Off the blog. [“Matter” cannot act “where” it is not.]

I am going to go off the blogging activity in general, and this blog in most particular, for some time. [And, this time round, I will keep my promise.]


The reason is, I’ve just received the shipment of a book which I had ordered about a month ago. Though only about 300 pages in length, it’s going to take me weeks to complete. And, the book is gripping enough, and the issue important enough, that I am not going to let a mere blog or two—or the entire Internet—come in the way.


I had read it once, almost cover-to-cover, some 25 years ago, while I was a student in UAB.

Reading a book cover-to-cover—I mean: in-sequence, and by that I mean: starting from the front-cover and going through the pages in the same sequence as the one in which the book has been written, all the way to the back-cover—was quite odd a thing to have happened with me, at that time. It was quite unlike my usual habits whereby I am more or less always randomly jumping around in a book, even while reading one for the very first time.

But this book was different; it was extraordinarily engaging.

In fact, as I vividly remember, I had just idly picked up this book off a shelf from the Hill library of UAB, for a casual examination, had browsed it a bit, and then had began sampling some passage from nowhere in the middle of the book while standing in an library aisle. Then, some little time later, I was engrossed in reading it—with a folded elbow resting on the shelf, head turned down and resting against a shelf rack (due to a general weakness due to a physical hunger which I was ignoring [and I would have have to go home and cook something for myself; there was none to do that for me; and so, it was easy enough to ignore the hunger]). I don’t honestly remember how the pages turned. But I do remember that I must have already finished some 15-20 pages (all “in-the-order”!) before I even realized that I had been reading this book while still awkwardly resting against that shelf-rack. …

… I checked out the book, and once home [student dormitory], began reading it starting from the very first page. … I took time, days, perhaps weeks. But whatever the length of time that I did take, with this book, I didn’t have to jump around the pages.


The issue that the book dealt with was:

[Instantaneous] Action at a Distance.

The book in question was:

Hesse, Mary B. (1961) “Forces and Fields: The concept of Action at a Distance in the history of physics,” Philosophical Library, Edinburgh and New York.


It was the very first book I had found, I even today distinctly remember, in which someone—someone, anyone, other than me—had cared to think about the issues like the IAD, the concepts like fields and point particles—and had tried to trace their physical roots, to understand the physical origins behind these (and such) mathematical concepts. (And, had chosen to say “concepts” while meaning ones, rather than trying to hide behind poor substitute words like “ideas”, “experiences”, “issues”, “models”, etc.)

Twenty-five years later, I still remain hooked on to the topic. Despite having published a paper on IAD and diffusion [and yes, what the hell, I will say it: despite claiming a first in 200+ years in reference to this topic], I even today do find new things to think about, about this “kutty” [Original: IITM lingo; English translation: “small”] topic. And so, I keep returning to it and thinking about it. I still am able to gain new insights once in an odd while. … Indeed, my recent ‘net search on IAD (the one which led to Hesse and my buying the book) precisely was to see if someone had reported the conceptual [and of course, mathematical] observation which I have recently made, or not. [If too curious about it, the answer: looks like, none has.]


But now coming to Hesse’s writing style, let me quote a passage from one of her research papers. I ran into this paper only recently, last month (in July 2017), and it was while going through it that I happened [once again] to remember her book. Since I did have some money in hand, I did immediately decide to order my copy of this book.

Anyway, the paper I have in mind is this:

Hesse, Mary B. (1955) “Action at a Distance in Classical Physics,” Isis, Vol. 46, No. 4 (Dec., 1955), pp. 337–353, University of Chicago Press/The History of Science Society.

The paper (it has no abstract) begins thus:

The scholastic axiom that “matter cannot act where it is not” is one of the very general metaphysical principles found in science before the seventeenth century which retain their relevance for scientific theory even when the metaphysics itself has been discarded. Other such principles have been fruitful in the development of physics: for example, the “conservation of motion” stated by Descartes and Leibniz, which was generalized and given precision in the nineteenth century as the doctrine of the conservation of energy; …

Here is another passage, once again, from the same paper:

Now Faraday uses a terminology in speaking about the lines of force which is derived from the idea of a bundle of elastic strings stretched under tension from point to point of the field. Thus he speaks of “tension” and “the number of lines” cut by a body moving in the field. Remembering his discussion about contiguous particles of a dielectric medium, one must think of the strings as stretching from one particle of the medium to the next in a straight line, the distance between particles being so small that the line appears as a smooth curve. How seriously does he take this model? Certainly the bundle of elastic strings is nothing like those one can buy at the store. The “number of lines” does not refer to a definite number of discrete material entities, but to the amount of force exerted over a given area in the field. It would not make sense to assign points through which a line passes and points which are free from a line. The field of force is continuous.

See the flow of the writing? the authentic respect for the intellectual history, and yet, the overriding concern for having to reach a conclusion, a meaning? the appreciation for the subtle drama? the clarity of thought, of expression?

Well, these passages were from the paper, but the book itself, too, is similarly written.


Obviously, while I remain engaged in [re-]reading the book [after a gap of 25 years], don’t expect me to blog.

After all, even I cannot act “where” I am not.


A Song I Like:

[I thought a bit between this song and another song, one by R.D. Burman, Gulzar and Lata. In the end, it was this song which won out. As usual, in making my decision, the reference was exclusively made to the respective audio tracks. In fact, in the making of this decision, I happened to have also ignored even the excellent guitar pieces in this song, and the orchestration in general in both. The words and the tune were too well “fused” together in this song; that’s why. I do promise you to run the RD song once I return. In the meanwhile, I don’t at all mind keeping you guessing. Happy guessing!]

(Hindi) “bheegi bheegi…” [“bheege bheege lamhon kee bheegee bheegee yaadein…”]
Music and Lyrics: Kaushal S. Inamdar
Singer: Hamsika Iyer

[Minor additions/editing may follow tomorrow or so.]

 

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See, how hard I am trying to become an Approved (Full) Professor of Mechanical Engineering in SPPU?—3

I was looking for a certain book on heat transfer which I had (as usual) misplaced somewhere, and while searching for that book at home, I accidentally ran into another book I had—the one on Classical Mechanics by Rana and Joag [^].

After dusting this book a bit, I spent some time in one typical way, viz. by going over some fond memories associated with a suddenly re-found book…. The memories of how enthusiastic I once was when I had bought that book; how I had decided to finish that book right within weeks of buying it several years ago; the number of times I might have picked it up, and soon later on, kept it back aside somewhere, etc.  …

Yes, that’s right. I have not yet managed to finish this book. Why, I have not even managed to begin reading this book the way it should be read—with a paper and pencil at hand to work through the equations and the problems. That was the reason why, I now felt a bit guilty. … It just so happened that it was just the other day (or so) when I was happily mentioning the Poisson brackets on Prof. Scott Aaronson’s blog, at this thread [^]. … To remove (at least some part of) my sense of guilt, I then decided to browse at least through this part (viz., Poisson’s brackets) in this book. … Then, reading a little through this chapter, I decided to browse through the preceding chapters from the Lagrangian mechanics on which it depends, and then, in general, also on the calculus of variations.

It was at this point that I suddenly happened to remember the reason why I had never been able to finish (even the portions relevant to engineering from) this book.

The thing was, the explanation of the \delta—the delta of the variational calculus.

The explanation of what the \delta basically means, I had found right back then (many, many years ago), was not satisfactorily given in this book. The book did talk of all those things like the holonomic constraints vs. the nonholonomic constraints, the functionals, integration by parts, etc. etc. etc. But without ever really telling me, in a forth-right and explicit manner, what the hell this \delta was basically supposed to mean! How this \delta y was different from the finite changes (\Delta y) and the infinitesimal changes (\text{d}y) of the usual calculus, for instance. In terms of its physical meaning, that is. (Hell, this book was supposed to be on physics, wasn’t it?)

Here, I of course fully realize that describing Rana and Joag’s book as “unsatisfactory” is making a rather bold statement, a very courageous one, in fact. This book is extraordinarily well-written. And yet, there I was, many, many years ago, trying to understand the delta, and not getting anywhere, not even with this book in my hand. (OK, a confession. The current copy which I have is not all that old. My old copy is gone by now (i.e., permanently misplaced or so), and so, the current copy is the one which I had bought once again, in 2009. As to my old copy, I think, I had bought it sometime in the mid-1990s.)

It was many years later, guess some time while teaching FEM to the undergraduates in Mumbai, that the concept had finally become clear enough to me. Most especially, while I was going through P. Seshu’s and J. N. Reddy’s books. [Reflected Glory Alert! Professor P. Seshu was my class-mate for a few courses at IIT Madras!] However, even then, even at that time, I remember, I still had this odd feeling that the physical meaning was still not clear to me—not as as clear as it should be. The matter eventually became “fully” clear to me only later on, while musing about the differences between the perspective of Thermodynamics on the one hand and that of Heat Transfer on the other. That was some time last year, while teaching Thermodynamics to the PG students here in Pune.

Thermodynamics deals with systems at equilibria, primarily. Yes, its methods can be extended to handle also the non-equilibrium situations. However, even then, the basis of the approach summarily lies only in the equilibrium states. Heat Transfer, on the other hand, necessarily deals with the non-equilibrium situations. Remove the temperature gradient, and there is no more heat left to speak of. There does remain the thermal energy (as a form of the internal energy), but not heat. (Remember, heat is the thermal energy in transit that appears on a system boundary.) Heat transfer necessarily requires an absence of thermal equilibrium. … Anyway, it was while teaching thermodynamics last year, and only incidentally pondering about its differences from heat transfer, that the idea of the variations (of Cov) had finally become (conceptually) clear to me. (No, CoV does not necessarily deal only with the equilibrium states; it’s just that it was while thinking about the equilibrium vs. the transient that the matter about CoV had suddenly “clicked” to me.)

In this post, let me now note down something on the concept of the variation, i.e., towards understanding the physical meaning of the symbol \delta.

Please note, I have made an inline update on 26th December 2016. It makes the presentation of the calculus of variations a bit less dumbed down. The updated portion is clearly marked as such, in the text.


The Problem Description:

The concept of variations is abstract. We would be better off considering a simple, concrete, physical situation first, and only then try to understand the meaning of this abstract concept.

Accordingly, consider a certain idealized system. See its schematic diagram below:

mechanicalengineering_1d_cov

 

 

 

 

There is a long, rigid cylinder made from some transparent material like glass. The left hand-side end of the cylinder is hermetically sealed with a rigid seal. At the other end of the cylinder, there is a friction-less piston which can be driven by some external means.

Further, there also are a couple of thin, circular, piston-like disks (D_1 and D_2) placed inside the cylinder, at some x_1 and x_2 positions along its length. These disks thus divide the cylindrical cavity into three distinct compartments. The disks are assumed to be impermeable, and fitting snugly, they in general permit no movement of gas across their plane. However, they also are assumed to be able to move without any friction.

Initially, all the three compartments are filled with a compressible fluid to the same pressure in each compartment, say 1 atm. Since all the three compartments are at the same pressure, the disks stay stationary.

Then, suppose that the piston on the extreme right end is moved, say from position P_1 to P_2. The final position P_2 may be to the left or to the right of the initial position P_1; it doesn’t matter. For the current description, however, let’s suppose that the position P_2 is to the left of P_1. The effect of the piston movement thus is to increase the pressure inside the system.

The problem is to determine the nature of the resulting displacements that the two disks undergo as measured from their respective initial positions.

There are essentially two entirely different paradigms for conducting an analysis of this problem.


The “Vector Mechanics” Paradigm:

The first paradigm is based on an approach that was put to use so successfully by Newton. Usually, it is called the paradigm of vector analysis.

In this paradigm, we focus on the fact that the forced displacement of the piston with time, x(t), may be described using some function of time that is defined over the interval lying between two instants t_i and t_f.

For example, suppose the function is:
x(t) = x_0 + v t,
where v is a constant. In other words, the motion of the piston is steady, with a constant velocity, between the initial and final instants. Since the velocity is constant, there is no acceleration over the open interval (t_i, t_f).

However, notice that before the instant t_i, the piston velocity was zero. Then, the velocity suddenly became a finite (constant) value. Therefore, if you extend the interval to include the end-instants as well, i.e., if you consider the semi-closed interval [t_i, t_f), then there is an acceleration at the instant t_i. Similarly, since the piston comes to a position of rest at t = t_f, there also is another acceleration, equal in magnitude and opposite in direction, which appears at the instant t_f.

The existence of these two instantaneous accelerations implies that jerks or pressure waves are sent through the system. We may model them as vector quantities, as impulses. [Side Exercise: Work out what happens if we consider only the open interval (t_i, t_f).]

We can now apply Newton’s 3 laws, based on the idea that shock-waves must have begun at the piston at the instant t = t_i. They must have got transmitted through the gas kept under pressure, and they must have affected the disk D_1 lying closest to the piston, thereby setting this disk into motion. This motion must have passed through the gas in the middle compartment of the system as another pulse in the pressure (generated at the disk D_1), thereby setting also the disk D_2 in a state of motion a little while later. Finally, the pulse must have got bounced off the seal on the left hand side, and in turn, come back to affect the motion of the disk D_2, and then of the disk D_1. Continuing their travels to and fro, the pulses, and hence the disks, would thus be put in a back and forth motion.

After a while, these transients would move forth and back, superpose, and some of their constituent frequencies would get cancelled out, leaving only those frequencies operative such that the three compartments are put under some kind of stationary states.

In case the gas is not ideal, there would be damping anyway, and after a sufficiently long while, the disks would move through such small displacements that we could easily ignore the ever-decreasing displacements in a limiting argument.

Thus, assume that, after an elapse of a sufficiently long time, the disks become stationary. Of course, their new positions are not the same as their original positions.

The problem thus can be modeled as basically a transient one. The state of the new equilibrium state is thus primarily seen as an effect or an end-result of a couple of transient processes which occur in the forward and backward directions. The equilibrium is seen as not a primarily existing state, but as a result of two equal and opposite transient causes.

Notice that throughout this process, Newton’s laws can be applied directly. The nature of the analysis is such that the quantities in question—viz. the displacements of the disks—always are real, i.e., they correspond to what actually is supposed to exist in the reality out there.

The (values of) displacements are real in the sense that the mathematical analysis procedure itself involves only those (values of) displacements which can actually occur in reality. The analysis does not concern itself with some other displacements that might have been possible but don’t actually occur. The analysis begins with the forced displacement condition, translates it into pressure waves, which in turn are used in order to derive the predicted displacements in the gas in the system, at each instant. Thus, at any arbitrary instant of time t > t_i (in fact, the analysis here runs for times t \gg t_f), the analysis remains concerned only with those displacements that are actually taking place at that instant.

The Method of Calculus of Variations:

The second paradigm follows the energetics program. This program was initiated by Newton himself as well as by Leibnitz. However, it was pursued vigorously not by Newton but rather by Leibnitz, and then by a series of gifted mathematicians-physicists: the Bernoulli brothers, Euler, Lagrange, Hamilton, and others. This paradigm is essentially based on the calculus of variations. The idea here is something like the following.

We do not care for a local description at all. Thus, we do not analyze the situation in terms of the local pressure pulses, their momenta/forces, etc. All that we focus on are just two sets of quantities: the initial positions of the disks, and their final positions.

For instance, focus on the disk D_1. It initially is at the position x_{1_i}. It is found, after a long elapse of time (i.e., at the next equilibrium state), to have moved to x_{1_f}. The question is: how to relate this change in x_1 on the one hand, to the displacement that the piston itself undergoes from P_{x_i} to P_{x_f}.

To analyze this question, the energetics program (i.e., the calculus of variations) adopts a seemingly strange methodology.

It begins by saying that there is nothing unique to the specific value of the position x_{1_f} as assumed by the disk D_1. The disk could have come to a halt at any other (nearby) position, e.g., at some other point x_{1_1}, or x_{1_2}, or x_{1_3}, … etc. In fact, since there are an infinity of points lying in a finite segment of line, there could have been an infinity of positions where the disk could have come to a rest, when the new equilibrium was reached.

Of course, in reality, the disk D_1 comes to a halt at none of these other positions; it comes to a halt only at x_{1_f}.

Yet, the theory says, we need to be “all-inclusive,” in a way. We need not, just for the aforementioned reason, deny a place in our analysis to these other positions. The analysis must include all such possible positions—even if they be purely hypothetical, imaginary, or unreal. What we do in the analysis, this paradigm says, is to initially include these merely hypothetical, unrealistic positions too on exactly the same footing as that enjoyed by that one position which is realistic, which is given by x_{1_f}.

Thus, we take a set of all possible positions for each disk. Then, for each such a position, we calculate the “impact” it would make on the energy of the system taken as a whole.

The energy of the system can be additively decomposed into the energies carried by each of its sub-parts. Thus, focusing on disk D_1, for each one of its possible (hypothetical) final position, we should calculate the energies carried by both its adjacent compartments. Since a change in D_1‘s position does not affect the compartment 3, we need not include it. However, for the disk D_1, we do need to include the energies carried by both the compartments 1 and 2. Similarly, for each of the possible positions occupied by the disk D_2, it should include the energies of the compartments 2 and 3, but not of 1.

At this point, to bring simplicity (and thereby better) clarity to this entire procedure, let us further assume that the possible positions of each disk forms a finite set. For instance, each disk can occupy only one of the positions that is some -5, -4, -3, -2, -1, 0, +1, +2, +3, +4 or +5 distance-units away from its initial position. Thus, a disk is not allowed to come to a rest at, say, 2.3 units; it must do so either at 2 or at 3 units. (We will thus perform the initial analysis in terms of only the integer positions, and only later on extend it to any real-valued positions.) (If you are a mechanical engineering student, suggest a suitable mechanism that can ensure only integer relative displacements.)

The change in energy E of a compartment is given by
\Delta E = P A \Delta x,
where P is the pressure, A is the cross-sectional area of the cylinder, and \Delta x is the change in the length of the compartment.

Now, observe that the energy of the middle compartment depends on the relative distance between the two disks lying on its sides. Yet, for the same reason, the energy of the middle compartment does depend on both these positions. Hence, we must take a Cartesian product of the relative displacements undergone by both the disks, and only then calculate the system energy for each such a permutation (i.e. the ordered pair) of their positions. Let us go over the details of the Cartesian product.

The Cartesian product of the two positions may be stated as a row-by-row listing of ordered pairs of the relative positions of D_1 and D_2, e.g., as follows: the ordered pair (-5, +2) means that the disk D_1 is 5 units to the left of its initial position, and the disk D_2 is +2 units to the right of its initial position. Since each of the two positions forming an ordered pair can range over any of the above-mentioned 11 number of different values, there are, in all, 11 \times 11 = 121 number of such possible ordered pairs in the Cartesian product.

For each one of these 121 different pairs, we use the above-given formula to determine what the energy of each compartment is like. Then, we add the three energies (of the three compartments) together to get the value of the energy of the system as a whole.

In short, we get a set of 121 possible values for the energy of the system.

You must have noticed that we have admitted every possible permutation into analysis—all the 121 number of them.

Of course, out of all these 121 number of permutations of positions, it should turn out that 120 number of them have to be discarded because they would be merely hypothetical, i.e. unreal. That, in turn, is because, the relative positions of the disks contained in one and only one ordered pair would actually correspond to the final, equilibrium position. After all, if you conduct this experiment in reality, you would always get a very definite pair of the disk-positions, and it this same pair of relative positions that would be observed every time you conducted the experiment (for the same piston displacement). Real experiments are reproducible, and give rise to the same, unique result. (Even if the system were to be probabilistic, it would have to give rise to an exactly identical probability distribution function.) It can’t be this result today and that result tomorrow, or this result in this lab and that result in some other lab. That simply isn’t science.

Thus, out of all those 121 different ordered-pairs, one and only one ordered-pair would actually correspond to reality; the rest all would be merely hypothetical.

The question now is, which particular pair corresponds to reality, and which ones are unreal. How to tell the real from the unreal. That is the question.

Here, the variational principle says that the pair of relative positions that actually occurs in reality carries a certain definite, distinguishing attribute.

The system-energy calculated for this pair (of relative displacements) happens to carry the lowest magnitude from among all possible 121 number of pairs. In other words, any hypothetical or unreal pair has a higher amount of system energy associated with it. (If two pairs give rise to the same lowest value, both would be equally likely to occur. However, that is not what provably happens in the current example, so let us leave this kind of a “degeneracy” aside for the purposes of this post.)

(The update on 26 December 2016 begins here:)

Actually, the description  given in the immediately preceding paragraph was a bit too dumbed down. The variational principle is more subtle than that. Explaining it makes this post even longer, but let me give it a shot anyway, at least today.

To follow the actual idea of the variational principle (in a not dumbed-down manner), the procedure you have to follow is this.

First, make a table of all possible relative-position pairs, and their associated energies. The table has the following columns: a relative-position pair, the associated energy E as calculated above, and one more column which for the time being would be empty. The table may look something like what the following (partial) listing shows:

(0,0) -> say, 115 Joules
(-1,0) -> say, 101 Joules
(-2,0) -> say, 110 Joules

(2,2) -> say, 102 Joules
(2,3) -> say, 100 Joules
(2,4) -> say, 101 Joules
(2,5) -> say, 120 Joules

(5,0) -> say, 135 Joules

(5,5) -> say 117 Joules.

Having created this table (of 121 rows), you then pick each row one by and one, and for the picked up n-th row, you ask a question: What all other row(s) from this table have their relative distance pairs such that these pairs lie closest to the relative distance pair of this given row. Let me illustrate this question with a concrete example. Consider the row which has the relative-distance pair given as (2,3). Then, the relative distance pairs closest to this one would be obtained by adding or subtracting a distance of 1 to each in the pair. Thus, the relative distance pairs closest to this one would be: (3,3), (1,3), (2,4), and (2,2). So, you have to pick up those rows which have these four entries in the relative-distance pairs column. Each of these four pairs represents a variation \delta on the chosen state, viz. the state (2,3).

In symbolic terms, suppose for the n-th row being considered, the rows closest to it in terms of the differences in their relative distance pairs, are the a-th, b-th, c-th and d-th rows. (Notice that the rows which are closest to a given row in this sense, would not necessarily be found listed just above or below that given row, because the scheme followed while creating the list or the vector that is the table would not necessarily honor the closest-lying criterion (which necessarily involves two numbers)—not at least for all rows in the table.

OK. Then, in the next step, you find the differences in the energies of the n-th row from each of these closest rows, viz., the a-th, b-th, c-th and c-th rows. That is to say, you find the absolute magnitudes of the energy differences. Let us denote these magnitudes as: \delta E_{na} = |E_n - E_a|\delta E_{nb} = |E_n - E_b|\delta E_{nc} = |E_n - E_c| and \delta E_{nd} = |E_n - E_d|.  Suppose the minimum among these values is \delta E_{nc}. So, against the n-th row, in the last column of the table, you write the value \delta E_{nc}.

Having done this exercise separately for each row in the table, you then ask: Which row has the smallest entry in the last column (the one for \delta E), and you pick that up. That is the distinguished (or the physically occurring) state.

In other words, the variational principle asks you to select not the row with the lowest absolute value of energy, but that row which shows the smallest difference of energy from one of its closest neighbours—and these closest neighbours are to be selected according to the differences in each number appearing in the relative-distance pair, and not according to the vertical place of rows in the tabular listing. (It so turns out that in this example, the row thus selected following both criteria—lowest energy as well as lowest variation in energy—are identical, though it would not necessarily always be the case. In short, we can’t always get away with the first, too dumbed down, version.)

Thus, the variational principle is about that change in the relative positions for which the corresponding change in the energy vanishes (or has the minimum possible absolute magnitude, in case the positions form a discretely varying, finite set).

(The update on 26th December 2016 gets over here.)

And, it turns out that this approach, too, is indeed able to perfectly predict the final disk-positions—precisely as they actually are observed in reality.

If you allow a continuum of positions (instead of the discrete set of only the 11 number of different final positions for one disk, or 121 number of ordered pairs), then instead of taking a Cartesian product of positions, what you have to do is take into account a tensor product of the position functions. The maths involved is a little more advanced, but the underlying algebraic structure—and the predictive principle which is fundamentally involved in the procedure—remains essentially the same. This principle—the variational principle—says:

Among all possible variations in the system configurations, that system configuration corresponds to reality which has the least variation in energy associated with it.

(This is a very rough statement, but it will do for this post and for a general audience. In particular, we don’t look into the issues of what constitute the kinematically admissible constraints, why the configurations must satisfy the field boundary conditions, the idea of the stationarity vs. of a minimum or a maximum, i.e., the issue of convexity-vs.-concavity, etc. The purpose of this post—and our example here—are both simple enough that we need not get into the whole she-bang of the variational theory as such.)

Notice that in this second paradigm, (i) we did not restrict the analysis to only those quantities that are actually taking place in reality; we also included a host (possibly an infinity) of purely hypothetical combinations of quantities too; (ii) we worked with energy, a scalar quantity, rather than with momentum, a vector quantity; and finally, (iii) in the variational method, we didn’t bother about the local details. We took into account the displacements of the disks, but not any displacement at any other point, say in the gas. We did not look into presence or absence of a pulse at one point in the gas as contrasted from any other point in it. In short, we did not discuss the details local to the system either in space or in time. We did not follow the system evolution, at all—not at least in a detailed, local way. If we were to do that, we would be concerned about what happens in the system at the instants and at spatial points other than the initial and final disk positions. Instead, we looked only at a global property—viz. the energy—whether at the sub-system level of the individual compartments, or at the level of the overall system.


The Two Paradigms Contrasted from Each Other:

If we were to follow Newton’s method, it would be impossible—impossible in principle—to be able to predict the final disk positions unless all their motions over all the intermediate transient dynamics (occurring over each moment of time and at each place of the system) were not be traced. Newton’s (or vectorial) method would require us to follow all the details of the entire evolution of all parts of the system at each point on its evolution path. In the variational approach, the latter is not of any primary concern.

Yet, in following the energetics program, we are able to predict the final disk positions. We are able to do that without worrying about what all happened before the equilibrium gets established. We remain concerned only with certain global quantities (here, system-energy) at each of the hypothetical positions.

The upside of the energetics program, as just noted, is that we don’t have to look into every detail at every stage of the entire transient dynamics.

Its downside is that we are able to talk only of the differences between certain isolated (hypothetical) configurations or states. The formalism is unable to say anything at all about any of the intermediate states—even if these do actually occur in reality. This is a very, very important point to keep in mind.


The Question:

Now, the question with which we began this post. Namely, what does the delta of the variational calculus mean?

Referring to the above discussion, note that the delta of the variational calculus is, here, nothing but a change in the position-pair, and also the corresponding change in the energy.

Thus, in the above example, the difference of the state (2,3) from the other close states such as (3,3), (1,3), (2,4), and (2,2) represents a variation in the system configuration (or state), and for each such a variation in the system configuration (or state), there is a corresponding variation in the energy \delta E_{ni} of the system. That is what the delta refers to, in this example.

Now, with all this discussion and clarification, would it be possible for you to clearly state what the physical meaning of the delta is? To what precisely does the concept refer? How does the variation in energy \delta E differ from both the finite changes (\Delta E) as well as the infinitesimal changes (\text{d}E) of the usual calculus?


Note, the question is conceptual in nature. And, no, not a single one of the very best books on classical mechanics manages to give a very succinct and accurate answer to it. Not even Rana and Joag (or Goldstein, or Feynman, or…)

I will give my answer in my next post, next year. I will also try to apply it to a couple of more interesting (and somewhat more complicated) physical situations—one from engineering sciences, and another from quantum mechanics!

In the meanwhile, think about it—the delta—the concept itself, its (conceptual) meaning. (If you already know the calculus of variations, note that in my above write-up, I have already supplied the answer, in a way. You just have to think a bit about it, that’s all!)


An Important Note: Do bring this post to the notice of the Officially Approved Full Professors of Mechanical Engineering in SPPU, and the SPPU authorities. I would like to know if the former would be able to state the meaning—at least now that I have already given the necessary context in such great detail.

Ditto, to the Officially Approved Full Professors of Mechanical Engineering at COEP, esp. D. W. Pande, and others like them.

After all, this topic—Lagrangian mechanics—is at the core of Mechanical Engineering, even they would agree. In fact, it comes from a subject that is not taught to the metallurgical engineers, viz., the topic of Theory of Machines. But it is taught to the Mechanical Engineers. That’s why, they should be able to crack it, in no time.

(Let me continue to be honest. I do not expect them to be able to crack it. But I do wish to know if they are able at least to give a try that is good enough!)


Even though I am jobless (and also nearly bank balance-less, and also cashless), what the hell! …

…Season’s greetings and best wishes for a happy new year!


A Song I Like:

[With jobless-ness and all, my mood isn’t likely to stay this upbeat, but anyway, while it lasts, listen to this song… And, yes, this song is like, it’s like, slightly more than 60 years old!]

(Hindi) “yeh raat bhigee bhigee”
Music: Shankar-Jaikishan
Singers: Manna De and Lata Mangeshkar
Lyrics: Shailendra


[E&OE]

Miscellaneous: books to read, a new QM journal, the imposter syndrome, the US presidential elections

While my mood of not wanting to do anything in particular still continues (and also, there is no word yet on the job-related matters, including on whether I might qualify as a Professor of Mechanical Engineering in SPPU or not), there are a few quick things that I may as well note.

Updates on 17th, 18th and 22nd Nov. 2016: See my English translation[s] of the song, at the end of the post.


Books to Read:

First, the books to read. Here are a few books on my to-read list:

  1. Sean Carroll, “The Big Picture” [^]. I have been browsing through Sean’s blog-posts since before the time the book was published, and so have grown curious. I don’t have the money to buy it, right now, but once I get the next job, I sure plan to buy it. Here is the review in NY Times [^]. And, here is a latest review, written by a software engineer (whose link appeared in Sean’s twitter feed (I don’t myself use my Twitter account, but sometimes do check out the feeds of others via browser))[^]. Judging from his posts, I do know that Sean writes really well, and I would certainly want to check out this book, eventually.
  2. Roger Penrose, “Fashion, Faith, and Fantasy in the New Physics of the Universe,” [^]. This is the latest offering by Penrose. Sometimes I simply type “quantum physics” in Google, and then, in the search results, I switch the tab over to “news.” I came to know of this book via this route, last week, when I ran into this review [^].
  3. Roger Schlafly, “How Einstein Ruined Physics: Motion, Symmetry and Revolution in Science,” [^]. Here is a review [^], though my curiosity about the book rests not on the review but on two things: (i) what I had thought of Einstein myself, as far back as in early 1990s, while at UAB (hint: Schlafly’s thesis wouldn’t be out of bounds for me), and (ii) my reading the available portions of the book at Google Books. …This book has been on my “to-read” list for quite some time, but somehow it keeps slipping off. … Anyway, to be read, soon after I land a job…

A New QM Journal:

A new journal has arrived on the QM scene: [^]. Once again, I got to know of it through the “news” tab in a Google search on “quantum physics”, when I took this link [^].

It’s an arXiv-overlay journal. What it means is that first you submit your paper to arXiv. … As you know, getting something published at arXiv carries a pretty low bar (though it is not zero, and there have been some inconsistencies rarely reported about improper rejections even at arXiv). It’s good to bring your work to the notice of your peers, but it carries no value in your academic/research publications record, because arXiv is not a proper journal as such. … Now, if your work is good, you want to keep it open-access, but you don’t want to pay for keeping it open-access, and, at the same time, you also want to have the credentials of a proper journal publication to your credit, you have a solution, in the form of this arXiv-overlay journal. You send the link to your arXiv-published paper to them. If their editorial board finds it fitting the standards and purpose of their journal, they will include it.

The concept originated, I guess, with Timothy Gowers [^] and others’ efforts, when they started a maths journal called “Discrete Analysis.” At least I do remember reading about it last year [^]. Here is Gowers’ recent blog post reflecting on the success of this arXiv-overlay journal [^]. Here is what Nature had to report about the movement a few months ago [^].

How I wish there were an arXiv for engineering sciences too.

Especially in India, there has been a proliferation of bad journals: very poor quality, but they carry an ISSN, and they are accepted as journals in the Indian academia. I don’t have to take names; just check out the record of most any engineering professor from outside the IISc/IIT system, and you will immediately come to know what I mean.

At the same time, for graduate students, especially for the good PhD students who happen to lie outside the IIT system (there are quite a few such people), and for that matter even for MTech students in IITs, finding a good publication venue sometimes is difficult. Journal publications take time—1 or 2 years is common. Despite its size, population, or GDP, India hardly has any good journals being published from here. At the same time, India has a very large, sophisticated, IT industry.

Could this idea—arXiv-overlay journal—be carried into engineering space and in India? Could the Indian IT industry help in some ways—not just technical assistance in creating and maintaining the infrastructure, but also by way of financial assistance to do that?

We know the answer already in advance. But what the hell! What is the harm in at least mentioning it on a blog?


Just an Aside (re. QM): I spent some time noting down, on my mental scratch-pad, how QM should be presented, and in doing so, ended up with some rough outlines of  a new way to do so. I will write about it once I regain enough levels of enthusiasm.


The Imposter Syndrome:

It seems to have become fashionable to talk of the imposter syndrome [^]. The first time I read the term was while going through Prof. Abinandanan’s “nanopolitan” blog [^]. Turns out that it’s a pretty widely discussed topic [^], with one write-up even offering the great insight that “true imposters don’t suffer imposter syndrome” [^]. … I had smelled, albeit mildly, something like a leftist variety of a dead rat here… Anyway, at least writing about the phenomenon does seem to be prevalent among science-writers; here is a latest (H/T Sean Carroll’s feed) [^]…

Anyway, for the record: No, I have not ever suffered from the imposter syndrome, not even once in my life, nor do I expect to do so in future.

I don’t think the matter is big enough for me to spend any significant time analyzing it, but if you must (or if you somehow do end up analyzing it, for whatever reasons), here is a hint: In your work, include the concept of “standards,” and ask yourself just one question: does the author rest his standards in reason and reality, or does he do so in some people—which, in case of the imposter syndrome, would be: the other people.

Exercise: What (all) would stand opposite in meaning to the imposter syndrome? Do you agree with the suggestion here [^]?


The US Presidential Elections: Why are they so “big”? should they be?

Recently, I made a comment at Prof. Scott Aaronson’s blog, and at that time, I had thought that I would move it here as a separate post in its own right. However, I don’t think I have the energy right now, and once it returns, I am not sure if it will not get lost in the big stack of things to do. Anyway, here is the link [^]. … As I said, I am not interested much—if at all—in the US politics, but the question I dealt with was definitely a general one.


Overall, though, my mood of boredom continues… Yaawwwnnnn….


A Song I Like:

(Hindi) “seene mein jalan…”
Lyrics: Shahryar
Music: Jaidev
Singer: Suresh Wadkar

[Pune today is comparable to the Bombay of 1979 1978—but manages to stay less magnificent.]


Update on 2016.11.17: English translation of the song:

For my English blog-readers: A pretty good translation of the lyrics is available at Atul’s site; it is done by one Sudhir; see here [^]. This translation is much better than the English sub-titles appearing in this YouTube video [^] which comes as the first result when you google for this song. …

I am not completely happy with Sudhir’s translation (on Atul’s site) either, though it is pretty good. At a couple of places or so, it gives a slightly different shade of meaning than what the original Urdu words convey.

For instance, in the first stanza, instead of

“Just for that there is a heart inside,
one searches a pretext to be alive,”

it should be something like:

“just because there is a heart,
someone searches (i.e., people search) for an excuse which can justify its beating”

Similarly, in the second stanza,  instead of:

“what is this new intensity of loneliness, my friend?”,

a more accurate translation would be:

“what kind of a station in the journey of loneliness is this, my friends?”.

The Urdu word “manzil” means: parts of the Koran, and then, it has also come to mean: a stage in a journey, a station, a destination, or even a floor in a multi-storied building. But in no case does it mean intensity, as such. The underlying thought here is something like this: “loneliness is OK, but look, what kind of a lonely place it is that I have ended up in, my friends!” And the word for “friend” appears in the plural, not the singular. The song is one of a silent/quiet reflection; it is addressed to everyone in general and none in particular.

… Just a few things like that, but yes, speaking overall, Sudhir’s translation certainly is pretty good. Much better than what I could have done purely on my own, and in any case, it is strongly recommended. … The lyrics are an indispensable part of the soul of this song—in fact, the song is so damn well-integrated, all its elements are! So, do make sure to see Sudhir’s translation, too.


Update on 2016.11.18: My own English translation:

I have managed to complete my English translation of the above song. Let me share it with you. I benefitted a great deal from Sudhir’s translation and notes about the meanings of the words, mentioned in the note above, as well as further from “ek fankaar” [^]. My translation tries to closely follow not only the original words but also their sequence. To maintain continuity, the translation is given for the entire song as a piece.

First, the original Hindi/Urdu words:

seene mein jalan aankhon mein toofaan sa kyun hai
is shehar mein har shakhs pareshaan saa kyun hai

dil hai to dhadakne ka bahaanaa koi dhoondhe
patthar ki tarah behis-o-bejaan sa kyun hai

tanahaai ki ye kaun si manzil hai rafeeqon
ta-hadd-e-nazar ek bayaabaan saa kyon hai

kyaa koi nai baat nazar aati hai ham mein
aainaa hamen dekh ke hairaan sa kyon hai

Now, my English translation, with some punctuation added by me [and with further additions in the square brackets indicating either alternative words or my own interpolations]:

Why is there jealousy in the bosom; a tempest, as it were, in the eyes?
In this city, every person—why does it seem as if he were deeply troubled [or harassed]?

[It’s as if] Someone has a heart, so he might go on looking for an alibi [or a pretext] to justify [keeping it] beating
[But] A stone, as if it were that, why is it so numb and lifeless [in the first place]?

What kind of a station in the journey of the solitude is this, [my noble] friends?
Right to the end of the sight, why is there [nothing but] a sort of a total desolation?

Is there something new that has become visible about me?
The mirror, looking at me, why does it seem so bewildered [or perplexed]?

Update on 22nd Nov. 2016: OK, just one two more iterations I must have; just a slight change in the second [and the first [, and the third]] couplet[s]. (Even if further improvements would may be possible, I am now going to stop my iterations right here.):

Why is there jealousy in the bosom; a tempest, as it were, in the eyes?
In this city, every silhouette [of a person]—why does it seem as if he were deeply troubled [or harassed]?

[It’s as if] A heart, one does have, and so, someone might go on looking for an alibi [or a pretext] to justify [keeping it] beating
[But] A stone, as if it were that, why is it so numb and lifeless [in the first place]?

What kind of a station in the journey of the solitude is this, [my noble] friends?
[That] Right to the end of the sight, why is there [nothing but] a sort of a total desolation?

Is there something new that has become visible about me?
The mirror, looking at me, why does it seem so bewildered [or perplexed]?

 


[E&OE]

 

Free books on the nature of mathematics

Just passing along a quick tip, in case you didn’t know about it:

Early editions of quite a few wonderful books concerning history and nature of mathematics have now become available for free downloading at archive.org. (I hope they have checked the copyrights and all):

Books by Prof. Morris Kline:

  1. Mathematics in Western Culture (1954) [^]
  2. Mathematics and the Search for Knowledge (1985) [^]
  3. Mathematics and the Physical World (1959) [^] (I began Kline’s books with this one.)

Of course, Kline’s 3-volume book, “Mathematical Thought from Ancient to Modern Times,” is the most comprehensive and detailed one. However, it is not yet available off archive.org. But that hardly matters, because the book is in print, and a pretty inexpensive (Rs. ~1600) paperback is available at Amazon [^]. The Kindle edition is just Rs. 400.

(No, I don’t have Kindle. Neither do I plan to buy one. I will probably not use it even if someone gives it to me for free. I am sure I will find someone else to pass it on for free, again! … I don’t have any use for Kindle. I am old enough to like my books only the old-fashioned way—the fresh smell of the paper and the ink included. Or, the crispiness of the fading pages of an old one. And, I like my books better in the paperback format, not hard-cover. Easy to hold while comfortably reclining in my chair or while lying over a sofa or a bed.)

Anyway, back to archive.org.

Prof. G. H. Hardy’s “A Mathematician’s Apology,” too, has become available for free downloading [^]. It’s been more than two decades since I first read it. … Would love to find time to go through it again.

Anyway, enjoy! (And let me know if you run into some other interesting books at archive.org.)

* * * * *   * * * * *   * * * * *

A Song I Like:
(Hindi) “chain se hum ko kabhie…”
Music: O. P. Nayyar
Singer: Asha Bhosale
Lyrics: S. H. Bihari

Incidentally, I have often thought that this song was ideally suited for a saxophone, i.e., apart from Asha’s voice. Not just any instrument, but, specifically, only a saxophone. … Today I searched for, and heard for the first time, a sax rendering—the one by Babbu Khan. It’s pretty good, though I had a bit of a feeling that someone could do better, probably, a lot better. Manohari Singh? Did he ever play this song on a sax?

As to the other instruments, though I often do like to listen to a flute (I mean the Indian flute (“baansuri”)), this song simply is not at all suited to one. For instance, just listen to Shridhar Kenkare’s rendering. The entire (Hindi) “dard” gets lost, and then, worse: that sweetness oozing out in its place, is just plain irritating. At least to me. On the other hand, also locate on the ‘net a violin version of this song, and listen to it. It’s pathetic. … Enough for today. I have lost the patience to try out any piano version, though I bet it would sound bad, too.

Sax. This masterpiece is meant for the sax. And, of course, Asha.

[E&OE]