquids cold and hot liquids hot? How does it know?

 

[Stephen Tashi]

In this article (https://arxiv.org/pdf/1203.2736.pdf), the authors begin by quoting Poincare...

The quote from Poincare is:

The very enunciation of the principle of least action is objectionable. To move from one point to another, a material molecule, acted on by no force, but compelled to move on a surface, will take as its path the geodesic line, i.e., the shortest path . This molecule seems to know the point to which we want to take it, to foresee the time it will take to reach it by such a path, and then to know how to choose the most convenient path. The enunciation of the principle presents it to us, so to speak, as a living and free entity. It is clear that it would be better to replace it by a less objectionable enunciation, one in which, as philosophers would say, final effects do not seem to be substituted for acting causes.

Since most of the congregation dismisses the notion of "physical meaning" as being utterly vague, perhaps we should digress to discussing the ideas in that paper. I interpret Poincare's remarks as saying that the motion of a particle is governed by physical laws that , at time t, do not require knowing conditions in the distant future in order to predict what happens at time t + dt. So a principle of least action that is based on predicting what happens to a particle at time t+dt based on considering possibilities that can happen throughout a time interval, including times long after t and places the particle will not travel may be mathematically correct, but it must be a consequence of physical laws that determine conditions at time t + dt without being a function of conditions in the distant future.

That view is plausible in classical physics. For example, if we imagine a particle affected by a force that we can vary "at random" both in time and space, then, in retrospect, must its path realize an extrema of action over all possible paths?

 

[Dale]

Since most of he congregation dismisses the notion of "physical meaning" as being utterly vague, perhaps we should digress to discussing the ideas in that paper.

That seems good to me. Thanks for posting the relevant quote

I interpret Poincare's remarks as saying that the motion of a particle is governed by physical laws that , at time t, do not require knowing conditions in the distant future in order to predict what happens at time t + dt.

That is my take on his view also. However, I would put a different take on it. In physics we write our laws as second order differential equations. Since the laws are second order you need two pieces of information. They could be position and velocity or they could be position now and position later. It isn’t that the system knows the future, it is just that knowing the initial position and the current velocity is the same thing as knowing the initial position and the final position.

 

[anorlunda]

That seems good to me. Thanks for posting the relevant quote

That is my take on his view also. However, I would put a different take on it. In physics we write our laws as second order differential equations. Since the laws are second order you need two pieces of information. They could be position and velocity or they could be position now and position later. It isn’t that the system knows the future, it is just that knowing the initial position and the current velocity is the same thing as knowing the initial position and the final position.

That sounds equivalent to locality. But we only allow Lagrangians that obey locality to calculate the action, so PLA can not contradict locality.

 

[crastinus]

So, I found a mathematician at San Jose State arguing that PLA must be derived and not fundamental. Here are a few quotes and the link:

His statement of what he thinks the issue is:
“How can a noncognizant system find the path of least time or least action? There is in mathematics what is known as Pontryagin's Maximum Principle which says that the path which maximizes a particular function over a time period is the one that maximizes a related function at each instant. This related function must have some physical reality and the instant-by-instant maximization is the real explanation for how the systems evolves and it is only incidentally that the overall function on the interval of time is maximized.”

His summary of his own view:
“The relationship between the interval optimization and instant-by-instant optimization works both ways, but the instant-by-instant evolution of a physical system is fundamental and it is the minimization of action which is derived.”

He concludes:
“A physical system moves in the path that minimizes least action because it moves at each instant according to criteria which results in it incidentally minimizing action. That instantaneous criterion can be represented in the system moving in the "direction" of maximum reduced force, force divided by mass. It is this instant-by-instant dynamics which is fundamental. Treating the Principle of Least Action as fundamental is misleading.”

I have not reproduced the key mathematical parts of his argument here. Please see his personal site for details: http://www.applet-magic.com/minprin.htm

It seems to me that he is not arguing that PLA is meaningless, though, whatever we mean by that. He seems to just be arguing that we should consider it differently than is usually done.

So, even if one rejects his account, what he’s done here doesn’t do anything to show PLA itself is somehow “purely mathematical”.

 

[crastinus]

How about, instead of talking about the physical meaning of some mathematical expression (admittedly rather vague), we ask whether a mathematical expression follows from some more generalized mathematical theory or whether observation must play a key role in the development of the expression? In the current case, the question would then be: Does PLA follow from some more general mathematical expression or is it something that can only be discovered?

It seems clear to me that PLA is a discovery about physical systems, not just a corollary that follows from a theorem.

 

[fresh_42]

This is - in my opinion - straight away nonsense. Of course you can transform the principle of least action into mathematics, but that doesn't solve the question. Why should a mathematical system behave like this? It only changes the platform of the question, nothing more. To me this is as asking why it's easier to lift a weight using a lever and find the answer in a length. The length itself doesn't carry any information without the physics it describes. Physics comes first. Mathematics is only the paint in which the picture is drawn, not the artist.

 

[anorlunda]

A physical system moves in the path that minimizes least action because it moves at each instant according to criteria which results in it incidentally minimizing action.

That is a distinction without meaning. Use your infinitesimal step as having end points A and B. Prove the PLA for that. Now add a second step B to C. and do likewise, now you have a double step A to C. PLA applies no matter how big or small the step.

PLA should not be confused with differentiation. It is not an approximation. To find a slope accurately, you need infinitesimal steps. Larger steps lead to approximate answers. That is not analogous to PLA with larger steps.

 

[Buckethead]

What could the physical meaning of PLA even be?

From a laypersons point of view (which may or may not be useful to the OP), I think the answer is yes. The reason being that PLA describes what nature does and if nature didn't do it, PLA would not be valid. It is a direct description of the goings on in nature hence it has physical meaning.

At the suggestion of someone in this forum I watched all of Feynman's lectures and bought and read the book on QED. It was wonderful! I've really learned something new.

Having read the book it seems one could argue that PLA is not fundamental because QED just explains it better. So it seems to me QED is closer and better at allowing us to understand what nature is doing, such as why a light beam reflects off a mirror the way it does. It still only tells us what nature is doing and not why it is doing it, but why nature does certain things (as has been pointed out to me many times in this forum) is not knowable.

So to me, both PLA and QED have physical meaning. Just my lay opinion for what it's worth.

 

[PeterEnders]

The PLA is a 'principle', not a 'fundamental law' or 'axiom' like Newton's axioms. It is usually equivalent to the equation(s) of motion. Now, in many cases, the equation of motion is not fundamental. Example 1: Newton's equation of motion, force = mass x acceleration. It does not enter Newton's axioms. Example 2: The standard (D'Alembert's) wave equation. It is not suitable to represent Huygens' principle.
Nevertheless, the PLA is at least a most powerful tool. For instance, special-relativistic equations can be written down covariantly (manifest invariantly) in a straightforward manner. More generally, its tight connection with the symmetry of the problem under consideration makes the generalization of equations much more easy.​

 

[Stephen Tashi]

A discussion of "The" Principle of Least Action will get confusing if there is more than one principle of least action. Considering only classical mechanics, what assumptions must hold before "a" principle of least action applies? (One can argue subjectively about whether such assumptions make a principle of least action "fundamental" or "not fundamental", but what the assumptions are should be an objective topic.)

That is a distinction without meaning. Use your infinitesimal step as having end points A and B. Prove the PLA for that. Now add a second step B to C. and do likewise, now you have a double step A to C. PLAl applies no matter how big or small the step.

That's plausible in a case when a principle of least action applies over a long time interval. However, are there cases when it doesn't? I'm thinking of a classical analog of a "delayed choice" experiment where we vary the force on a particle as a function of space and time and try to trick it into taking a path that is not a stationary path in the action (or where "action" is undefined). For example: "Ah-hah, you started out going due North, but should have stayed where your were for 3 seconds and then taken a Northeasterly trajectory."

 

[Mark Harder]

I wonder if the PLA is equivalent to the principle of determinism in dynamics. Let a particle follow a trajectory. Determinism says that at each instant, the momentum and location of the particle are a direct consequence of the previous momentum and location. The trajectory as a whole may be viewed as a sequence of instants, each of which follows from its immediate predecessor, according to the applicable laws of motion. But the trajectory is also given by the PLA. What is the relationship between the PLA description of a trajectory and the deterministic model? I don't know enough physics to establish this connection, but I once asked an instructor if the equivalence was true and he replied in the affirmative, without elaboration.

As an example of a purely mathematical justification of the PLA, one might prove such as it is implied by the fact that the equations of motion are second order ODEs with boundary conditions (i.e. f = m⋅d² r/dt² ), a mathematical proposition implied by another proposition that follows from yet more mathematical statements of physical dynamics.

 

[Bultaco]

What is the criteria used to judge if something has “physical meaning”? The lack of consensus on the answer is probably more due to a lack of consensus about “physical meaning” than about the principle of least action.

I'm sure you meant to say "criterion"! :-)