Does the Principle of Least Action Have a Physical Meaning?

[zdcyclops]

Much ado about nothing. Math is not physics. A set of equations is not physics, it is math. The math is used to describe everything that exists, i.e. can be observed. Do some observed objects obey the PLA? If so then PLA has a physical meaning. Meanings are a much harder subject.

 

[crastinus]

After much more research, I have actually found some literature in which the differences between those who see PLA as having meaning in some sense and those who don't are discussed.

Chris Gray (at the University of Guelph, who wrote the scholarpedia.org article on PLA here) reviews Coopersmith's recent book on PLA, The Lazy Universe (Oxford University Press, 2017), for the American Journal of Physics 86(5):395-398 · May 2018. He says this about PLA's meaning (or, as he says, "justification", which is not quite the same):

"Some (including the author of The Lazy Universe, p. 194) see the justification of Hamilton’s principle in the Feynman path integral of quantum mechanics, which implies Hamilton’s principle in the classical limit. Others feel any justification of classical action principles should be done within the framework of classical mechanics [T. Toffoli, “What is the Lagrangian Counting?” Int. J. Theor. Phys. 42, 363 (2003)]. Many others do not think that the question is relevant and no underlying a priori physical principle exists: the action is not always a minimum, and besides, it is just mathematics since any equation can be reformulated as a variational statement [E. Gerjuoy, A. Rau, and L. Spruch, “A unified formulation of the construction of variational principles,” Rev. Mod. Phys. 55, 725 (1983)]."

Anyway, it seems that this answers my original question about what reasons some might give for thinking PLA does not have a physical meaning (or whatever we want to call it).

 

[FactChecker]

So do they think it's just luck that it gives the right answer so often? If there is no physical basis and any equation can be formulated in those terms, then it seems like a strange run of luck.

 

[Stephen Tashi]

After much more research, I have actually found some literature in which the differences between those who see PLA as having meaning in some sense and those who don't are discussed.

You found https://arxiv.org/pdf/1203.2736.pdf several posts ago and that article distinguishes two different concepts for "the" principle of least action. Rather than huff-and-puff about about a principle that is not clearly defined, we could distinguish between those two possibilities.

 

[Dale]

any equation can be reformulated as a variational statement

This is interesting. Did he describe any details about this or list a reference in the footnotes?

I was not aware of that, but I could see how it might work.

 

[PeterEnders]

As a schoolboy, I was highly excited about the fact, that so many physical content can be formulated in the most simple form \delta S = 0. Much later I learned, that Lagrange's and Hamilton's representations of classical mechanics treat position and velocity/momentum variables on equal footing. This has led to Gibbs' paradox. The solution is Newton's and Euler's axiomatic of classical mechanics as well as Euler's insight, that the (then often considered to be a theleological reasoning) PLA and the causal description by the equation of motion are equivalent.
Here, not the _second_-order in time equation 'force = mass x acceleration' is fundamental (and is not written down in Newton's 'Principia', indeed), but the _first_-order in time equations
dx = v dt; dv = F/m dt

 

[FactChecker]

any equation can be reformulated as a variational statement

I think this sounds more significant and profound than it really is. It wouldn't surprise me at all if for any arbitrary equation one can find a mathematical problem that that equation solves. The real question would be whether those problems have anything to do with physics.

 

[bobob]

What could the physical meaning of PLA even be?

Well here is one physical meaning. Consider Newton's first law about uniform motion in a straight line? How do you define a straight line? You are assuming that you know what that means, but that basically turns out to be the path an object in uniform motion travels, so it's a bit circular without something more. The alternative of defining it as the shortest distance between two points is one that comes from an action principle that has a definite physical meaning.

 

[atyy]

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?

The PLA does not have physical meaning in the sense that it is not always unique. for example, the (classical) Einstein field equations can be derived from several different actions.

https://arxiv.org/abs/gr-qc/9305011
http://www.phy.olemiss.edu/~luca/Topics/gr/action.html
http://www.phy.olemiss.edu/~luca/Topics/gr/action_types.html
http://www.phy.olemiss.edu/~luca/Topics/gr/action_first.html
http://www.phy.olemiss.edu/~luca/Topics/gr/action_vielbein.html

 

[Dr. Courtney]

If a model makes predictions that are experimentally verifiable, then it has a physical meaning.

But once explanations are found to be equivalent, then it is a beauty contest which one is more fundamental.

Hamiltonian, Lagrangian, and Newtonian descriptions of classical mechanics are equivalent; therefore, there is not a completely scientific way of distinguishing one as "more fundamental." It's a beauty contest.

Likewise, wave and matrix formulations of quantum mechanics are equivalent.

 

[atyy]

This is interesting. Did he describe any details about this or list a reference in the footnotes?

I was not aware of that, but I could see how it might work.

I'm not sure to what extent this is true, but one can search using "inverse problem of the calculus of variations".

https://www.sciencedirect.com/science/article/abs/pii/0020722584900260
Enzo Tonti, Variational formulation for every nonlinear problem

https://bookstore.ams.org/memo-98-473/
Ian Anderson & Gerard Thompson, The Inverse Problem of the Calculus of Variations for Ordinary Differential Equations

 

[Dale]

one can search using "inverse problem of the calculus of variations".

Thanks, that’s perfect!

 

[Auto-Didact]

The vagueness of the term 'physical meaning' doesn't imply that no such thing exists, it may just mean that it is difficult to simply put into words. In many branches of academia there is a widely used method for solving such issues, namely doing a Delphi method panel study among a group of recognized experts in order to let the issue come to a natural resolution.

"Some (including the author of The Lazy Universe, p. 194) see the justification of Hamilton’s principle in the Feynman path integral of quantum mechanics, which implies Hamilton’s principle in the classical limit. Others feel any justification of classical action principles should be done within the framework of classical mechanics [T. Toffoli, “What is the Lagrangian Counting?” Int. J. Theor. Phys. 42, 363 (2003)]. Many others do not think that the question is relevant and no underlying a priori physical principle exists: the action is not always a minimum, and besides, it is just mathematics since any equation can be reformulated as a variational statement [E. Gerjuoy, A. Rau, and L. Spruch, “A unified formulation of the construction of variational principles,” Rev. Mod. Phys. 55, 725 (1983)]."

This discussion on the physical content of some equation directly reminds me of what Feynman discussed in his lectures Vol 2, Chp 25-6 (link here, scroll down to part 25-6). I'd copy and paste it here, but its a bit long and I'm not in the mood for converting equations to LaTeX, so just read the entire bit there.

The gist of this seems to be that any equation (or even notation) which experts agree upon on not having any physical content can sometimes seen to still have some as yet unknown physical content (as opposed to purely mathematical content) when viewed from some other point of view. This other point of view can be something like the viewpoint used in another branch of mathematics, perhaps even some mathematical branch unknown to the expert(s) in question but which directly relates to the physics in question possibly even in a novel experimentally unexplored manner; it of course goes without saying that this has happened very often in the course of the history of physics. In other words, whether some piece of mathematics does or does not have any physical content is often a historical path dependent statement.

 

[quickAndLucky]

Most strongly interacting conformal field theories have no Lagrangian description. If you know the S-Matrix you have completely specified the theory and there isn't a need for a Lagrangian or action. If you can construct realistic field theories without the PLA you might argue that the PLA is not "fundamental" and thus lacks physical meaning.

 

[Dr. Courtney]

Most strongly interacting conformal field theories have no Lagrangian description. If you know the S-Matrix you have completely specified the theory and there isn't a need for a Lagrangian or action. If you can construct realistic field theories without the PLA you might argue that the PLA is not "fundamental" and thus lacks physical meaning.

Could one likewise argue, "If you can construct realistic theories of mechanics without Newtonian forces, one might argue that Newtonian forces are not "fundamental" and thus lack physical meaning?

If so, how are we going to teach intro physics?

 

[martinbn]

The PLA does not have physical meaning in the sense that it is not always unique. for example, the (classical) Einstein field equations can be derived from several different actions.

I have always wondered why physicists prefer actions/Lagrangians over the equations! At the end of the day arn't the equations that matter?

 

[Dr. Courtney]

The PLA does not have physical meaning in the sense that it is not always unique. for example, the (classical) Einstein field equations can be derived from several different actions.

Potentials are not unique either. Does this mean that they do not have a physical meaning?

Physical meaning comes from the predictions made by a model. Being the only model to make those predictions is not a requirement for physical meaning.

 

[sophiecentaur]

Could one likewise argue, "If you can construct realistic theories of mechanics without Newtonian forces, one might argue that Newtonian forces are not "fundamental" and thus lack physical meaning?

If so, how are we going to teach intro physics?

But, just imagine: someone might invent an entirely different way of describing 'concrete' events in a non Newtonian way which would make sense (perhaps even better sense) to a class of eleven year olds. Bigger paradigm changes have taken place in the past - what a thought!

 

[Dr. Courtney]

But, just imagine: someone might invent an entirely different way of describing 'concrete' events in a non Newtonian way which would make sense (perhaps even better sense) to a class of eleven year olds. Bigger paradigm changes have taken place in the past - what a thought!

You missed the point. I am disproving the notion that alternate models demonstrate that a given model is not physical. If one accepts the notion that an accurate model needs to be unique to have a physical meaning, then as soon as a phenomenon has two equally accurate models, then neither one of them has a physical meaning. (Or picking which one has the physical meaning is a subjective beauty contest rather than an objective scientific endeavor.) "Making sense" to eleven year olds is a subjective criteria. Paradigm shifts in science come from making better predictions, not by making the same predictions in a way that is more aesthetically pleasing or by making more sense to eleven year olds.

 

[anorlunda]

I am disproving the notion that alternate models demonstrate that a given model is not physical. If one accepts the notion that an accurate model needs to be unique to have a physical meaning, then as soon as a phenomenon has two equally accurate models, then neither one of them has a physical meaning.

That can be problematical. Are Newton/Lagrangian/Hamiltonian approaches different models or different ways to express the same model? Can we make a precise definition of "model?"

 

[Dr. Courtney]

That can be problematical. Are Newton/Lagrangian/Hamiltonian approaches different models or different ways to express the same model? Can we make a precise definition of "model?"

I view them as different models which happen to be completely equivalent in terms of testable predictions. (As are matrix and wave quantum mechanics). Newtonian mechanics knows nothing of kinetic energy, potential energy, or action. Does the introduction of kinetic energy, potential energy, and action in later formulations mean forces have no physical meaning? Or do the forces in Newtonian mechanics mean the kinetic energy, potential energy, and action have no meaning in later formulations?

A scientific model is a set of principles that can be used to make testable predictions.

Whether we realize that two models are equivalent does not change their validity or physical meaning. Their physical meaning and validity rest in their ability to make accurate predictions about experiments, not in their uniqueness from other models (either generally, or for specific experiments in question).

 

[Auto-Didact]

A model is any mathematical description of some phenomenon, usually capturable by some set of differential equations or something mathematically equivalent to that.

The mathematics of Newtonian, Lagrangian and Hamiltonian mechanics are different physical theories but deeply interrelated mathematical procedures, wherein which calculations of a model of the same phenomenon based on these procedures will always lead to the same answers, i.e. the physical theories are in some sense mathematically equivalent. This would be mysterious were it not that they are essentially different aspects of the same theory, namely mechanics, and that the mathematics of mechanics has all these different properties which can each be seperately exploited and further understood independently on their own terms.

The physical content however, i.e. what is intrinsically psychologically implied to the physicist by the mathematics regarding the phenomenon and other analogous phenomenon, of the three approaches are quite distinct and these distinctions can themselves suggest different ways of modifying concepts of the theory in order to generalize to some grander theory; in theoretical science, these modifications and generalizations can lead to very different kinds of theories, which aren't necessarily equivalent anymore to some other theories after such a modification is made compared to the situation before the modification was made where the theories were equivalent.

Feynman, having as usual already thought about these issues extensively, explains this marvelously in a matter of minutes here:

 

[atyy]

Potentials are not unique either. Does this mean that they do not have a physical meaning?

Physical meaning comes from the predictions made by a model. Being the only model to make those predictions is not a requirement for physical meaning.

Yes, that's exactly what I was thinking of. It is quite conventional to say within classsical electrostatics that potentials themselves are not physically meaningful, because they are not unique. Rather it is the potential difference that is physically meaningful.

One can choose to use other language, but this choice is very reasonable.

 

[sophiecentaur]

I am disproving the notion that alternate models demonstrate that a given model is not physical.

Paradigm shifts in science come from making better predictions, not by making the same predictions in a way that is more aesthetically pleasing or by making more sense to eleven year olds.

I was rather assuming that the necessity of Physicality would apply to individuals with only Concrete Cognitive skills. From this

If so, how are we going to teach intro physics?

I thought you were thinking along the same lines.
My view of the term 'physical' is that an idea is readily accepted in terms of already familiar and very concrete terms. By definition, a Physical explanation excludes factors that are new to the learner. One has only to examine one's own appreciation of Physics (and a lot of other stuff, of course) to realize that 'Physical' is mostly synonymous with 'familiar'.
Physical descriptions include "Nature abhors a vacuum" and 'the music of the spheres' and we have advanced beyond them by using notions of pressure and gravitational laws which would have been very non-physical at one time.

 

[Dr. Courtney]

Another example: the choice of a coordinate system is part of a physical model from the earliest physics courses. Dropping a rock from tower. Is y = 0 m the starting position or the ending position? The choice impacts the model, and while the two models are equivalent (make the same predictions), they are different models.

At some point (early, I hope) those who practice physics realize that there are no right or wrong choices of coordinate systems, though the math is sometimes easier or harder as a result of our choices. But the fact that the choice of coordinate system is not _unique_ does not mean that the choice of coordinate system (and the resulting mathematical models) do not have physical meaning. Of course, the choice of coordinate system has a physical meaning.

 

[atyy]

Another example: the choice of a coordinate system is part of a physical model from the earliest physics courses. Dropping a rock from tower. Is y = 0 m the starting position or the ending position? The choice impacts the model, and while the two models are equivalent (make the same predictions), they are different models.

At some point (early, I hope) those who practice physics realize that there are no right or wrong choices of coordinate systems, though the math is sometimes easier or harder as a result of our choices. But the fact that the choice of coordinate system is not _unique_ does not mean that the choice of coordinate system (and the resulting mathematical models) do not have physical meaning. Of course, the choice of coordinate system has a physical meaning.

Yes, that too is another example where it is quite conventional to say that coordinates are physically meaningless. So in the same sense that potentials and coordinates are meaningless, a non-unique action is physically meaningless. It is the underlying invariant geometric object that is physically meaningful.

As before, one can choose different language, but this is quite conventional.

 

[Orodruin]

Yes, that too is another example where it is quite conventional to say that coordinates are physically meaningless. So in the same sense that potentials and coordinates are meaningless, a non-unique action is physically meaningless. It is the underlying invariant geometric object that is physically meaningful.

As before, one can choose different language, but this is quite conventional.

You can get rid of the "physically meaningless" part of a potential by specifying equivalence classes of potentials. Same thing goes for gauge choices and actions. I would not consider an action meaningless, but rather say that it contains meaningful (and some meaningless) stuff.

 

[gmax137]

I'm still stuck on what "more fundamental" and "physically meaningful" mean to you guys. Are these the same as, "what is really happening"? As in, "what is going on inside the watch to make it work"?

I have pretty much resigned myself to the idea that that question is outside the purview. If you have, as Dr. Courtney describes, two models that give the same predictions, we might expect the 11th grader to ask, "yes so which model is true?" We don't expect the physicist to ask that question, because s/he knows there is no way to answer.

 

[Orodruin]

I'm still stuck on what "more fundamental" and "physically meaningful" mean to you guys. Are these the same as, "what is really happening"?

To me, definitely not. To me something "physically meaningful" is something that tells you something (a priori) measurable about a system.

 

[sophiecentaur]

Of course, the choice of coordinate system has a physical meaning.

To you, possibly but that may not apply to everyone. It's not an "of course" matter. The direction of +y is a small problem for someone new to the SUVAT equations but is the "physical" appreciation of Up, when a ball is thrown up or Down when it is released from a tower, really Physical? But, once you have started using SUVAT, you have left the physical behind and you just accept the outcome and then believe the answer to the equations will tell you where you are and how fast you're going. What happens then is that you turn the suvat results into a 'physical' idea in your head. You made a mental jump to increase your set of Physical things.
At the other end, if you use a Hamiltonian approach to a problem, it will deliver an up / down / how far answer too, which is also Physical It will tell you whether or not you will land safely on the Moon, which is also Physical. The Maths is just a black box which is divorced from the Physical 'till the other end of the process.

 

[Orodruin]

The choice impacts the model, and while the two models are equivalent (make the same predictions), they are different models.

I disagree. If two models make the same predictions, then they are by definition the same model. A model in physics is not the mathematics behind the predictions, it is the sum of its testable predictions. If two mathematical frameworks lead to the same testable predictions, then they are for all purposes (apart from computational) to be regarded as the same model. You cannot tell them apart by empirical observation.

 

[Auto-Didact]

I disagree. If two models make the same predictions, then they are by definition the same model. A model in physics is not the mathematics behind the predictions, it is the sum of its testable predictions. If two mathematical frameworks lead to the same testable predictions, then they are for all purposes (apart from computational) to be regarded as the same model. You cannot tell them apart by empirical observation.

This is simply incorrect, for it completely ignores:
a) that physics is an incomplete science, i.e. all statements made in physics are uncertain too some extent and in order to progress in theoretical physics we must actively remain uncertain by keeping open the possibility that any experimentally verified statement may merely be an approximation which can be corrected or improved by further theoretical guesswork and inductive reasoning,
b) that we are humans that think about theories in our minds and our minds simply are not perfect logical deductive machines, i.e. intuition is an actual part of physics reasoning and it is not reducible to deductive logic; this fact alone already ensures that theoretical physics clearly is not completely reducible to formal mathematics and in contrary to the opinion of many scientists and mathematicians, this is not a bug, it is a feature.

The point is that mathematical equivalence does not imply psychological equivalence. Feynman adresses this exact point in the video I posted earlier, but here it is again in text:

Suppose you have two theories, A and B, which look completely different psychologically, with different ideas in them and so on, but that all the consequences that are computed from each are exactly the same, and both agree with experiment. The two theories, although they sound different at the beginning, have all consequences the same, which is usually easy to prove mathematically by showing that the logic from A and B will always give corresponding consequences. Suppose we have two such theories, how are we going to decide which one is right? There is no way by science, because they both agree with experiment to the same extent. So two theories, although they may have deeply different ideas behind them, may be mathematically identical, and then there is no scientific way to distinguish them.

However, for psychological reasons, in order to guess new theories, these two things may be very far from equivalent, because one gives a man different ideas from the other. By putting the theory in a certain kind of framework you get an idea of what to change. There will be something, for instance, in theory A that talks about something, and you will say, ‘I’ll change that idea in here’. But to find out what the corresponding thing is that you are going to change in B may be very complicated it may not be a simple idea at all. In other words, although they are identical before they are changed, there are certain ways of changing one which looks natural which will not look natural in the other. Therefore psychologically we must keep all the theories in our heads, and every theoretical physicist who is any good knows six or seven different theoretical representations for exactly the same physics. He knows that they are all equivalent, and that nobody is ever going to be able to decide which one is right at that level, but he keeps them in his head, hoping that they will give him different ideas for guessing.

 

[cosmik debris]

I disagree. If two models make the same predictions, then they are by definition the same model. A model in physics is not the mathematics behind the predictions, it is the sum of its testable predictions. If two mathematical frameworks lead to the same testable predictions, then they are for all purposes (apart from computational) to be regarded as the same model. You cannot tell them apart by empirical observation.

I'm not sure I agree with this. A model is a story with its associated mathematical description. I might model relativity using the usual Einstein method or use the LET (Lorentz Ether Theory) model. The story and the mathematics may be different different but the predictions of each model are the same.

Cheers

 

[quickAndLucky]

Could one likewise argue, "If you can construct realistic theories of mechanics without Newtonian forces, one might argue that Newtonian forces are not "fundamental" and thus lack physical meaning?

If so, how are we going to teach intro physics?

If there were some realistic theories of mechanics that could not be represented using forces then I would argue that forces might not be fundamental. But even if forces weren't fundamental they could still be practical to use in in most cases and definitely worth teaching.

 

[Dr. Courtney]

Of course, the choice of coordinate system has a physical meaning.

To you, possibly but that may not apply to everyone. It's not an "of course" matter.

Perhaps some may disagree, but I think most would agree that a heliocentric model of the solar system has a different physical meaning than a geocentric model.

Further a choice of a coordinate system in which a system is separable gives much more insight that one in which the motion cannot be separated. Take the hydrogen atom in an electric field. Sure, at the end of long and difficult calculations, solving the problem in spherical coordinates may give the same answer as solving it in a coordinate system where the system separates. But a coordinate system in which the system separates also gives the very physical insight that the system is separable. Separation of variable and the associated constants of motion do have physical meanings.

As a teacher, I've been very successful convincing students that models in different coordinate systems are equivalent in that they make the same testable predictions. But it's a hard sell that a model that requires calculus is the same as a model that can be solved with only algebra or that a model that requires numerical integration on a computer is the same as a model that can be solved in a few lines of pencil and paper. A clever choice of coordinate system is part of the model insofar as it is often the difference between whether or not a problem is solvable with the available tools. (Likewise, a clever choice of a basis set in atomic and molecular physics is often the difference between whether a problem is solvable with the available computing power.)

I disagree. If two models make the same predictions, then they are by definition the same model. A model in physics is not the mathematics behind the predictions, it is the sum of its testable predictions. If two mathematical frameworks lead to the same testable predictions, then they are for all purposes (apart from computational) to be regarded as the same model. You cannot tell them apart by empirical observation.

That's certainly a possible definition of model, but I'd prefer to say that the two models were equivalent, but not the same. Consider the hydrogen atom. It is separable in several coordinate systems, including spherical, parabolic, and semi-parabolic. But add a uniform electric field (Stark effect), and not it is only separable in parabolic and semi-parabolic coordinates. Are two models really the same if one handles a specific perturbation much easier than another?