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The Meaning of the Concept of Potential in Mechanics (and in Physics)

If someone knows of books/articles dealing with the meaning of the concept of potential in physics (or concerning the physical bases underlying the energy methods of mechanics) then I would very much appreciate getting to know about these.

Please note, when I say physical bases, I mean physical bases---not "simpler/prior mathematical notions/procedures, very easy to work out." Thus, my query is for material that is primarily conceptual, not mathematical. (As an aside: Mathematical material on this topic is so easy to get that, speaking metaphorically, a stone's throw would yield a dozen references if not 1200. ... But I was talking about treatment that is not exclusively mathematical. Essentially, a counterbalance to Lagrange is what I was looking for.)

Also note, by potential, I do not mean the limited context of electromagnetism (EM) alone. Indeed, if you ask me, energy methods are far more valuable in mechanics than in EM primarily because the (statically) indeterminate case is so easy to run into, in mechanics. The momentum approach isn't, therefore, most convenient.   

I have already browsed through Lanczos (The Variational Principles of Mechanics) and find it helpful. Just the right sort of book, even though if I were to have the material to write this book, I wouldn't present it in the order that he does. ... Anyway, apart from this book, is there any other source? That's the question I have here.

I might as well mention here that for my purpose here, Goldstein (Classical Mechanics) has been a big let down (both in terms of the contents as well as their ordering) and so has been Weinstok (Calculus of Variations). I remember having browsed very rapidly through Morse and Feschback a few years back, but without finding anything directly useful in this context.

So, there. Any indicators/links other than Lanczos would be very much appreciated. If there aren't any, I guess I might myself write up a research article on this topic.

Thanks in advance for any links/references.


kaushik das's picture

I found the following book interesting. I hope that you will find it interesting too.

Simitses G J 1990 Dynamic Stability of Suddenly Loaded Structures (New York: Springer-Verlag)

Dear Kaushik,

Since Google books doesn't have a preview or list the contents of this book, could you pl. drop a line as to how the book succeeds in conveying the physics of potential? Thanks in advance.

Otherwise, nice to get to know you and your interests, but then, as it happens, even my interests ultimately have only a finite intellectual bandwidth! ...



Serdar Goktepe's picture


Take a look at the books

The thermomechanics of nonlinear irreversible behaviors: An introduction by Maugin and

Foundations of Potential Theory by Kellogg at Google books.

The latter might not be the one you are looking for. 




Dear Serdar,

Thanks. Running through the table of contents, the second one (by Kellogg) seems to be in line with what I was looking for.



m_rahman's picture


I am not very clear about what concretely you are looking for. Anyway, you might wish to consult the book "Elements of Natural Philosophy" by Thomson and Tait (especially articles 500 t0 512).

Hope this is of help.

M. Rahman

Dear Rahman,

Thanks for pointing out what I anticipate to be a gem of a book. (I've just downloaded it off, and can't wait to go through it.) I hope it has some comments on how they were thinking about CoV and potential theory about a century back---esp., how Lord Kelvin was looking at it!

Very difficult to get a handle on what I am looking for concretely (unless I begin to write down something on it), but here are a few (absolutely off-the-cuff) indicators:

(1) A XII standard kid learns that energy is of fundamental importance and a unifying concept in physics, and that there are two forms of it in mechanics---kinetic and potential. (Probably 90%+ of them would recall these as 1/2 mv^2 and mgh, respectively, for whatever it is that the formulae are worth).

This same kid comes to engineering, does a first year course on applied mechanics (of rigid bodies), and gets to know about the principle of virtual work.

Right next year, without any prior intro, we tell him that energy methods are of fundamental importance in mechanics of solids and structures. The examples taken are linear whereby it's difficult to believe that a more complicated procedure helps. Anyway, he takes you on trust, and begins to wonder: Potential is what? Voltage, mgh, "bring" a charge "from" infinity and "measure" work done, etc... And, amazingly, it works in solid mechanics; how?

Can you point out anything which tells him the how?

Ok, if not him, at least to his teacher?

The last was my question!

(ii) Another way to look at it (and I am trying to keep this entire post short and sweet, small and beautiful---and so, it's pretty vague):

I remember the following: Leibnitz had "vis viva" right in the 17th century, if not, at least by the early  18th. That's nothing but our kinetic energy gone off by a factor of two. Then, I remember having read in Enc. Brit. that potential theory began with Gauss's work. That puts it decades before the 19th century even began. And I know that Joule was still searching for a word like "energy" when he measured the mechanical equivalent of heat---roughly, mid-19th century. That was, decades *after* stress was known to be a tensor (in essential terms though not with modern notation. In any case, Helmholtz was still calling energy and force interchangeably around mid-19th century. ... Enough of it. (And I have omitted Laplace's work, etc.)

Enough evidence that there is a lot more to potential than potential energy. If so, why do we tell the student that energy is fundamental? In which sense do we mean it? And why don't we tell the student that there is a lot more about potential functions than what is meant by potential energy? Or the potential as used in the context of EM fields (a decidedly late 19th century development)?

Another point. How do you square off the basic ideas of Calculus of Variation and potential? After all, if there is not a potential function, can you even imagine CoV to apply? If so, what were Euler and Lagrange doing almost a century before the concept of energy got formulated?

Has someone integrated the different ideas by all these men coming from different periods of history (i.e. different basic ideas of physical abstractions) and offered a consistent account of just what the concept of potential mean?

That, generally, was the question I had, and that's why I found Lanczos to be the best among the available accounts. (How I wish Morris Kline had written to a greater detail about these matters.)

(iii) Finally, the most important question: How do we conceptually tie up (a) momentum conservation (Newton's) and energy conservation (Libniz') to each other. (Apart from just repeating that they are integrals in time and space; can we state something more physical about them?) (b) kinetic energy and potential energy (c) KE and potential functions as for vector fields (d) KE and PE and potential functions, as for tensor fields...


As to my comment in the original post above that I was looking for a counterbalance to Lagrange: 

It's obvious that this whole issue reeks of confusions. And, most all of such confusions wouldn't have arisen in the first place if only people had at least attempted to identify (and then, to teach) the physical meaning of the basic concepts they use in mathematics. Instead, people have been acting as if replacing physics by mathematics takes care of it all. Lagrange is a particularly glaring example of that trend in physics and mechanics: When he published his famous treatise, he boasted of the fact that not a single diagram was to be found in his book. But for the prestige that gets accorded to such statements, esp. in academia, the boast, actually, is worthless.

(A visible analog from the programming world is what they write for the obfuscated programming contests---it too has no conceptual isolations, has symbols literally overloaded with meaning (particularly, that specific values stand also for condensed algorithmic logic), has poor layout, and so on and so forth. And, no software producer takes them seriously. But when you come to academia and physics and mechanics research, exactly opposite happens! ... The matter is esp. unfortunate because CoV is crucial to all of fundamental physics---after all, QED is built that way!)


Given the confusion, the relative paucity of references, and the fundamental importance of the concepts, one can be sure that at least one research article is definitely called for. That is, if a good source doesn't exist already. And (to the best of my resources and knowledge) I am afraid, it doesn't.




Here is a relevant passage I found in Chapter 4 of George W. Collin, II,'s book: "The Foundations of Celestial Mechanics" (book available for free from his site here):

"We have seen how the solution of any classical mechanics problem is first one of determining the equations of motion. These then must be solved in order to find the motion of the particles that comprise the mechanical system. In the previous chapter, we developed the formalisms of Lagrange and Hamilton, which enable the equations of motion to be written down as either a set of n second order differential equations or 2n first order differential equations depending on whether one chooses the formalism of Lagrange or Hamilton. However, in the methods developed, the Hamiltonian required knowledge of the Lagrangian, and the correct formulation of the Lagrangian required knowledge of the potential through which the system of particles moves. Thus, the development of the equations of motion has been reduced to the determination of the potential; the rest is manipulation. In this way the more complicated vector equations of motion can be obtained from the far simpler concept of the scalar field of the potential."

[Italics mine.]

He speaks of classical mechanics, but one can also add quantum mechanics. Schrodinger's equation is formulated in terms of a Hamiltonian, and Feynman's QED is a path-integral (i.e. variational) approach.

Of course I do not necessarily agree with every position possibly implied in Collin's above passage. For instance, I don't think that the concept of a scalar potential field itself is as simple as he seems to imply. Nope, it isn't, even though most all physicists treat it as if it were simple enough that it could be taken as a God-given. (The concept does lead to simpler mathematical manipulations in certain types of problems; this does not make the concept itself any simpler; it certainly isn't simple enough to be axiomatic.)

But yes, the passage does bring out the crucial fundamental importance that the concept of potential has in fundamental physics, and therefore, also in all of practical engineering.

Coming to practical engineering, for example, the only physical basis ever put forth for the FE procedure (despite all the inaccuracies and discontinuities in gradient fields) is that it involves variational principles. In other words, potential theory. 

But, can any physicist/FE analyst tell a succint physical account of what the concept of potential itself means---what are its physical referents, how it comes into being, what kind of physical (as in contrast to philosophical) context it assumes, from what other referents (or, more generally, alternative viewpoints) is the idea best contrasted? Et cetera.

If you cannot tell the physical basis, then, why should anyone take FE seriously? How different is your position from those pre-Snell physicists who could tabulate the variations of angles of refraction and perhaps also do a polynomial fit to the data but not come up with the proper law: a ratio of the sines. Remember, we give it a sines-fit and not a polynomial fit precisely because sines lead to an explanation (in terms of wave theory) that is consistent with all the wider conceptual context. Sans such a physical conceptual basis, both polynomial and sines-ratio fits would look equally nice, equally acceptable---and decidedly anti-cognition!

So, not just numerical accuracy (and error and convergence measures) but also conceptual basis and integrations are crucially important concerns when it comes to FE---no matter how flippantly today's researchers treat the latter (and yesterday's researchers had been, right since the inception of the method).

I do think that it is possible to answer the above questions I raised concerning the physical meaning of potential, and that I have something (probably new and in any case definitely independently arrived at ideas) to tell about these. ... Feel free to suggest me a proper avenue to express myself :)

(In the meanwhile, I would be offline for a few days.)


PS: (Suggest, only if you have your actual affiliation(s)/credentials put up here at your iMechanica profile---i.e. are real.) 

PPS: (As an aside, dismissing anything non-mathematically-symbolic as "philosophical" might be momentarily convenient, but often serves only to expose the speaker's ignorance. On more than one occasion, I could easily tell how two different supposedly "philosophical" explanations actually lead to two different kinds of mathematics---but the listener/reader had already made up his mind. Both philosophy and physics have their own place---not just mathematics, number-crunching, and smoothly regurgitating symbols at a short notice!)


Just to add,

I would really appreciate receiving suggestions as to what journal I might send my abovementioned paper... 

The one comment that I have heard again and again regarding my PhD research was actually a question. The question didn't touch upon any of the contents of my paper(s). It instead was: Why did I publish my results only at conferences but not in journals. (An informal explanation document for my choices is available on my blog; I will insert a link to it here later on.)

But more interestingly, and more seriously, I now find it strange that the same/similar people don't at all respond if I turn back and ask them to suggest a journal or two, right in advance!... They don't recommend me even a single journal...

Is the whole idea just putting me down, or what? [Not impossible.] Or is it that people know that there does not exist a single reputed journal that would accept such an article... Is that the reason? [Not impossible, again!]

Why this curious absence of replies?

I am serious. Since the topic like above is so rare (else, one would have found references already), it is next to impossible for one to get to know the appropriate journals... Any suggestions?

Elsevier representatives? Other journal publishing companies/organizations? Open source publishing and/or champions? Others?

Or is the whole idea to deny a chance for any such papers to gain respectability, through the device of looking the other way? Is that the whole idea/game?

If you are truly interested in publishing such a paper, post a preprint on here and people may

suggest an appropriate journal. Of course you run the  risk of somebody stealing your ideas but I

think that risk is quite slim. Just my 2 cents.




If you consider yourself to be one of the "people," then feel free to advise me a suitable journal by going through any one of the paper numbers 1 through 5 to be found at my Web site here. ... As most every reputed person here seems to know (better than me), there always is enough of extra material that a journal article (or more of them) can be written even if a conference publication has been published already.

If not, then please state, very briefly, the reason why you consider the risk to be both slim and by indirect indication worth the rewards if the man to lose the basic idea is me and the people involved are other than you and one of the countries to benefit from those ideas is that of your education and current affiliation.

I don't mind receiving your reply on this aside. With that said, I would also like to reiterate that I would like the focus of this thread to go back to the original intent explained in adequate detail via my multiple postings above.

Your work is a bit outside my area of expertise so I don't really feel comfortable suggesting a specific journal. However perhaps you can browse the American Physical Society offerings and see if any of those fit your topics?

Also I don't know if your reply was meant to be argumentative but it sort of seemed that way. I was just trying to be helpful and you can take that for what it is worth. I have no intention of stealing your work as I little interest in the areas you are pursuing. I realize that you seem to have some sort  of vendetta against American schools but I don't think it is fair or reasonable for you to hold this above the head of every single member of these institutions just because you had  bad experiences with a few of them.



APS... One consideration: Going by statistical measures (such as the relative frequency) concerning what all they do think is fit to publish, they wouldn't be my first choice. Another consideration: But, they are prestigious/reputed, and are published by/in a superpower. Putting it (and many other things) together: They seem OK. Overall: My choice would be (i) Nature, (ii) Science, (iii) APS, strictly in that order. Another noteworthy is the Foundations of Physics, esp. since the time that I noticed that Gerard ’t Hooft's team has allowed both Shahriar Afshar's and Hrvoje Nikolić's articles to get published in their journal. So, to me, the best possibility seems to be a combination of Nature + Foundations articles.... For some of the articles, not all. For instance, the paper no. 1 is relevant to both quantum and continuum mechanics. I would have appreciated if you were to notice that. (Any author, I suppose, is delighted if he is understood.) And replies are awaited for those articles too... Apart from the presently proposed article (on the physical meaning of the concept of potential---a topic of simultaneous relevance to all: physics, mechanics and engineering)

I will not address everything else you raise but...

Argumentative? Yes. Aren't we all supposed to be---i.e. in a rational way? Is not this one important way that Reason operates? "I have no intention of stealing your work" See above, e.g. "the people involved are other than you." Also, please notice, they don't yet have a watertight 1:1 distribution mechanism in journals---it always is 1:many.... "I realize... vendetta against American..." FYI, your "realization" isn't a real-ization.

Overall, I would appreciate it if people (esp. those in the USA) read my posts more carefully. And if their replies were more specific---i.e. if they took care to make it appear below the specific post to which the replies referred. (I expect to see some significant number of replies that violate this advise to appear in iMechanica in near future---Internet!)

I am adding one more post below to reemphasise the topic of the thread---to take its emphasis away from easy hints/allegations (like the said "vendetta"). ... I will continue doing so as many times as is necessary!

This thread is really about the following:

1. The original post:
The Meaning of the Concept of Potential in Mechanics (and in Physics)

2. A Clarification

3. Some additional clarifications

4. An additional explanation, by way of a contrast to the movement in this simultaneously active thread.

Replies from others, say those from people from the Top 3 or Top 5 American schools are, I believe, both merited and awaited. Needless to add, well-thought out replies from any quarters are encouraged. ... But then, there is something to be said for the Top 3 or Top 5 schools, isn't it? For instance, when these schools award tenures, don't they consider the candidate's Top 3 / Top 5 education/experience?... So, there is something to be said about the Top 3/ Top 5 schools... Why do they refrain from commenting here---in a thread concerning things such as what this one does. (At this point, please follow the links given in this comment).


I've long searched for how to reconcile the confusion with potential energy, virtual work, and so on.  I first learned of CoV and the hanging chain problem 8 years ago in an undergraduate course on theoretical mechanics at Penn State University, where it was presented in explicit Cartesian form (i.e. y(x)).  As far as I can recall, the next problem on the board was the uniformly loaded chain, with the claim that minimizing length squared was the same as minimizing length.

It turned out this wasn't true (, and I set out to find the source of the problem.  Since the shape is a solution of the equations of equilibrium, I used a free body diagram to derive parametric equilibrium equations.  This lead to finding the functional to minimize to obtain the correct equilibrium equations (ref. master thesis on formfinding, and, thus, the correct shape.

Since then, I have figured out how to extend the technique to the form-finding of uniformly stressed membranes of constant thickness.  The geometric construction of the functional whose minimization yields the equations of static equilibrium is illustrated in the extended abstract of my presentation at (ref. eCommons, structural morphology).  In principle, a similar construction should suffice to derive a potential energy functional based on a linearized strain tensor.

My research seems to suggest that potential does not have a physical meaning and is little more than a mathematical trick to satisfy equilibrium in the form of a minimization problem.  Perhaps I'm mistaken (cf. stability, snap-through buckling, thermomechanical coupling, plasticity, et Al).

Hi Dave,

Thanks for your comment! (And for the link to that nice n funny problem...)

The view of potential that you express above is not only the predominant one, in my (limited) reading thus far, it is the only one that has ever been expressed explicitly---if at all. Many authors keep it only implicit. (Calling potential an attribute or measure of a field and then defining the field as a primarily mathematical idea doesn't take us any farther either, and that's another popular variation on the same basic idea...) 

Of course, I disagree with this view, and am trying to formulate a new one... A physical one. But it's a complicated matter, and guess I will be able to do it only one step at a time. But the basic idea is clear: If you have five units of something, there are five objects of that kind actually existing in physical reality. Five doesn't exist in reality; five mangoes and five bananas and five people (and so on) do. One only has to identify what. Similarly, pressure field doesn't exist apart from the fluid that has that property. Similarly, potential...

More, later. 


arash_yavari's picture

Dear Ajit:

I'm not sure if this is directly related to your question but you may want to have a look at the following beautifully written book by Richard Feynman: The Character of Physical Law.

I agree with you that the concept of potential in mechanics is not completely clear. As others have already pointed out, one wonders how these seemingly different energy-related concepts, e.g. balance of energy, principle of virtual work, etc. are related. The other issue is starting with linearized elasticity and it that case the connection with the nonlinear theory can be easily lost.

Writing an expository article is a good idea, in my opinion. You may want to consider American Journal of Physics, which is education oriented. You may also want to look at some related articles there.



Dear Arash,

Thank you very much for all your points, so well put, and sorry for the delay in my reply...

I had just skimmed through Character of Physical Law in a bookstore many years back and it is time to carefully go through it now. ... I sure will pick it up in the immediate future and let you (and all iMechanicians) know what I think of it... But, gathering from what he says in his special/extra lecture on the action principle in his famous CalTech Lectures book, and relating it to the whats and hows of his QED theory (given in his small book on QED for the layman), I already know that I am going to disagree with him... Indeed, I think Feynman never did advocate a physical idea for potential (and any of the related concepts). For this reason, permitting myself to paraphrase Newton (who paraphrased Aristotle), let me say: "Amicus Feynman [, Green, Hamilton, Lagrange, Euler, Leibniz, Fermat et al], sed magis amica Veritas."

I think it should be easier to get the idea of potential right, first for vector fields and then the stress/strain tensor ones... I think it won't be anytime soon that I would be able to do justice, in a detailed way, to nonlinearity, esp. in the tensor fields. However, let me also hasten to add that, from what I anticipate now, nonlinearity---whether constitutive or differential---wouldn't be as core an issue as reversibility and dissipation would be. (Conservative-ness, the way I currently understand it, is primarily important out of the reversibility i.e. non-dissipation considerations; not the other way around.) Further, within the two nonlinearities, when it comes to identifying the basic physical meaning, differential nonlinearity would be far more challenging than the constitutive. I guess I won't be able to cover all such issues right in the first article.

AJP is a very fitting suggestion. Edwin Taylor (MIT) has written a series articles advocating the action viewpoint in it. So, especially for the idea of potential in the context of vector fields, AJP should be a fitting place to publish.

But how about elasticity? That was my concern. It was for this topic, that I was looking forward to suggestions. Which of the specifically mechanics-/elasticity-/solid mechanics  journals would be fitting. Due to rarity of such publications (if any) it is here that I am not at all sure which journal(s) would be best fitting... As such, I still solicit advice on this matter (esp., from the people from the Top 3 schools---whatever the reader's definition of it.)

Thanks, once again, and regards,


arash_yavari's picture

Dear Ajit:

I would say both ZAMP and Journal of Elasticity would be appropriate.



Dear Arash,

Thank you so much for making specific suggestions. I will keep these in mind when I come to submitting the article(s).



Temesgen Markos's picture

Hi Ajit,

The idea of a potentials and energy had bothered me for some time. What happens in most texts is people would derive a potential for a system in a conservative force field, starting with Newton's laws, and then before you know they would be using it in cases where the forces are not conservative. You will be left asking "but why?".

To quote from Marion and Thornton's classical dynamics text, "We have become so enamored with these conservation theorems that we have elevated them to the status of laws and we have come to insist that they be valid in any physical theory, even those that apply to situations in which Newtonian mechanics is not valid, as, for example, in the interaction of moving charges in quantum-mechanical systems. We do not actually have conseration laws in such situations, but rather conservation postulates that we force on the theory. ... We therefore extend the usual concept of energy ... to satisfy our preconceived notion that energy must be conserved. This may seem an arbitrary and drastic step to take, but nothing, it is said, succeeds as success, and these conservation "laws" have been the most successful set of principles in physics.

This is the sort of honesty, I think, should be presented to students on their first encounter with potentials and energy methods.  

Dear Temesgen,

Thanks for providing the excerpts and sharing your thoughts. However, I suppose I have some additions and disagreements to make.

I have not read Marion and Thornton. After browsing through the description and Table of Contents at, I hazard some guesses here. ... I think that as an engineering/applied mechanician, the point that you would want to make concerning non-conservative forces and what the authors state in the above passage are, or rather, should be, two orthogonal issues.

In the excerpts you quote, the authors seem only to be emphasizing the notion that conservation is a postulate... Now, asking myself why modern American physics authors like them might want to emphasize this "postulate-ness" in the context of the basic/fundamental classical physics (the topics which they cover), the only plausible answer or guess that I can think of is that their intellectual position must have proceeded from the influence of certain irrational philosophical schools such as the logical positivism. In any event, I would be very surprised if these authors ever meant to challenge, e.g., Noether's theorem (see here and here).

In contrast, the sort of situations which you might have in mind, the sort of situations which we engineers/applied mechanicians often run into (right in our first two years of undergraduate studies) are those wherein the phenomenon of non-conservation has to be taken as a fundamental given (fundamental, in applied sciences). Such situations are, for example, plasticity and fracture in mechanics of materials, turbulent loss of energy in fluids, etc. Here, by "a fundamental given," I mean that both the phenomena we concern ourselves with and our working abstractions themselves are such that non-conservative fields must be treated almost at par with the conservative fields. The direct objects of our study are *basically* messy (at the engineering level of abstractions) so to speak.

A situation parallel to the abovementioned does not arise in the fundamental theories of physics, on the other hand. ... Think about it this way... Despite the Second Law of Thermodynamics, QM is famously known to be a linear theory. How come? Essentially, the only answer possible is this: During the division of work (i.e. the choices exercised during scoping of different abstractions to have) physicists choose to keep the fundamental theory (QM) simple, and lump the effects of non-conservation and all into that one single issue (whose more complex manifestations they happily ignore most of the times in their own work): boundary conditions.

... Now, if you ask me, this is a sound choice, speaking epistemologically. To put it in direct terms: Who would want to bother with friction in fundamental QM? And why on earth? Isn't it better handled at a higher level, say, via molecular interactions such as the van der Waals forces and all? ... That's the nature of the subdivision of work (and abstractions) involved. I am perfectly fine with such a conceptual division. It affords the required economy of thought in our basic theories.

So, overall I do think that it's a sound policy to treat the physical facts corresponding with Noether's mathematical theorem as fundamental truths rather than as mere postulates.

On the other hand, yes, when mechanics are once again being formulated and used at the macro-scale, then friction has to be treated more or less at a level that is just one level up from the most fundamental level (the way, e.g., Newton treated this phenomenon of friction). This, too, is fine.

So, where is the real trouble? ... The trouble occurs when authors get so enamoured by the power of the energetic physics (the action principle, CoV, the Lagrangian, the Hamiltonian etc.) that they fail to circumscribe its scope. They fail to highlight the limited nature of its effectiveness while dealing with the situations such as the motion of a striker on a carrom board (or that of the real billiards balls on a real table, or the real cricket ball moving in air complete with frictional drag, etc.) The problem lies *there*. The problem lies in not mentioning the limitations of the energetic physics in treating *such* situations... But this issue is not at all well-served by characterizing conservation as a mere "postulate." Indeed, the real issue here has not even been addressed thereby at all!

Indeed, the way I think about it (and I still haven't reached any good level of clarity but since this is an informal forum with just the right sort of informality to it, I may share this), I think that conservation is all about going from the local to the global... (If you don't get what I mean by that, fine... Let it be. I am not very clear on this notion either! But I hope to fix up my ideas in future.)

All in all, IMHO, conservation is a physical fact, not a postulate---but only in the basic physical theory, not in the applied continuum mechanics wherein the limitations of the energetic approach must be brought out directly and highlighted appropriately.

One final point, a non-technical kind of an aside. On several such points of pedagogy, the position that I find most reasonable is that the author may perfectly be honest (and most times they indeed are) and yet he may unwittingly commit such errors (e.g. not delimiting abstractions correctly, etc.). So, I don't think it's an issue pertaining to honesty, primarily. It's not an issue of ethics, primiarily, but of epistemology. The issue rather is of an author's integrative links failing despite all good and honest intentions, due to the corruptive influences of a bad epistemology (i.e. fundamental philosophy) in the culture in general. Merely being honest won't be enough; people would also have to have good epistemological ideas being made available to them (and people must also have visible concrete examples wherein good epistemological ideas actually succeed in practice).

... Good comment, Temesgen, quite thought provocating... (And I am sure you will readily agree at least with this particular point of mine, won't you :) )

Martha J Lindeman's picture

Hi Ajit,

I found your blog because I also am working on understanding the physical reality of potentials, particularly within the context of how physicists' mental models have hindered the reconciliation of subdomains of physics. (My PhD work mathematically modeled how people create and process information, and for 20+ years I have consulted in technology projects where I created the user-interaction design by integrating the mental models of project stakeholders.)

On 9/2 you wrote, "I already know that I am going to disagree with him... Indeed, I think Feynman never did advocate a physical idea for potential (and any of the related concepts)."

There a long paragraph in Feynman's lecture on the vector potential (Lectures, II-15-12) that probably will surprise you.  There Feynman intensely complains that physicists "have repeatedly said that the vector potential had no direct physical significance" and thus they focused on only the magnetic and electric fields.  He argues that bias exists even though it was "obvious from the day it was written" that the "vector potential appears in the wave equation of quantum mechanics (called the Schroedinger equation)".   He closes his half-page paragraph with the sentence, "It is interesting that something like this can be around for thirty years but, because of certain prejudices of what is and is not significant, continues to be ignored." Thus Feynman does strongly advocate a physical idea for the vector potential.

Best wishes with your work!I would be interested in learning more about it.



Hello Martha,

1. Interesting comment about mental models... BTW, where did you do your PhD (dept/guide)? What kind of mathematics did you find suitable to study the problem you mention for your PhD?

2. Thanks for pointing out the excerpt from Feynman.

My reading on EM is far too limited as of today, but I am catching up. ... But anyway, it is limited enough that I can't immediately put his comment concerning those 30 years in any meaningful context. [Indeed, I haven't looked up the excerpt in Feynman Lectures after you mentioned it either! [If you would believe me: too busy to pick this topic up!!][Hope you don't find it insulting, but rather than delay the reply to this post indefinitely, I thought it would be OK to write one without looking up Feynman Lectures...]]

The main reason I don't think Feynman was very strong on advocating a physical idea for potential is because in his Lectures, he goes to great lengths to explain (and, uncharacteristically, to add to student pains) to emphasize the notion that EM theory is, in its essence, mathematical and abstract. He seems to go to extreme in that notion. He emphasizes, time and again, that there is no way anyone could establish a physical theory consisting of some concrete mechanism for it... He takes issue with the "gears" etc. "theory" of EM, where he seems out to impress the student that no physical interpretation can in principle be asked for. [An attempt like mine has to be seen as a failure of Feynman's considerable pursuasive skills.]

... Indeed, throughout Lectures v. II, he follows not an inductive but a deductive approach, first presenting the vector calculus identities and thereafter deducing every experimental fact (the laws empirically found by Coulomb, Ampere, Biot-Savart, Faraday et al.) from those identities, thus turning both the historical sequence and the only logically possible inductive development, right on its head.... For all his reputation, he has never been helpful in learning EM to me; Resnick & Halliday (or Stanton, or Zemansky, or...) were far superior (though they all too often fell short).

And, as far as I recollect from memory (without actually re-opening the v II of the Lectures, though I will do so soon), Feynman does not find it necessary to advance a physical theory even for the scalar potential let alone for the vector one. Please correct me on this count if I am wrong. That is, pl. show me the specific place where he explains a specific physical mechanism, operating in reality, to which the mathematical abstraction of the electrostatic potential corresponds. As far as I know, there is none of that sort in Feynman.

The reason Feynman cannot explain or assign a physical meaning for even the scalar potential is because he didn't have one. Otherwise, he was both talkative and bold enough to have said something about it during his long teaching career. And one probable reason he didn't have one could be because... Well, let me keep polemics based on mere guesses aside...

But still, since you touch on QM, let me add one final observation. If Feynman were to have a theory of the sort you say he did, he wouldn't talk (almost gleefully) about a "particle" that "arrives" to present "from" "future" and undergoes collision. But he does! Is this a respect for physical mechanisms? For a physical theory?

So, there.

Thanks for your compliments too (your next comment). Hope you will come back on both the above two points.




Martha J Lindeman's picture

Hi Ajit,

First, to answer your questions about my degree.  I received my PhD from Harvard University, 1985, as Martha J. Gordon.  My major general exam was in cognitive psychology (with a good-sized portion of the exam on issues in artificial intelligence), and also two minor general exams, one in psychobiology and one in language (specifically word creation and meaning).  My major advisor was William K. Estes, with Stephen Kosslyn, Steven Pinker, and Roger Brown for the minor generals.  (All four of them have their own Wikipedia pages if you want to know more. Bill's Wikipedia entry is very bare --also see

The mathematics for my research and dissertation varied depending on whether I was creating or evaluating models of human information processing.  I have not calculated eigenvectors and eigenvalues in years, but I did a lot of that back then -- one major issue in the comparison of the ability of the mathematical models (two of mine and some created by others) to predict the human data was whether the relationships among stimuli features were linear or non-linear.  I also looked at whether the relationship math was a function of the degree of similarity among category exemplars.

I had to write and code many of my statistical analyses because they included comparisons at the trial and group of trials levels rather than just across an entire experiment or set of human participants.  In addition, the level of precision I needed required that I replace some of the math in the UNIX kernal with code that provided more precision (and thus less rounding error). My 'makefiles' got really interesting sometimes!

Please note that I did not say Feynman had a theory that provided a physical meaning for the potentials.  Rather, I said he complained intensely that the importance of the potentials for physical reality had been ignored.

You note in your reply that Feynman starts with the potentials and then uses a deductive rather than an inductive approach to teach electromagnetism.  Feynman's starting with the potentials parallels both Maxwell's and Einstein's approaches, as opposed to those of Heaviside who called Maxwell's focus on the potential "metaphysics", "unmanageable", and "useless and treacherous".  Thus there has been some vitriolic descriptions of a focus on potentials that goes back much further than the 30 years Feynman complains about in the section I quoted.

If you have not read it, you might find Maxwell's definitions interesting.  He defined the vector potential that exists at any spatial point of the field in two ways (Treatise Vol. 2: Article 604):  (1) “the electromagnetic momentum at that point” as “the time-integral of the electromagnetic intensity which would be produced at that point by the sudden removal of all the currents from the field, and (2) as identical with the vector-potential of magnetic induction.  So in essence he redefined the vector potential of magnetic induction as electromagnetic momentum.

Feynman (Section II-15-4) defines a "real" field as "a set of numbers we specify in such a way that what happens at a point depends only on the numbers at that point" (his italics). He also notes that the gauge invariance of the A field is irrelevant to whether or not it is real.  The contrast he then sets up is "whether the vector potential is a proper 'real' field for describing magnetic effects, or whether it is just a useful mathematical tool." He then makes some remarks on the mathematical usefulness of the vector potential A. He then states that he introduced the vector potential A "because it does have an important physical significance." (his italics) The next Section (15-5) discuss how the vector potential A relates to quantum mechanics, and it contains the paragraph I quoted in my first post.

From everything I have read (including, if I remember correctly, Feynman's statement that Einstein's field equation for general relativity can be derived from the potentials), Feynman appears to follow Maxwell's belief that the potential "may even be called the fundamental quantity in the theory of electromagnetism."  (Treatise Vol 2: Article 540)

Hope this is useful!


P.S. I am still searching for where I read Feynman's statement about the general relativity field equation.  I can only remember that it was a copy on white paper, which makes me think it was in an article rather than in a book. Does anyone know that source?


Hi Martha,

1. Wow!  ... On second thoughts, make it wow^n... I call myself a consultant in computational mechanics, but haven't used any "infinite" (or indefinite- or high-) precision numerical library yet, and you, as a *psychology* student, had used it some 20+ years ago!... Enough to give us engineers an inferiority complex (is that the term here?) ... I might use a higher precision library when I come to using those iterative solution or computational geometry libraries in near future.

2. I think it will be quite some time before I develop all those informal but dense clusters of readings and ideas with which I would later on begin placing the past EM researchers' remarks in context.

3. But I can comment on Feynman's definition of what constitutes a field.

Feynman's definition of a field, too, is mathematical in nature---not physical. How come? Because, physically, there is no such a thing as a point phenomenon, whether you speak in reference to the mechanics of particles or of fields. What we routinely consider as a point property is nothing but actually a line, surface or volume property (a flux or a density) taken in the limit of the vanishing size for the defining element. (Many aspects related to this idea were nicely covered in a thread started by Rui Huang at iMechanica here.)

Another point. One must always remember that numbers are by themselves meaningless unless the context identifies, at least implicitly, the physical quantity of which these numbers are a measure, and the standard (i.e. physical units) being used. So, numbers must be separated from *physical* quantities. Seems mundane, but consider an implication, in reference to what Feynman says.

The "number" at a point might stand as a measure of something physical that has happened locally near that point. Or, perhaps, it might also represent an end-result of a physical interaction with something else existing at some other place (a second line, surface, or volume element...). Merely looking at the points and their numbers, you cannot tell which of these two possibilities exist. You must have (or implicitly assume) a *physical* theory which will allow you to assign a meaning to that number in one of those several possible ways. It could a stricly local phenomenon, or an integrated result of a transient propagational process (another possibility), or a result of an instantaneous action-at-a-distance (yet another possibility). Mere mathematical definition of "one number for each point" won't give you enough grounds to settle the question one way or the other. And that, precisely, is the broad pattern by which an overemphasis on mathematics serves only to erode physics. Sometimes, people can end up talking ridiculous things in a serious context---something "arrives" "from" future!

3. Finally, if potential is indeed a fundamental, how does it relate to the perceived concretes? What kind of physical mechanisms explain the link? This is a question for which, as you seem to agree, Feynman had no answer. 

4. Of course your comments are useful...

Also, it's nice getting to know you. ... I like people who are able to think about physics with modalities other than the routine "shut up and calculate!" approach....

I might add that visualization is one of the things I like. For me, it is a major way to understand a theory or to approach problems. ... I had written something about visualization (in a different context) on my personal blog last year (see here). (In the '80s, I used to paint, too!) The point with visualization isn't that it's good because it's the right-brain thing to do (which I very much doubt anyway), or because Einstein did it, or because at perceptual level reality is apprehended visually, or because some philosopher said that the soul thinks in terms of images. None of these. My reason is different. When you begin to visualize, you can let go of the burden of the immediate calculations, and thereby, you can have your mental resources freed up to begin to appreciate the larger structure of the theory or the problem. ... If only Feynman were to realize that he can't visualize a point (a geometric entity with no extension), he wouldn't have stopped at definining field as point-phenomena. He would have gone further and explicitized the physics of potential... But, for a physical concept he regarded as fundamental, he stopped only at its mathematics. End result? (Even) He missed it---the physical meaning of potential!

5. Finally, let me point out a book involving EM and potential theory that I am sure you will find interesting. See here; I very much welcome your comments on that thread too!

Martha J Lindeman's picture

Thanks, Ajit, for your great response -- you made my day!  I have being exploring all of this on my own for so long that I had forgotten had much fun it is to talk with someone else about it!

I will follow the links and do the reading asap -- probably later this week.  I have 'taken off' from consulting work to work on what I am calling 'Quantum Interval Mechanics'.  There are very specific reasons for each of the three words.  Today I will only provide some background and briefly describe what I think about quantum points.

I think a summary of my relevant background would clarify the context in which I write. I mentioned my PhD work, but its context goes deeper.  Although the official title of one of my minor general exams was "language", it was a subset of psycholinguistics that focused on how people create and use word meanings  ( The other minor general exam could have been labeled several different names, such as psychobiology, behavioral neuroscience, or cognitive neuroscience (I would have to go back and look at the questions on my exam to know which label is best.  Just for fun, see My major focus on cognitive psychology pulled the minors and other things together to understand and model the cognitive processes of category formation, which is the foundational cognitive process.  Without category formation, every incoming perceptual stimuli would be novel (i.e., it would not relate to any previous experience) and there would be no learning!

My PhD work built on my undergrad work at Ohio State, when I came back as a sophomore after being out of school for 16 years.  My first attempt at college, in which I started a double major in math and physics, ended when my Grandmother became ill. I only had one semester of calculus and engineering physics, but it was enough to keep me interested in it since then.  When I came back to school in 1977, my honors advisor helped me arrange an "Interdisciplinary Honors Contract" as my major that let me pull together the courses I wanted to take in various domains into a coherent and approved course of study on the mind-body relationship.  In 1980 when I went to Harvard that was not a topic to be discussed or even considered, so I designed the three general exams to do it without telling anyone that was the ultimate focus.

I have also wandered into other fields to broaden my learning about reality. For example, I used the information gained in an undergrad part-time job as an editor of molecular-biology publications to write and win an undergraduate research grant in molecular biology.  I wanted to learn more about the fundamental categories of biology. (Unfortunately when I got to Argonne Labs for my internship, I discovered the mice I had planned to use did not have the genetic history I had been told, so they did not have the DNA I needed and that research was not completed.) To the other extreme, I wanted to study non-physical aspects of reality. Thus, I have a Certificate in Theological Studies from Bethel Seminary, where for one of my required papers I wrote on the physics of miracles using scripture and secular sources.  That paper is important only because it convinced me that the electromagnetic spectrum is the fundamental object of physical reality.

In summary, when I choose the words I write, I choose them to convey my unique thoughts in terms of the standard definitions.  However, my son has laughed and said at times I need a translator (because I attempt to be so precise Foot in mouth). I always appreciate reasoned feedback about my attempts to communicate, for both the words and their combined content.  Hopefully, we will have fun! Laughing

Now to my definition of a physical point, which is based on Elmore and Heald's Physics of Waves, p. 12ff. The physical reality of the electromagnetic spectrum is comprised of complex waves.  Any complex wave can be represented by a radius vector of length A that"rotates clockwise in time and counterclockwise in space." Physics is currently primarily concerned with the projection of that rotating vector on the real axis as the physical wave. Thus we deal differently with the 'imaginary' part of the wave because we define "i" as the square root of -1 rather than as a 90-degree rotation.  (Gauss said complex numbers should be called "lateral units", with positive numbers as "direct units" and negative numbers as "inverse units". Maor, e: The Story of a Number, p. 164)

Theoretical physics has struggled for more than a century to be free of the bias of human perceptual models--unfortunately, that bias and its related mental models still have an impact.  For example, I believe it was a perceptual bias that caused Minkowski to introduce complex numbers into Einstein's interval equation.  Consequently, Kip Thorne notes in Black Holes & Time Warps that workers in general relativity switch between a 4D and 3+1D model of spacetime depending the type of problem.  What if we apply a 3+1D model to other domains such as complex waves?

What if we give up the perceptual bias that material particles must exist continuously at the quantum level?  What if a quantum particle only exists twice per cycle of a complex wave, only when the radius vectors for the spatial wave and the temporal wave are coincident in spacetime and thus the tips of the radius vectors have the same point coordinates? Modeling a physically real point as the interaction of space and time components of one complex wave achieves several goals:  (1) it resolves the complementarity issue, (2) it provides propagation as waves and interaction as particles, (3) it explains why particles can tunnel through barriers, (4) it can be used in a description of the zero-point energy of the vacuum, and (5) some others I am still considering how to describe with mathematics.

Planck's constant equals the unit charge and unit magnetic flux of two unit electromagnetic waves (|search_for=elecmag_in! and click on symbol to show equation ). There are numerous complex waves passing through any point in spacetime on their journeys within the universe. The aggregation of those points create physical reality and its fields.  What are your thoughts about this model?

Thanks again for your post!  


P.S.  In this post I have collapsed levels within what Einstein calls "stratification of the scientific system" (Later Years, p. 63). If you wish to see it, I have a stratification showing all levels between Planck's constant and the EBDH fields. You might be interested because the levels are specifically defined by the EM potentials.

P.P.S. The only picture I have for my profile is two years old -- so imagine the brownish red mixed with two years of gray hair and cut to shoulder length!

Dear Martha,

1. Regarding your ideas for a QM model that you describe: I think it would be best if you could write a new thread on this topic, perhaps attaching a PDF document of a preliminary version of your paper in that thread. With the description given above, it's difficult to see what the idea is. You might have something there, but I fail to see what it is. ... Indeed, you should post those thoughts regardless of what my further points in this reply are.

2. I gather a few things, some philosophic and others physics-related, that I must spell out right away. 

Philosophically, I consider perception (of entities in reality) to be the ultimate source of all knowledge. Taken to the field of physics this principle implies that no conceptual-level construct (e.g. a mathematical idea or equation or imagination) may logically precede the perceptually evident physical reality. In other words, you cannot "explain" percepts---though this is a cardinal (and oft-repeated) error of modern physics. 

Unfortunately, I "sense" from your above writeup that you are about to embark on yet another variant of precisely the same error.

Consider just one statement: "Modeling a physically real point as the interaction of space and time components of one complex wave..."

I disagree with it on multiple counts: (i) To repeat: There is no such a thing as a "physically real point." A point is a *mathematical* abstraction. What is physical real are physical objects, and they all always exist with extension---else you couldn't have perceived them (neither could you have derived the concept of space). You cannot conceive of a physical object without extension---and that is what a point means. (ii) If you wish to modify it to: "modeling a phenomenon at a point as the intersection..." then you are somewhat better off by stating first that "Let us assume that a complex wave physically exists." This is better but still in error. After all, if it is me (and not Feynman or Lewis Little), the first thing that would cross my mind is: "Wave? What is it that waves? Please identify." (iii) I don't at all get what you mean by the "interaction of the space and time components of one complex wave." Please explain (in that separate, dedicated thread---you can always put a link back to this discussion): if it is one wave, any interaction of its two parts is simply going to alter that wave itself, and so, it will no longer obey the basic equation for it which you assume to begin with. In such a case, it will not be a basic or a starting point of a physical theory. How do you address this apparent circularity?

Another statement: "There are numerous complex waves passing through any point in spacetime on their journeys within the universe. The aggregation of those points create physical reality and its fields."

Nothing creates physical reality. The physical reality is what it is. ... I have neither the time nor the space nor the inclination to get into the subtleties here, but if you wish, please see Objectivist literature on this point, to which I agree. A particularly good starting point would be Peikoff's "Objectivism: The Philosophy of Ayn Rand"---the sections on primacy of existence etc, and the references therein. What I am going to state here is not a substitute for that reference, but just an indication: I believe that science has to accept the existence of an objective reality. In particular, no content of human mind (such as a thought, or mathematical construct,) precedes that reality.

3. If I already find so many points that are objectionable, then why do I suggest to you to write a post and/or a paper?

A few reasons:
(i) Sometimes, it so happens that there is some good idea in a paper even though the paper's main or primary theme is not acceptable. Unless the paper is written and put out for reading, there is no way to tell even the secondary good aspects of it.
(ii) Writing a paper focuses your thoughts. On a freer medium, for instance, you can touch on basics of QM, then relativity, black holes, psychobiology... That makes the writing lose the focus, which can only be gained by writing a paper.... Might be shocking to you, but I am not at all interested in Einstein's relativity theory, don't care a bit about black/warm/white holes, or dark matter for that matter, and don't really wish to address popular science press through my research. There are some interesting problems at the basic level of even QM, and I have some new ideas about them, and that's about all for me. Sorry if I have let you down, but that's the way it is. So, if you write a paper with a QM focus, I could at least begin dealing with it, even if I may disagree with it in the end.
(iii) Writing the paper will force you to pick up a QM modeling situation. Here, I would suggest that you could consider the single-quantum double-slit interference experiment---how your idea works out for that experiment... A few reasons: (a) The weight of Feynman's reputation will work for you if you pick that up. After all, it was he who (rightly) singled out that experiment out as containing the basic strange character of the quantum phenomena. (b) Many physicists (and I myself) have found it to be both very beautiful and very testing a situation. (c) After The Feynman Lectures, almost one generation of physicists by now has got used to thinking of basic QM in its terms---so, all their thoughts and insights might come in handy to help you if you address that. (No physicist made himself available to listen to me despite my best efforts before and since publication of my papers, but it's a theoretical possibility that they just might, and that's what you might grant that particular community.)

Also one last suggestion, if you care for one: The one question (the most weighty in today's physics culture) that any new QM theorist would face is: does your theory predict anything new? No matter how silly this question might be, it does actually take precedence over the other (twin) question: does your theory correspond with physical reality (as observed by the sum totality of all and the finest experimentation) and does it remain consistent with the other knowledge? ... Since today's physicist is going to ask you the first question first (and, in all likelihood, if he has any reputation whatsoever, never bring himself to raise the second question), it would help if you keep that particular aspect in mind while writing your paper.

OK. That's about it. I will sure comment once you post something on a new thread. And, given the archiving nature of iMechanica, it's OK to post preliminary thoughts too... As Zhigang once said, there is no confusion here as to who said what when. And, you don't lose the opportunity to publish the same thoughts in a journal later on. So, make use of it. The PDF format will only help you focus... (Indeed, this thread itself, as you can easily see, was to let me do some loud thinking and to solicit suggestion for the best places to publish the eventual papers, on the topic of physical meaning of potential.)

All the best.


Martha J Lindeman's picture

Hi Ajit,

Thanks for your reasoned comments as feedback. We completely agree in that I realized yesterday that if the discussion continued it should be transferred to somewhere other than this blog. Also thanks for the suggestion about writing a paper, but I will continue to spend my time writing my thoughts for a different audience. So for closure I have included here brief responses that I thought you might find relevant.

Best wishes in your work and in getting your papers published!


  1. I agree with objectivism that physical reality exists independent of consciousness and objective knowledge can be gained from concept formation and logical thinking. However, you state that you “consider perception (of entities in reality)to be the ultimate source of all knowledge.” As a cognitive psychologist, I know that perceptions are based on learned mental models that either help or hinder discovery and understanding of physical reality. For example, your visual perception of your surroundings hides the reality that your imagination fills in the blank spot on each retina where the optic nerve connects.

  2. From a perceptual viewpoint, a “point” is anything that has a perceptually disregarded internal structure. For example, a garden hose viewed from sufficient distance is simply a line. The key concepts are “inside” and “outside”, and each perceiver chooses whether to disregard an object's internals. For example, an auto mechanic needs to understand the internals of an engine whereas a driver does not. In physics, the concepts 'center of gravity' and 'center of mass' are based on the physical reality of a 'point'.

  3. The figure-ground principle is a very strong subconscious principle that determines what a person is perceiving. A fundamental question in physics is what is 'figure' and what is 'ground'--the whole "aether" question was based on the perceptual belief that light (electromagnetic) waves require a physical medium for transmission through spacetime. In contrast, I believe that light waves and spacetime are synonyms, which is why “c” as the ratio of space to time is a constant across frames of reference. Thus there is no medium necessary for photons to 'wave'. 



Martha J Lindeman's picture

I explored your website -- congratulations on completing your PhD!


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