An interesting problem from the classical mechanics of vibrations

Update on 18 June 2017:
Added three diagrams depicting the mathematical abstraction of the problem; see near the end of the post. Also added one more consideration by way of an additional question.


TL;DR: A very brief version of this post is now posted at iMechanica; see here [^].


How I happened to come to formulate this problem:

As mentioned in my last post, I had started writing down my answers to the conceptual questions from Eisberg and Resnick’s QM text. However, as soon as I began doing that (typing out my answer to the first question from the first chapter), almost predictably, something else happened.

Since it anyway was QM that I was engaged with, somehow, another issue from QM—one which I had thought about a bit some time ago—happened to now just surface up in my mind. And it was an interesting issue. Back then, I had not thought of reaching an answer, and even now, I realized, I had not very satisfactory answer to it, not even in just conceptual terms. Naturally, my mind remained engaged in thinking about this second QM problem for a while.

In trying to come to terms with this QM problem (of my own making, not E&R’s), I now tried to think of some simple model problem from classical mechanics that might capture at least some aspects of this QM issue. Thinking a bit about it, I realized that I had not read anything about this classical mechanics problem during my [very] limited studies of the classical mechanics.

But since it appeared simple enough—heck, it was just classical mechanics—I now tried to reason through it. I thought I “got” it. But then, right the next day, I began doubting my own answer—with very good reasons.

… By now, I had no option but to keep aside the more scholarly task of writing down answers to the E&R questions. The classical problem of my own making had begun becoming all interesting by itself. Naturally, even though I was not procrastinating, I still got away from E&R—I got diverted.

I made some false starts even in the classical version of the problem, but finally, today, I could find some way through it—one which I think is satisfactory. In this post, I am going to share this classical problem. See if it interests you.


Background:

Consider an idealized string tautly held between two fixed end supports that are a distance L apart; see the figure below. The string can be put into a state of vibrations by plucking it. There is a third support exactly at the middle; it can be removed at will.

 

 

 

Assume all the ideal conditions. For instance, assume perfectly rigid and unyielding supports, and a string that is massive (i.e., one which has a lineal mass density; for simplicity, assume this density to be constant over the entire string length) but having zero thickness. The string also is perfectly elastic and having zero internal friction of any sort. Assume that the string is surrounded by the vacuum (so that the vibrational energy of the string does not leak outside the system). Assume the absence of any other forces such as gravitational, electrical, etc. Also assume that the middle support, when it remains touching the string, does not allow any leakage of the vibrational energy from one part of the string to the other. Feel free to make further suitable assumptions as necessary.

The overall system here consists of the string (sans the supports, whose only role is to provide the necessary boundary conditions).

Initially, the string is stationary. Then, with the middle support touching the string, the left-half of the string is made to undergo oscillations by plucking it somewhere in the left-half only, and immediately releasing it. Denote the instant of the release as, say t_R. After the lapse of a sufficiently long time period, assume that the left-half of the system settles down into a steady-state standing wave pattern. Given our assumptions, the right-half of the system continues to remain perfectly stationary.

The internal energy of the system at t_0 is 0. Energy is put into the system only once, at t_R, and never again. Thus, for all times t > t_R, the system behaves as a thermodynamically isolated system.

For simplicity, assume that the standing waves in the left-half form the fundamental mode for that portion (i.e. for the length L/2). Denote the frequency of this fundamental mode as \nu_H, and its max. amplitude (measured from the central line) as A_H.

Next, at some instant of time t = t_1, suppose that the support in the middle is suddenly removed, taking care not to disturb the string in any way in the process. That is to say, we  neither put in any more energy in the system nor take out of it, in the process of removing the middle support.

Once the support is thus removed, the waves from the left-half can now travel to the right-half, get reflected from the right end-support, travel all the way to the left end-support, get reflected there, etc. Thus, they will travel back and forth, in both the directions.

Modeled as a two-point BV/IC problem, assume that the system settles down into a steadily repeating pattern of some kind of standing waves.

The question now is:

What would be the pattern of the standing waves formed in the system at a time t_F \gg t_1?


The theory suggests that there is no unique answer!:

Here is one obvious answer:

Since the support in the middle was exactly at the midpoint, removing it has the effect of suddenly doubling the length for the string.

Now, simple maths of the normal modes tells you that the string can vibrate in the fundamental mode for the entire length, which means: the system should show standing waves of the frequency \nu_F = \nu_H/2.

However, there also are other, theoretically conceivable, answers.

For instance, it is also possible that the system gets settled into the first higher-harmonic mode. In the very first higher-harmonic mode, it will maintain the same frequency as earlier, i.e., \nu_F = \nu_H, but being an isolated system, it has to conserve its energy, and so, in this higher harmonic mode, it must vibrate with a lower max. amplitude A_F < A_H. Thermodynamically speaking, since the energy is conserved also in such a mode, it also should certainly be possible.

In fact, you can take the argument further, and say that any one or all of the higher harmonics (potentially an infinity of them) would be possible. After all, the system does not have to maintain a constant frequency or a constant max. amplitude; it only has to maintain the same energy.

OK. That was the idealized model and its maths. Now let’s turn to reality.


Relevant empirical observations show that only a certain answer gets selected:

What do you actually observe in reality for systems that come close enough to the above mentioned idealized description? Let’s take a range of examples to get an idea of what kind of a show the real world puts up….

Consider, say, a violinist’s performance. He can continuously alter the length of the vibrations with his finger, and thereby produce a continuous spectrum of frequencies. However, at any instant, for any given length for the vibrating part, the most dominant of all such frequencies is, actually, only the fundamental mode for that length.

A real violin does not come very close to our idealized example above. A flute is better, because its spectrum happens to be the purest among all musical instruments. What do we mean by a “pure” tone here? It means this: When a flutist plays a certain tone, say the middle “saa” (i.e. the middle “C”), the sound actually produced by the instrument does not significantly carry any higher harmonics. That is to say, when a flutist plays the middle  “saa,” unlike the other musical instruments, the flute does not inadvertently go on to produce also the “saa”s from any of the higher octaves. Its energy remains very strongly concentrated in only a single tone, here, the middle “saa”. Thus, it is said to be a “pure” tone; it is not “contaminated” by any of the higher harmonics. (As to the lower harmonics for a given length, well, they are ruled out because of the basic physics and maths.)

Now, if you take a flute of a variable length (something like a trumpet) and try very suddenly doubling the length of the vibrating air column, you will find that instead of producing a fainter sound of the same middle “saa”, the flute instead produces the next lower “saa”. (If you want, you can try it out more systematically in the laboratory by taking a telescopic assembly of cylinders and a tuning fork.)

Of course, really speaking, despite its pure tones, even the flute does not come close enough to our idealized description above. For instance, notice that in our idealized description, energy is put into the system only once, at t_R, and never again. On the other hand, in playing a violin or a flute we are continuously pumping in some energy; the system is also continuously dissipating its energy to its environment via the sound waves produced in the air. A flute, thus, is an open system; it is not an isolated system. Yet, despite the additional complexity introduced because of an open system, and therefore, perhaps, a greater chance of being drawn into higher harmonic(s), in reality, a variable length flute is always observed to “select” only the fundamental harmonic for a given length.

How about an actual guitar? Same thing. In fact, the guitar comes closest to our idealized description. And if you try out plucking the string once and then, after a while, suddenly removing the finger from a fret, you will find that the guitar too “prefers” to immediately settle down rather in the fundamental harmonic for the new length. (Take an electric guitar so that even as the sound turns fainter and still fainter due to damping, you could still easily make out the change in the dominant tone.)

OK. Enough of empirical observations. Back to the connection of these observations with the theory of physics (and maths).


The question:

Thermodynamically, an infinity of tones are perfectly possible. Maths tells you that these infinity of tones are nothing but the set of the higher harmonics (and nothing else). Yet, in reality, only one tone gets selected. What gives?

What is the missing physics which makes the system get settled into one and only one option—indeed an extreme option—out of an infinity of them of which are, energetically speaking, equally possible?


Update on 18 June 2017:

Here is a statement of the problem in certain essential mathematical terms. See the three figures below:

The initial state of the string is what the following figure (Case 1) depicts. The max. amplitude is 1.0. Though the quiescent part looks longer than half the length, it’s just an illusion of perception.:

Fundamental tone for the half length, extended over a half-length

Case 1: Fundamental tone for the half length, extended over a half-length

The following figure (Case 2) is the mathematical idealization of the state in which an actual guitar string tends to settle in. Note that the max. amplitude is greater (it’s \sqrt{2}) so  as to have the energy of this state the same as that of Case 1.

Case 2: Fundamental tone for the full length, extended over the full length

Case 2: Fundamental tone for the full length, extended over the full length

 

 

 

 

 

 

 

 

The following figure (Case 3) depicts what mathematically is also possible for the final system state. However, it’s not observed with actual guitars. Note, here, the frequency is half of that in the Case 1, and the wavelength is doubled. The max. amplitude for this state is less than 1.0 (it’s \dfrac{1}{\sqrt{2}}) so as to have this state too carry exactly the same energy as in Case 1.

Case 3: The first overtone for the full length, extended over the full length

Case 3: The first overtone for the full length, extended over the full length

 

 

 

 

 

 

 

 

Thus, the problem, in short is:

The transition observed in reality is: T1: Case 1 \rightarrow Case 2.

However, the transition T2: Case 1 \rightarrow Case 3 also is possible by the mathematics of standing waves and thermodynamics (or more basically, by that bedrock on which all modern physics rests, viz., the calculus of variations). Yet, it is not observed.

Why does only T1 occur? why not T2? or even a linear combination of both? That’s the problem, in essence.

While attempting to answer it, also consider this : Can an isolated system like the one depicted in the Case 1 at all undergo a transition of modes?

Enjoy!

Update on 18th June 2017 is over.


That was the classical mechanics problem I said I happened to think of, recently. (And it was the one which took me away from the program of answering the E&R questions.)

Find it interesting? Want to give it a try?

If you do give it a try and if you reach an answer that seems satisfactory to you, then please do drop me a line. We can then cross-check our notes.

And of course, if you find this problem (or something similar) already solved somewhere, then my request to you would be stronger: do let me know about the reference!


In the meanwhile, I will try to go back to (or at least towards) completing the task of answering the E&R questions. [I do, however, also plan to post a slightly edited version of this post at iMechanica.]


Update History:

07 June 2017: Published on this blog

8 June 2017, 12:25 PM, IST: Added the figure and the section headings.

8 June 2017, 15:30 hrs, IST: Added the link to the brief version posted at iMechanica.

18 June 2017, 12:10 hrs, IST: Added the diagrams depicting the mathematical abstraction of the problem.


A Song I Like:

(Marathi) “olyaa saanj veli…”
Music: Avinash-Vishwajeet
Singers: Swapnil Bandodkar, Bela Shende
Lyrics: Ashwini Shende

 

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See, how hard I am trying to become an Approved (Full) Professor of Mechanical Engineering in SPPU?—4

In this post, I provide my answer to the question which I had raised last time, viz., about the differences between the \Delta, the \text{d}, and the \delta (the first two, of the usual calculus, and the last one, of the calculus of variations).


Some pre-requisite ideas:

A system is some physical object chosen (or isolated) for study. For continua, it is convenient to select a region of space for study, in which case that region of space (holding some physical continuum) may also be regarded as a system. The system boundary is an abstraction.

A state of a system denotes a physically unique and reproducible condition of that system. State properties are the properties or attributes that together uniquely and fully characterize a state of a system, for the chosen purposes. The state is an axiom, and state properties are its corollary.

State properties for continua are typically expressed as functions of space and time. For instance, pressure, temperature, volume, energy, etc. of a fluid are all state properties. Since state properties uniquely define the condition of a system, they represent definite points in an appropriate, abstract, (possibly) higher-dimensional state space. For this reason, state properties are also called point functions.

A process (synonymous to system evolution) is a succession of states. In classical physics, the succession (or progression) is taken to be continuous. In quantum mechanics, there is no notion of a process; see later in this post.

A process is often represented as a path in a state space that connects the two end-points of the staring and ending states. A parametric function defined over the length of a path is called a path function.

A cyclic process is one that has the same start and end points.

During a cyclic process, a state function returns to its initial value. However, a path function does not necessarily return to the same value over every cyclic change—it depends on which particular path is chosen. For instance, if you take a round trip from point A to point B and back, you may spend some amount of money m if you take one route but another amount n if you take another route. In both cases you do return to the same point viz. A, but the amount you spend is different for each route. Your position is a state function, and the amount you spend is a path function.

[I may make the above description a bit more rigorous later on (by consulting a certain book which I don’t have handy right away (and my notes of last year are gone in the HDD crash)).]


The \Delta, the \text{d}, and the \delta:

The \Delta denotes a sufficiently small but finite, and locally existing difference in different parts of a system. Typically, since state properties are defined as (continuous) functions of space and time, what the \Delta represents is a finite change in some state property function that exists across two different but adjacent points in space (or two nearby instants in times), for a given system.

The \Delta is a local quantity, because it is defined and evaluated around a specific point of space and/or time. In other words, an instance of \Delta is evaluated at a fixed x or t. The \Delta x simply denotes a change of position; it may or may not mean a displacement.

The \text{d} (i.e. the infinitesimal) is nothing but the \Delta taken in some appropriate limiting process to the vanishingly small limit.

Since \Delta is locally defined, so is the infinitesimal (i.e. \text{d}).

The \delta of CoV is completely different from the above two concepts.

The \delta is a sufficiently small but global difference between the states (or paths) of two different, abstract, but otherwise identical views of the same physically existing system.

Considering the fact that an abstract view of a system is itself a system, \delta also may be regarded as a difference between two systems.

Though differences in paths are not only possible but also routinely used in CoV, in this post, to keep matters simple, we will mostly consider differences in the states of the two systems.

In CoV, the two states (of the two systems) are so chosen as to satisfy the same Dirichlet (i.e. field) boundary conditions separately in each system.

The state function may be defined over an abstract space. In this post, we shall not pursue this line of thought. Thus, the state function will always be a function of the physical, ambient space (defined in reference to the extensions and locations of concretely existing physical objects).

Since a state of a system of nonzero size can only be defined by specifying its values for all parts of a system (of which it is a state), a difference between states (of the two systems involved in the variation \delta) is necessarily global.

In defining \delta, both the systems are considered only abstractly; it is presumed that at most one of them may correspond to an actual state of a physical system (i.e. a system existing in the physical reality).

The idea of a process, i.e. the very idea of a system evolution, necessarily applies only to a single system.

What the \delta represents is not an evolution because it does not represent a change in a system, in the first place. The variation, to repeat, represents a difference between two systems satisfying the same field boundary conditions. Hence, there is no evolution to speak of. When compressed air is passed into a rubber balloon, its size increases. This change occurs over certain time, and is an instance of an evolution. However, two rubber balloons already inflated to different sizes share no evolutionary relation with each other; there is no common physical process connecting the two; hence no change occurring over time can possibly enter their comparative description.

Thus, the “change” denoted by \delta is incapable of representing a process or a system evolution. In fact, the word “change” itself is something of a misnomer here.

Text-books often stupidly try to capture the aforementioned idea by saying that \delta represents a small and possibly finite change that occurs without any elapse of time. Apart from the mind-numbing idea of a finite change occurring over no time (or equally stupefying ideas which it suggests, viz., a change existing at literally the same instant of time, or, alternatively, a process of change that somehow occurs to a given system but “outside” of any time), what they, in a way, continue to suggest also is the erroneous idea that we are working with only a single, concretely physical system, here.

But that is not the idea behind \delta at all.

To complicate the matters further, no separate symbol is used when the variation \delta is made vanishingly small.

In the primary sense of the term variation (or \delta), the difference it represents is finite in nature. The variation is basically a function of space (and time), and at every value of x (and t), the value of \delta is finite, in the primary sense of the word. Yes, these values can be made vanishingly small, though the idea of the limits applied in this context is different. (Hint: Expand each of the two state functions in a power series and relate each of the corresponding power terms via a separate parameter. Then, put the difference in each parameter through a limiting process to vanish. You may also use the Fourier expansion.))

The difference represented by \delta is between two abstract views of a system. The two systems are related only in an abstract view, i.e., only in (the mathematical) thought. In the CoV, they are supposed as connected, but the connection between them is not concretely physical because there are no two separate physical systems concretely existing, in the first place. Both the systems here are mathematical abstractions—they first have been abstracted away from the real, physical system actually existing out there (of which there is only a single instance).

But, yes, there is a sense in which we can say that \delta does have a physical meaning: it carries the same physical units as for the state functions of the two abstract systems.


An example from biology:

Here is an example of the differences between two different paths (rather than two different states).

Plot the height h(t) of a growing sapling at different times, and connect the dots to yield a continuous graph of the height as a function of time. The difference in the heights of the sapling at two different instants is \Delta h. But if you consider two different saplings planted at the same time, and assuming that they grow to the same final height at the end of some definite time period (just pick some moment where their graphs cross each other), and then, abstractly regarding them as some sort of imaginary plants, if you plot the difference between the two graphs, that is the variation or \delta h(t) in the height-function of either. The variation itself is a function (here of time); it has the units, of course, of m.


Summary:

The \Delta is a local change inside a single system, and \text{d} is its limiting value, whereas the \delta is a difference across two abstract systems differing in their global states (or global paths), and there is no separate symbol to capture this object in the vanishingly small limit.


Exercises:

Consider one period of the function y = A \sin(x), say over the interval [0,2\pi]; A = a is a small, real-valued, constant. Now, set A = 1.1a. Is the change/difference here a \delta or a \Delta? Why or why not?

Now, take the derivative, i.e., y' = A \cos(x), with A = a once again. Is the change/difference here a \delta or a \Delta? Why or why not?

Which one of the above two is a bigger change/difference?

Also consider this angle: Taking the derivative did affect the whole function. If so, why is it that we said that \text{d} was necessarily a local change?


An important and special note:

The above exercises, I am sure, many (though not all) of the Officially Approved Full Professors of Mechanical Engineering at the Savitribai Phule Pune University and COEP would be able to do correctly. But the question I posed last time was: Would it be therefore possible for them to spell out the physical meaning of the variation i.e. \delta? I continue to think not. And, importantly, even among those who do solve the above exercises successfully, they wouldn’t be too sure about their own answers. Upon just a little deeper probing, they would just throw up their hands. [Ditto, for many American physicists.] Even if a conceptual clarity is required in applications.

(I am ever willing and ready to change my mind about it, but doing so would need some actual evidence—just the way my (continuing) position had been derived, in the first place, from actual observations of them.)

The reason I made this special note was because I continue to go jobless, and nearly bank balance-less (and also, nearly cashless). And it all is basically because of folks like these (and the Indians like the SPPU authorities). It is their fault. (And, no, you can’t try to lift what is properly their moral responsibility off their shoulders and then, in fact, go even further, and attempt to place it on mine. Don’t attempt doing that.)


A Song I Like:

[May be I have run this song before. If yes, I will replace it with some other song tomorrow or so. No I had not.]

Hindi: “Thandi hawaa, yeh chaandani suhaani…”
Music and Singer: Kishore Kumar
Lyrics: Majrooh Sultanpuri

[A quick ‘net search on plagiarism tells me that the tune of this song was lifted from Julius La Rosa’s 1955 song “Domani.” I heard that song for the first time only today. I think that the lyrics of the Hindi song are better. As to renditions, I like Kishor Kumar’s version better.]


[Minor editing may be done later on and the typos may be corrected, but the essentials of my positions won’t be. Mostly done right today, i.e., on 06th January, 2017.]

[E&OE]

 

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]

See, how hard I am trying to become an Approved (Full) Professor of Mechanical Engineering in SPPU?—2

Remember the age-old decade-old question, viz.:

“Stress or strain: which one is more fundamental?”

I myself had posed it at iMechanica about a decade ago [^]. Specifically, on 8th March 2007 (US time, may be EST or something).

The question had generated quite a bit of discussion at that time. Even as of today, this thread remains within the top 5 most-hit posts at iMechanica.

In fact, as of today, with about 1.62 lakh reads (i.e. 162 k hits), I think, it is the second most hit post at iMechanica. The only post with more hits, I think, is Nanshu Lu’s, providing a tutorial for the Abaqus software [^]; it beats mine like hell, with about 5 lakh (500 k) hits! The third most hit post, I think, again is about sharing scripts for the Abaqus software [^]; as of today, it lags mine very closely, but could overtake mine anytime, with about 1.48 lakh (148 k) hits already. There used to be a general thread on Open Source FEM software that used to be very close to my post. As of today, it has fallen behind a bit, with about 1.42 lakh (142 k) hits [^]. (I don’t know, but there could be other widely read posts, too.)

Of course, the attribute “most hit” is in no fundamental way related to “most valuable,” “most relevant,” or even “most interesting.”

Yet, the fact of the matter also is that mine is the only one among the top 5 posts which probes on a fundamental theoretical aspect. All others seem to be on software. Not very surprising, in a way.

Typically, hits get registered for topics providing some kind of a practical service. For instance, tips and tutorials on software—how to install a software, how to deal with a bug, how to write a sub-routine, how to produce visualizations, etc. Topics like these tend to get more hits. These are all practical matters, important right in the day-to-day job or studies, and people search the ‘net more for such practically useful services. Precisely for this reason—and especially given the fact that iMechanica is a forum for engineers and applied scientists—it is unexpected (at least it was unexpected to me) that a “basically useless” and “theoretical” discussion could still end up being so popular. There certainly was a surprise about it, to me. … But that’s just one part.

The second, more interesting part (i.e., more interesting to me) has been that, despite all these reads, and despite the simplicity of the concepts involved (stress and strain), the issue went unresolved for such a long time—almost a decade!

Students begin to get taught these two concepts right when they are in their XI/XII standard. In my XI/XII standard, I remember, we even had a practical about it: there was a steel wire suspended from a cantilever near the ceiling, and there was hook with a supporting plate at the bottom of this wire. The experiment consisted of adding weights, and measuring extensions. … Thus, the learning of these concepts begins right around the same time that students are learning calculus and Newton’s  3 laws… Students then complete the acquisition of these two concepts in their “full” generality, right by the time they are just in the second- or third-year of undergraduate engineering. The topic is taught in a great many branches of engineering: mechanical, civil, aerospace, metallurgical, chemical, naval architecture, and often-times (and certainly in our days and in COEP) also electrical. (This level of generality would be enough to discuss the question as posed at iMechanica.)

In short, even if the concepts are so “simple” that UG students are routinely taught them, a simple conceptual question involving them could go unresolved for such a long time.

It is this fact which was (honestly) completely unexpected to me, at least at the time when I had posed the question.

I had actually thought that there would surely be some reference text/paper somewhere that must have considered this aspect already, and answered it. But I was afraid that the answer (or the reference in which it appears) could perhaps be outside of my reach, my understanding of continuum mechanics. (In particular, I knew only a little bit of tensor calculus—only that as given in Malvern, and in Schaum’s series, basically. (I still don’t know much more about tensor calculus; my highest reach for tensor calculus remains limited to the book by Prof. Allan Bower of Brown [^].)) Thus, the reason I wrote the question in such a great detail (and in my replies, insisted on discussing the issues in conceptual details) was only to emphasize the fact that I had no hi-fi tensor calculus in mind; only the simplest physics-based and conceptual explanation was what I was looking for.

And that’s why, the fact that the question went unresolved for so long has also been (actually) fascinating to me. I (actually) had never expected it.


And yes, “dear” Officially Approved Mechanical Engineering Professors at the Savitribai Phule Pune University (SPPU), and authorities at SPPU, as (even) you might have noticed, it is a problem concerning the very core of the Mechanical Engineering proper.


I had thought once, may be last year or so, that I had finally succeeded in nailing down the issue right. (I might have written about it on this blog or somewhere else.) But, still, I was not so sure. So, I decided to wait.

I now have come to realize that my answer should be correct.


I, however, will not share my answer right away. There are two reasons for it.

First, I would like it if someone else gives it a try, too. It would be nice to see someone else crack it, too. A little bit of a wait is nothing to trade in for that. (As far as I am concerned, I’ve got enough “popularity” etc. just out of posing it.)

Second, I also wish to see if the Officially Approved Mechanical Engineering Professors at the Savitribai Phule Pune University (SPPU)) would be willing and able to give it a try.

(Let me continue to be honest. I do not expect them to crack it. But I do wish to know whether they are able to give it a try.)

In fact, come to think of it, let me do one thing. Let me share my answer only after one of the following happens:

  • either I get the Official Approval (and also a proper, paying job) as a Full Professor of Mechanical Engineering at SPPU,
  • or, an already Officially Approved Full Professor of Mechanical Engineering at SPPU (especially one of those at COEP, especially D. W. Pande, and/or one of those sitting on the Official COEP/UGC Interview Panels for faculty interviews at SPPU) gives it at least a try that is good enough. [Please note, the number of hits on the international forum of iMechanica, and the nature of the topic, once again.]

I will share my answer as soon as either of the above two happens—i.e., in the Indian government lingo: “whichever is earlier” happens.


But, yes, I am happy that I have come up with a very good argument to finally settle the issue. (I am fairly confident that my eventual answer should also be more or less satisfactory to those who had participated on this iMechanica thread. When I share my answer, I will of course make sure to note it also at iMechanica.)


This time round, there is not just one song but quite a few of them competing for inclusion on the “A Song I Like” section. Perhaps, some of these, I have run already. Though I wouldn’t mind repeating a song, I anyway want to think a bit about it before finalizing one. So, let me add the section when I return to do some minor editing later today or so. (I certainly want to get done with this post ASAP, because there are other theoretical things that beckon my attention. And yes, with this announcement about the stress-and-strain issue, I am now going to resume my blogging on topics related to QM, too.)

Update at 13:40 hrs (right on 19 Dec. 2016): Added the section on a song I like; see below.


A Song I Like:

(Marathi) “soor maagoo tulaa mee kasaa? jeevanaa too tasaa, mee asaa!”
Lyrics: Suresh Bhat
Music: Hridaynath Mangeshkar
Singer: Arun Date

It’s a very beautiful and a very brief poem.

As a song, it has got fairly OK music and singing. (The music composer could have done better, and if he were to do that, so would the singer. The song is not in a bad shape in its current form; it is just that given the enormously exceptional talents of this composer, Hridaynath Mangeshkar, one does get a feel here that he could have done better, somehow—don’t ask me how!) …

I will try to post an English translation of the lyrics if I find time. The poem is in a very, very simple Marathi, and for that reason, it would also be very, very easy to give a rough sense of it—i.e., if the translation is to be rather loose.

The trouble is, if you want to keep the exact shade of the words, it then suddenly becomes very difficult to translate. That’s why, I make no promises about translating it. Further, as far as I am concerned, there is no point unless you can convey the exact shades of the original words. …

Unless you are a gifted translator, a translation of a poem almost always ends up losing the sense of rhythm. But even if you keep a more modest aim, viz., only of offering an exact translation without bothering about the rhythm part, the task still remains difficult. And it is more difficult if the original words happen to be of the simple, day-to-day usage kind. A poem using complex words (say composite, Sanskrit-based words) would be easier to translate precisely because of its formality, precisely because of the distance it keeps from the mundane life… An ordinary poet’s poem also would be easy to translate regardless of what kind of words he uses. But when the poet in question is great, and uses simple words, it becomes a challenge, because it is difficult, if not impossible, to convey the particular sense of life he pours into that seemingly effortless composition. That’s why translation becomes difficult. And that’s why I make no promises, though a try, I would love to give it—provided I find time, that is.


Second Update on 19th Dec. 2016, 15:00 hrs (IST):

A Translation of the Lyrics:

I offer below a rough translation of the lyrics of the song noted above. However, before we get to the translation, a few notes giving the context of the words are absolutely necessary.

Notes on the Context:

Note 1:

Unlike in the Western classical music, Indian classical music is not written down. Its performance, therefore, does not have to conform to a pre-written (or a pre-established) scale of tones. Particularly in the Indian vocal performance, the singer is completely free to choose any note as the starting note of his middle octave.

Typically, before the actual singing begins, the lead singer (or the main instrument player) thinks of some tone that he thinks might best fit how he is feeling that day, how his throat has been doing lately, the particular settings at that particular time, the emotional interpretation he wishes to emphasize on that particular day, etc. He, therefore, tentatively picks up a note that might serve as the starting tone for the middle octave, for that particular performance. He makes this selection not in advance of the show and in private, but right on the stage, right in front of the audience, right after the curtain has already gone up. (He might select different octaves for two successive songs, too!)

Then, to make sure that his rendition is going to come out right if he were to actually use that key, that octave, what he does is to ask a musician companion (himself on the stage besides the singer) to play and hold that note on some previously well-tuned instrument, for a while. The singer then uses this key as the reference, and tries out a small movement or so. If everything is OK, he will select that key.

All this initial preparation is called (Hindi) “soor lagaanaa.” The part where the singer turns to the trusted companion and asks for the reference note to be played is called (Hindi) “soor maanganaa.” The literal translation of the latter is: “asking for the tone” or “seeking the pitch.”

After thus asking for the tone and trying it out, if the singer thinks that singing in that specific key is going to lead to a good concert performance, he selects it.

At this point, both—the singer and that companion musician—exchange glances at each other, and with that indicate that the tone/pitch selection is OK, that this part is done. No words are exchanged; only the glances. Indian performances depend a great deal on impromptu variations, on improvizations, and therefore, the mutual understanding between the companion and the singer is of crucial importance. In fact, so great is their understanding that they hardly ever exchange any words—just glances are enough. Asking for the reference key is just a simple ritual that assures both that the mutual understanding does exist.

And after that brief glance, begins the actual singing.

Note 2:

Whereas the Sanskrit and Marathi word “aayuShya” means life-span (the number of years, or the finite period that is life), the Sanskrit and Marathi word “jeevan” means Life—with a capital L. The meaning of “jeevan” thus is something like a slightly abstract outlook on the concrete facts of life. It is like the schema of life. The word is not so abstract as to mean the very Idea of Life or something like that. It is life in the usual, day-to-day sense, but with a certain added emphasis on the thematic part of it.

Note 3:

Here, the poet is addressing this poem to “jeevan” i.e., to the Life with a capital L (or the life taken in its more abstract, thematic sense). The poet is addressing Life as if the latter is a companion in an Indian singing concert. The Life is going to help him in selecting the note—the note which would define the whole scale in which to sing during the imminent live performance. The Life is also his companion during the improvisations. The poem is addressed using this metaphor.

Now, my (rough) translation:

The Refrain:
[Just] How do I ask you for the tone,
Life, you are that way [or you follow some other way], and I [follow] this way [or, I follow mine]

Stanza 1:
You glanced at me, I glanced at you,
[We] looked full well at each other,
Pain is my mirror [or the reference instrument], and [so it is] yours [too]

Stanza 2:
Even once, to [my] mind’s satisfaction,
You [oh, Life] did not ever become my [true]  mate
[And so,] I played [on this actual show of life, just whatever] the way the play happened [or unfolded]

And, finally, Note 4 (Yes, one is due):

There is one place where I failed in my translation, and most any one not knowing both the Marathi language and the poetry of Suresh Bhat would.

In Marathi, “tu tasaa, [tar] mee asaa,” is an expression of a firm, almost final, acknowledgement of (irritating kind of) differences. “If you must insist on being so unreasonable, then so be it—I am not going to stop following my mind either.” That is the kind of sense this brief Marathi expression carries.

And, the poet, Suresh Bhat, is peculiar: despite being a poet, despite showing exquisite sensitivity, he just never stops being manly, at the same time. Pain and sorrow and suffering might enter his poetry; he might acknowledge their presence through some very sensitively selected words. And yet, the underlying sense of life which he somehow manages to convey also is as if he is going to dismiss pain, sorrow, suffering, etc., as simply an affront—a summarily minor affront—to his royal dignity. (This kind of a “royal” sense of life often is very well conveyed by ghazals. This poem is a Marathi ghazal.) Thus, in this poem, when Suresh Bhat agrees to using pain as a reference point, the words still appear in such a sequence that it is clear that the agreement is being conceded merely in order to close a minor and irritating part of an argument, that pain etc. is not meant to be important even in this poem let alone in life. Since the refrain follows immediately after this line, it is clear that the stress gets shifted to the courteous question which is raised following the affronts made by one fickle, unfaithful, even idiotic Life—the question of “Just how do I treat you as a friend? Just how do I ask you for the tone?” (The form of “jeevan” or Life used by Bhat in this poem is masculine in nature, not neutral the way it is in normal Marathi.)

I do not know how to arrange the words in the translation so that this same sense of life still comes through. I simply don’t have that kind of a command over languages—any of the languages, whether Marathi or English. Hence this (4th) note. [OK. Now I am (really) done with this post.]


Anyway, take care, and bye for now…


Update on 21st Dec. 2016, 02:41 AM (IST):

Realized a mistake in Stanza 1, and corrected it—the exchange between yours and mine (or vice versa).


[E&OE]

See, how hard I am trying to become a (Full) Professor of Mechanical Engineering in SPPU?

Currently, I am not only cashless but also jobless. That’s why, I try harder.

I am trying very hard to be a (Full) Professor of Mechanical Engineering, especially at the Savitribai Phule Pune University (or SPPU for short).

That’s right.

And that’s why, I have decided to adopt an official position whereby I abandon all my other research and study interests, especially those related to the mechanics of the quanta. Instead, I have officially decided to remain interested only in the official problems from the Mechanical Engineering discipline proper—not only for my studies, but also for my research interests.

… If only I were to have my first degree in Mechanical Engineering, instead of in Metallurgy! (It was some 37.5–33.5 years ago, with my decision to choose Metallurgy being from some 36.5 years ago.) … If only I were to choose Mechanical right back then, this problem wouldn’t have arisen today. …

Tch! …

…But, well, thinking of my first degree, its circumstances—where I got it from (COEP, the engineering college with the highest cut-off merit in the entire Maharashtra state), in what class (First Class with Distinction, the highest class possible), and, most crucially, for spending all my time at what place (The Boat Club)… You know, looking back some 3.5 decades later of all those circumstances—the circumstances of how I chose Metallurgy, back then, as I was sitting at the Boat Club… Hmmm… Boat Club. … Boat Club! Boat Club!!

It gives me some ideas.

So, to better support my current endeavors of becoming an Officially Approved Full Professor of Mechanical Engineering in SPPU, may be, I should solve some Mechanical Engineering problems related to boats. Preferably, those involving not just fluid mechanics, but also mechanisms and machine design—and vibrations! [Oh yes. I must not forget them! Vibrations are, Officially, a Mechanical Engineering topic. In fact even Acoustics. …]

Thinking along such lines, I then thought of one problem, and sort of solved it too. Though I am not going to share my answer with you, I certainly want to share the problem itself with you. (Don’t ask me for answers until I get the job as an Officially Approved Full Professor in Mechanical Engineering at SPPU.)

OK, so here we go.


The Problem Description:

Consider a boat floating on a stand-still lake. The boat has a very simple shape; it is in the shape of a rectangular parallelpiped (i.e., like a shoe-box, though not quite exactly like a punt).

In the plan (i.e. the top view), the boat looks like this:
mechanicalengineeringboat

 

 

 

 

 

As shown in the figure, at the centers of the front- and back-sides of the boat, there are two circular cylindrical cavities of identical dimensions, both being fitted with reciprocating pistons. These pistons are being driven by two completely independent mechanisms. The power-trains and the prime-movers are not shown in the diagram; in this analysis, both may be taken to be mass-less and perfectly rigid. However, the boat is assumed to have some mass.

We will try to solve for the simplest possible case: perfectly rigid boat walls (with some mass), perfectly rigid but mass-less pistons, complete absence of friction between the pistons and the cylinder walls, etc.

Assume also that both the boat and the lake water are initially stand-still, and that there are no other influences affecting the motions (such as winds or water currents).

Now, let’s put the pistons in oscillatory motions. In general, the frequencies of their oscillations are not equal. Let the frequency for the left- and right-side pistons be f_L and f_R Hz, respectively.

Problem 1:

Build a suitable Mechanical Engineering model, and predict how the boat would move, in each of the following three scenarios:

  • f_L = f_R
  • f_L > f_R
  • f_L < f_R

In each case, determine (i) whether the boat as a whole (i.e. its center of mass or CM) would at all undergo any motion at all or not, (ii) if yes, whether the motion of the CM would have an element of oscillations to it or not, and finally, (iii) whether the boat (i.e. its CM) would undergo a net displacement over a large number of pistons oscillations or not (i.e., the question asks whether the so-called “time-averaged” net displacement occurs in any one direction or not), and if yes, in which direction.

You may make other minor assumptions. For instance, in each of the above 3 cases, you may assume that at time t = 0, both the pistons are at their innermost positions, with each piston beginning its motion by pushing outwards. Also check out the effect of assuming, some other, suitable, values for the initial phases.

Though not at all necessary, if it will help you, you may perhaps consider the case where the higher frequency is an integer multiple of the lower frequency, e.g., in the second of the three cases, assume f_L = n f_R, where n \in \mathcal{N}. However, note that eventually, you are expected to solve the problem in the general case, the one in which the ratio of the frequencies may be any real number. The cases of practical interest may be where the ratio ranges from 0.0 to a real number up to, say, 2.67 or 3.14 (or, may be, 5.25).

Notice that nowhere thus far have we said that the oscillatory motion of the pistons would be SHM (i.e. simple harmonic). You may begin with an SHM, but as a further problem below illustrates, the piston motion may neither be simple-harmonic, nor even symmetrical in the to- and fro-directions.

On the fluid mechanics side: In your analysis, assume that the length of the boat is much, much greater than the stroke-lengths of the pistons. Essentially, we want to ensure that the water waves produced at one end do not significantly affect the local dynamics at the other end.

You may assume a highly simplified model for the fluid—the problem is not supposed to have a crucial bearing on what kind of a fluid you assume. I mean to say, we are not looking for so detailed a model that you would have to perform a CFD analysis. (That task, we will leave to the Naval Architecture engineers.) However, do make sure to note how your model behaves for an inviscid flow vs. for a viscous flow.

So, in short, the problem is to determine the nature of the motion of the boat, if there is any—i.e., to determine if its CM undergoes a net displacement in the time-averaged sense or not, and if yes, in which direction it occurs.

Problem 2:

Assume a relatively smaller stroke-length for one of the pistons, and repeat the problem.

Problem 3:

Assume that one of the frequencies is zero, which is as good as saying that the boat is fitted with only one cylinder-and-piston. Repeat the analysis.

Problem 4:

Continue to assume that one of the frequencies is zero. Now, also assume that the outward stroke of the moving piston happens faster than its inward stroke. Determine the nature of the motion, if any, for the CM of the boat.

Problem 5 (Optional):

Assuming that the prime mover outputs a uniform circular (or rotary) motion, design a suitable mechanism which will help implement the idea of having non-SHM motions—e.g., different stroke-times in the outward and inward directions. Conduct an informal (or a more formal, calculus-based) displacement-, velocity- and acceleration-analysis, if you wish.

Give it a thought whether this entire idea of transforming a circular motion to a nonuniform reciprocating motion can be done away with, thereby saving on energy—in real life, there is friction—using certain ideas from electrical engineering and electronics.

Ooops!

No, no, no! No!! Throw out that horrendous idea! I mean the very last one!!

We want to remain concerned only with the Mechanical Engineering Problems proper. That is the Official position I have adopted, remember?


That’s right. What I described above was, really, really, really only a Mechanical Engineering Problem.

It really, really, really has nothing to do with anything else such as electrical engineering or quantum physics.

[And if even Prof. Thanu Padmanabhan (IUCAA) does not know quantum physics (he told me so once, right in person), why should I be concerned with it, anyway?]

Anyway, so, Officially speaking, I made up this problem only because I want to become an Officially Approved Full Professor of Mechanical Engineering at SPPU.


If you are interested in some other Mechanical Engineering problems, especially on the fluids-thermal side, check out my recent posts on the Eco-Cooler, and see if you can take further the analysis given in them.

I myself had made a much more advanced engineering analysis right at that time, but I am not going to give it—or its results—until some time after I land and join the kind of job I am looking for—a Full Professor’s. (And I hope that you do have the sense to see that this is not a “prestige issue” on my part.)

The post having a preliminary (quantitative) fluids-thermal analysis is here [^], though the qualitative analysis of the problem begins with an earlier post, here [^].


[Guess the problem, as given, is enough for the time being. I may even come back and add one or two variations on the problem! But no guarantees.]

Update right on 2016.12.02: OK, here are a couple of minor variations. What happens if, when a piston comes to a rest at the extreme stroke, it continues staying idle for a while, before resuming its towards-the-center motion? What if the piston motion is such that the point of zero displacement does not occur exactly at the middle of its overall stroke-length?

I may post some further variations on the problem, or suggest alternative analogous problems, in future.

Currently, I am not just cashless but also jobless. That’s why, I try harder.


More, may be later. As to the Song I Like section, I don’t have anything playing at the back of my mind right away, so let me see if something strikes me by the time I come back tomorrow to give a final editing touch to this post. In that case, I will add this section; else, I will not!


[After the update right on 2016.12.02: I am done with this post now, and if there are any errors, I will let them stay. If you find the post confusing somewhere, please do drop me a line, though. Best, and take care.]

[E&OE]

 

Conservation of angular momentum isn’t [very] fundamental!

What are the conservation principles (in physics)?

In the first course on engineering mechanics (i.e. the mechanics of rigid bodies) we are taught that there are these three conservation principles: Conservation of: (i) energy, (ii) momentum, and (iii) angular momentum. [I am talking about engineering programs. That means, we live entirely in a Euclidean, non-relativistic, world.]

Then we learn mechanics of fluids, and the conservation of (iv) mass too gets added. That makes it four.

Then we come to computational fluid dynamics (CFD), and we begin to deal with only three equations: conservation of (i) mass, (ii) momentum, and (iii) energy. What happens to the conservation of the angular momentum? Why does the course on CFD drop it? For simplicity of analysis?

Ask that question to postgraduate engineers, even those who have done a specialization in CFD, and chances are, a significant number of them won’t be able to answer that question in a very clear manner.

Some of them may attempt this line of reasoning: That’s because in deriving the fluids equations (whether for a Newtonian fluid or a non-Newtonian one), the stress tensor is already assumed to be symmetrical: the shear stresses acting on the adjacent faces are taken to be equal and opposite (e.g. \sigma_{xy} = \sigma_{yx}). The assumed equality can come about only after assuming conservation of the angular momentum, and thus, the principle is already embedded in the momentum equations, as they are stated in CFD.

If so, ask them: How about a finite rotating body—say a gyroscope? (Assume rigidity for convenience, if you wish.) Chances are, a great majority of them will immediately agree that in this case, however, we have to apply the angular momentum principle separately.

Why is there this difference between the fluids and the finite rotating bodies? After all, both are continua, as in contrast to point-particles.

Most of them would fall silent at this point. [If not, know that you are talking with someone who knows his mechanics well!]


Actually, it so turns out that in continua, the angular momentum is an emergent/derivative property—not the most fundamental one. In continua, it’s OK to assume conservation of just the linear momentum alone. If it is satisfied, the conservation of angular momentum will get satisfied automatically. Yes, even in case of a spinning wheel.

Don’t believe me?

Let me direct you to Chad Orzel; check out here [^]. Orzel writes:

[The spinning wheel] “is a classical system, so all of its dynamics need to be contained within Newton’s Laws. Which means it ought to be possible to look at how angular momentum comes out of the ordinary linear momentum and forces of the components making up the wheel. Of course, it’s kind of hard to see how this works, but that’s what we have computers for.” [Emphasis in italics is mine.]

He proceeds to put together a simple demo in Python. Then, he also expands on it further, here [^].


Cool. If you think you have understood Orzel’s argument well, answer this [admittedly deceptive] question: How about point particles? Do we need a separate conservation principle for the angular momentum, in addition to that for the linear momentum at least in their case? How about the earth and the moon system, granted that both can be idealized as point particles (the way Newton did)?

Think about it.


A Song I Like:

(Hindi) “baandhee re kaahe preet, piyaa ke sang”
Singer: Sulakshana Pandit
Music: Kalyanji-Anandji
Lyrics: M. G. Hashmat

 

[E&OE]

Squeezing in a post before the 2015 gets over…

The first purpose of this post is to own up a few nasty things that I did. Recently I posted some nasty comments on iMechanica. I got as randomly nasty in them as I could.

My overwhelming mental state at that time was to show just a (mild) example of the “received” things, of what I have had to endure, for years. In fact what I had to endure has been far worse than mere comments on the ‘net, but I tried to keep it aside even in that nasty moment. … Yes, that’s right. I have resisted putting out nastiness, in response to that which I have gotten over years (for more than a decade-and-a-half!). I have not succeeded always, and this recent instance is one of that infrequent times I could not.

On the other hand, check the better side of my record at the same forum, I mean iMechanica: Hundreds of comments on more than two hundred threads.

Yes, I do regret my recent “response.” But if you ask me, the issue has gone beyond the considerations of justifiable-ness and otherwise. Not in the sense that moral principles don’t apply for such things (exchanges on the Internet), but in this sense: Let us change the chairs. I mean to say: Even if someone else in my position were to write ten-folds more such comments, and if I on the other hand were to be in a general observer’s position, then: the current state of the world is such that I would no longer have a right to expect any better coming off him. If anything else better were at all to come off him, I may or may not be grateful (it would depend on the specific value of that better thing to me). But I would certainly put it on account of his graciousness.

There.

All the same, I will sure try to improve my own record, and try to avoid such nastiness in future, esp. at iMechanica (a forum that has given me so much of intellectual satisfaction, and has extended so much friendliness). [No, if you ask me, the matter involves such bad context that I won’t include this resolve as a part of my NYR, even though I will, as I said, try even more to observe it.]


I also have been down with a bout of cold and cough for the past 2–3 days, now barely recovering, and therefore don’t expect to join in the New Year’s party anywhere.


My NYR remains as before (namely, to share my newer thoughts on QM). There is an addition in fact.

I have found that I can now resolve the issue: “Stress or strain: which one is more fundamental?” It is one of the most widely read threads at iMechanica (current count: 135,000+), and though a lot of knowledgeable and eminent mechanicians participated in it, at the natural cessation of any further real discussion several years ago, the matter had still remained unresolved [^].

I now have found a logic to take the issue to (what I think is) its definite resolution. I intend to share it in the new year. That’s my NYR no. 2 (the no. 1 being about QM). I am also thinking of writing a journal paper about this stress-strain issue—for no reason other than the fact it has gone unresolved for such a long time, despite such wide publicity. It clearly has gone beyond the stage of an informal discussion, and does deserve, IMO, a place in an archival journal. For the same reason, give me time—months, if I decide to include some simulations, or at least several weeks, if I decide to share only the bare logic, before I come back.

Yes, as usual, you can always ask me in person, and I could give the gist of my answer right on the fly. It’s only the aspect of writing down a proper archival journal paper that takes time.


A Song I Like:

It’s being dropped for this time round.

I cannot pick out which one of the poems of Mangesh Padgaonkar I love better. He passed away just yesterday, at a ripe age of 86.

Just like most any Marathi-knowing person of my age (and so many of other ages as well), I have had a deeply personal kind of an appeal for Mangesh Padgaonkar’s poetry. It’s so rich, so lovely, and yet so simple of language—and so lucid. He somehow had a knack to spot the unusual, the dramatic in a very commonplace circumstance, and bring it out lucidly, using exactly the right shade of some very lyrical words. At other times, he also had the knack to take something very astounding or dramatic but to put it in such simple (almost homely) sort of way, that even a direct dramatic statement would cause no real offence. (I here remember his “salaam.”) And, even if he always was quite modern in terms of some basic attitudes (try putting his “yaa janmaavara” as “nothing but the next” in a series of the poems expressing the received Indian wisdom, or compare his “shraavaNaata ghana neeLaa” with the best of any naturalistic poet), his poetry still somehow remained so deeply rooted in the Marathi culture. Speaking of the latter, yes, though he was modern, one could still very easily put him in the series of “bhaa. raa. taambe,” “baalakavee,” and others. Padgaonkar could very well turn out to be the last authentic exponent of the Marathi Enlightenment.

All in all, at least in my mind, he occupies the same place as that reserved for the likes of V. S. Khandekar and “kusumaagraj.” People like these don’t just point out the possibilities, in some indirect and subtle ways, they actually help you mould your own sense of what words like art and literature mean.

If I were to be my younger self, my only regret would be that he never received the “dynaanapeetha” award. Today, I both (i) know better, and (ii) no longer expect such things to necessarily come to a pass.

Anyway, here is a prayer that may his soul find “sadgati.”


Alright now, let me conclude.

Here is wishing you all the best for a happy and prosperous new year!

[May be another pass, “the next year”…]

[E&OE]

Understanding tensors (of engineering sciences)—part 2: yet another DIY experiment

I continue from my last post.

There is another simple DIY experiment that you can perform at home. The idea of this experiment had occurred quite some time back, but I had completely forgotten it. (I had forgotten it even while delivering lectures for the FEM courses which I taught in 2009 and 2012). Last night, I happened to recall the idea once again, and thought of immediately sharing it with you via this blog post.

A Fun DIY Experiment # 2:

Get that piece of men’s innerware which is known in India as the “banian” or “banyan,” and in American English as the “vest” [^].  (If not sure, check out the “aaraam kaa maamla” ads.) Basically, a banyan (at least these days) is a cotton garment like a T-shirt, but it’s bit smaller in size, and as an inner-ware, it is also meant to be more closely fitting to the body. That also makes it more easily stretchable, and therefore, better suited to our purposes. It’s also very inexpensive.

Start with a new (i.e. unused and never washed (i.e. never stretched/wrinkled)) banyan. The cloth should be easily stretchable. The fabric should be plain and simple, and without any special knitting pattern; e.g. no “self-stripes” etc. Cut it open and lay the cloth flat on a table. Mark a set of regular Cartesian grid-points on it with the help of an ink pen. You can easily make a bigger grid (say of the size 15 cm X 15 cm) at a regular spacing, say of 1 cm.

Lay the cloth flat and unstretched on the glass surface of a computer scanner (or even a Xerox machine), and obtain an image, say PH1. Next, with the help of a friend, stretch the cloth non-uniformly, by pulling unevenly along many directions. Make sure that the stretch is non-uniform but completely planar, and, of course, that there are no wrinkles. Scan it in this stretched state, and thus obtain the second image, PH2.

Advantages of this second experiment are easy to see: (i) As compared to the balloon rubber, is easier to lay the banyan cloth flat and without wrinkles. (ii) It is easier to stretch it in many directions. (iii) It is easier to mark out a regular grid—the regularity of the fabric of the cloth actually helps in ensuring regularity.

Also, even if you manage to get a good piece of a large rubber balloon, it should anyway be easier to obtain the image of a grid on it using an image scanner/Xerox machine rather than using a digital camera—the issues of having to maintain the same zoom and distance don’t arise.

Process the images as mentioned in the previous post, and keep them ready.

In the meanwhile, also consult the references mentioned in the last post, and make sure to go through the following concepts in particular: (i) position vector for a point-particle; (ii) displacement vector for a point-particle; (iii) the position vectors for an infinity of points in a continuum—i.e. the position vector field; (iv) the line segment, i.e., the relative position vector (i.e., the difference between two position vectors); (v) the translation and rotation of a line segment; (vi) the relative displacement vector of a line segment (i.e., the relative displacement vector of a relative position vector!); (vii) the rigid-body translation and rotation vs. the change of size and shape of a continuum body; (viii) the displacement gradient tensor at a point in a continuum; (ix) the rotation tensor at a point in a continuum body vs. its rigid-body rotation as a whole; (viii) the strain tensor at a point in a continuum body; etc. …

We will of course look into all these concepts—in fact, we will calculate the particular values that all these quantities assume in our simple experiment, using the basic data of the two images that our simple experiment generates. That will be our topic in the next post.

But before coming to it, let’s take a pause for a moment to recall what the purpose of this whole exercise is. It is: to know the physical meanings/correspondents of the mathematical concepts; to try and develop a proper hierarchical order the concepts; to develop a physical “feel” for the more abstract concepts involved. And, as far as the last is concerned (developing a physical feel for abstract concepts), there’s no substitute to realizing what the more concrete context of a given more abstract concept is. In understanding the proper context of mathematical concepts, there is no substitute to physical observation. That’s why, no matter how ridiculously simple these experiments might look like, do not skip the step of actually performing one of these two experiments.

And, BTW, in this series, more DIY experiments and fun ideas are going to follow.

More, later.

* * * * *   * * * * *   * * * * *

A Song I Like:

(Hindi) “ek ghar banaoonga, tere ghar ke saamane…”
Singers: Mohammad Rafi and Lata Mangeshkar
Music: S. D. Burman
Lyrics: Hasrat Jaipuri

[E&OE]

A little more on my research on the diffusion equation

Alright. Here we go again. … In my last post, I had mentioned a bit about how attending the recent ISTAM conference at Pune, had helped me recall my thoughts on the diffusion equation and all. In particular, I had mentioned in that post how I had discovered a Berkeley professor’s paper only after publishing my own paper (in ISTAM, 2006, held at Vishakhapattanam), and that I would revise my ISTAM paper and send it over to a journal.

The gentleman in question is Prof. T. N. Narasimhan. Unfortunately, I gather, he has passed away in 2011 [^][^]. Here is the group of his relevant papers:

Narasimhan, T. N. (1999) “Fourier’s heat conduction equation: history, influence, and connections,” Reviews of Geophysics, vol. 37, no. 1, pp. 151–172

Narasimhan, T. N. (2009) “The dichotomous history of diffusion,” Physics Today, July 2009, pp. 48–53

Narasimhan, T. N. (2009) “Laplace, Fourier, and stochastic diffusion,” arXiv:0912.2798, 13 Dec 2009

Narasimhan, T. N. (2010) “On physical diffusion and stochastic diffusion,” Current Science, vol. 98, no. 1, 10 January 2010, pp. 23–26

All these papers are available somewhere or the other on the ‘net. (Copy-paste the paper titles in a Google Scholar search, and you should get to the PDF files.)

The first paper is the most comprehensive among them. In this paper, Prof. Narasimhan discusses the historical context of the development of Fourier’s theory and its ramifications. The paper even gives a very neat (and highly comprehensive) table of the chronology of the developments related to the diffusion equation.

In this paper, unlike in so many others on the diffusion equation, he explicitly (even if only passingly) looks into the issue of the action-at-a-distance. However, the way he discusses this issue, it seems to me, he perhaps had entirely missed the crucial objections that are to be made against the idea of IAD—the same basic things which, in yet another context, lead people into believing in quantum entanglement in the sense they do. (And, how!) In fact, in this paper, Narasimhan does not really discuss any of those basic considerations concerning IAD. If so, then what is it that he discusses?

Narasimhan uses the term “action at a distance” (AD), and not the more clarifying instantaneous action at a distance (IAD). However, such a usage hardly matters. It’s true that if it’s just the AD that you take up for discussion, then it’s the issue of the absence of a mediating agency or a medium (or the premise of a contact-less transmission of momenta/forces) that you highlight, and not so much the instantaneity of the transmission, even if the latter has always been implied in any such a discussion. As people had observed right in the time of Newton himself, the assumption of instantaneity, of IAD, was there, built right at the base of his theory of gravitation, even if it was billed only as an AD theory back then. The difference between IAD and AD is more terminological in nature.

But then, Narasimhan is not even very explicit in his positions with respect to AD either. To get to his rather indirect remarks on the AD issue, first see his discussion related to Biot and the particles approach that he was trying (pp. 154, the 1999 paper).

Today, i.e., after the existence and acceptance of the kinetic molecular theory for more than a century, a modern reader would expect the author to say that someone who believes in, or at least is influenced by, a particles-based approach would naturally be following a local approach, and hence should be found on the side of denying the IAD. Instead, though the author does not explicitly take any position, from the way he phrases his lines, he seems to suggest that he thinks that someone who adopts a particles-based approach would have found AD to be natural. This contradiction was what I had found intriguing initially.

But then, soon enough, I figured out a plausible way in which the author’s thought-train might have progressed. He must have taken the gravitational interaction between n number of bodies as the paradigm of every AD theory, and therefore, must have come to associate (I)AD with any particles-based approach—in exact opposition to the local nature of the particles-based theories of the 19th century and the later techniques (e.g. LBM, SPH, etc.) of the 20th century. That can only be the reason why Narasimhan makes the AD-related comments the way he makes them, especially in reference to Biot’s work (pp. 154).

I would try to gather more historical material, and in any case, address this issue in my forthcoming paper. That’s what I meant when I said I would revise my paper and send it to a journal. I didn’t mean to say that I would be revising my position—I would be only clarifying it to a greater detail. My position is that a kinetic theoretical model i.e. a particles-based approach, the default way people interpret it, does not involve IAD.

Anyway, back to Narasimhan’s paper. Further on this issue, on page 155 of the same paper, the author states the following:

“Essentially, Fourier moved away from discontinuous bodies and towards continuous bodies. Instead of starting with the basic equations of action at a distance, Fourier took an empirical, observational approach to idealize how matter behaved macroscopically.”

[Bold emphasis mine]

In this passage, to be historically accurate, in place of: “action at a distance” the author should have said: “a discrete/particles-based approach;” and in place of: “an empirical, observational” he should have said: “a continuum-based” approach. After all, none, to my knowledge, has ever empirically or experimentally observed an infinite speed of heat transmission—none possibly could.

Now, of course, the Fourier theory does not really acquire its IAD nature because it’s a continuum theory. The reasons are different; however, that’s yet completely different point. See my ISTAM paper for more details.

Coming back to Narasimhan’s paper, of course, the above-mentioned flaws present in it are wholly minor. On the other hand, his paper carries excellent and comprehensive commentary on so many other important aspects, including the historical ones. Indeed, he is to be lauded and thanked for at least including the (I)AD issue in a paper on diffusion, despite being at Berkeley. …

These days, given the attitudes of the people at places like Berkeley, Stanford, MIT, etc., they would seem to carry this attitude towards my paper on diffusion: “Uh. But it all was already known; wasn’t it?” No, it was not. That precisely is (and has been) the point. Unless you have read my paper, what goes by being “known” would squarely consist of something like the following:

“There is IAD in Newton’s theory of gravity. And also, in the Fourier theory, though its effects are quantitatively negligible, and so, we can always neglect it in analysis and interpretation. So, there is this IAD in the partial differential diffusion equation, and this fact has always been known. And, lately, some attempts have been made to rectify this situation, e.g. the relativistic heat equation, but with limited success.”

The correct statement is:

“There is IAD in Newton’s theory of gravity. There also is IAD in the Fourier theory. But there is no IAD in the diffusion equation itself. Following the commonly accepted way of taking it, the kinetic molecular theory may be taken not to have any IAD in it. However, it would be easily possible to introduce IAD also into it. The relativistic wave equation is not at all relevant to this set of basic observations.”

There is quite a difference between the two sets of statements.

I am still going through Narasimhan’s other papers, but at least after a cursory look, these seem more like just restatements being made to different audiences of what essentially are the same basic positions.

Apart from it all, here are a few other papers, now on the Brownian movement side of it:

Hanggi, Peter and Marchesoni, Fabio (2005) “100 years of Brownian motion,” arXiv:cond-mat/0502053v1. 2 Feb. 2005

Chowdhury, Debashish (2005) “100 years of Einstein’s theory of Brownian motion: from pollen grains to protein trains,” arXiv:cond-mat/0504610v1. 24 April 2005

Gillespie, Daniel T. (1996) “The mathematics of Brownian motion and Johnson noise,” American Journal of Physics, vol. 64, no. 3, pp. 225–240

As you can see by browsing through all these papers, few people seem to have appreciated the aspect of IAD or locality, in these two theories. Perhaps, that’s a part of the reason why quantum conundrums continue to flourish. Yet, it is important to isolate this particular aspect, if we are to be clear concerning our fundamentals. As someone said, Fourier’s theory has by now become a part of the very culture of science. So deep is its influence. It’s time we stopped being nonchalant about it, and began re-examining its premises and implications.

Ok. Enough for today. If you are interested, go through these papers, and I will be back with some further comments on them. Hopefully soon. I anyway need to finish this paper. Without getting a couple of papers or so published in journals, I cannot guide PhD students. But, that doesn’t mean I will deliberately send this diffusion paper to a sub-standard or even a low impact journal. I will try to get it published in as high quality (but fitting sort of) a journal as possible. And, if you have any suggestions as to which journal I should send this diffusion paper, please do not hesitate in dropping me a line.

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I still remain jobless.

However, on a whim, for this post, I decided to add the “A Song I Like” section.

The selection, as usual, is more or less completely random. How random? Well, here is the story about the selection of this particular song, this time around…

As it so happens, sometimes, you “get” or “catch” some tune right in the morning, and then it stays with you for the whole day. The harder you try to get it out of your mind, the more lightly but more firmly it keeps returning to you throughout the day. It doesn’t even have to be a good tune; it simply keeps returning back. That’s what happened with this song, though, the song happens to be a better one. The song happened to so fleetingly alight on my mind in a recent short journey, that I had not realized that I was silently humming it almost halfway down in that journey. (I was not playing any song/music at that time.) So, even though there is another Lata number (“yeh kaun aayaa”) from the same movie (“saathee”) which I perhaps would have chosen if I were deliberately to make a selection, in view of the lightness with which it had come to me—almost as if entirely by itself—I decided to keep this particular song (“mere jeevan saathee”). (BTW, I haven’t seen the movie, and as usual, the video and other aspects don’t count.) Another point. Neither of these two songs looks like it was composed by Naushad—and, in my books, that’s a plus. When truly in his elements, Naushad feels—to me at least—too traditional, perhaps a bit too melancholic, and, what’s the word… too conforming and invention-less?… Yes, that’s it. He feels too much of a conformist and too invention-less, as far as I am concerned. (Even if some tunes of his might have actually been inventive or of high quality, they follow the groove of the traditional song composition, the traditional guideposts so faithfully, that upon listening to the song, he feels invention-less, anyway. And then, I have my own doubts as to how many times he actually was being inventive, anyway!) Alright. Here is that song—an exception for Naushad, as far as I am concerned. And then, Lata, as usual, takes what is only a first class tune, and manages to take it to an altogether different, higher plane, imparting it with, say, a distinction class:

A Song I Like:
(Hindi) “mere jeevan saathee, kalee thee main to pyaasee…”
Singer: Lata Mangeshkar
Music: Naushad
Lyrics: Majrooh Sultanpuri

[May be, a minor editing is due, though I would not spend much time on it when I return.]

[E&OE]

Can an Infinitesimal Have Parts?

Context and Motivation:

The title question of this post has been lingering in my mind for quite some time—actually, years (nay, decades). Some two decades ago or so, I thought I had reached some good understanding of it. But then, some of the discussion at a recent iMechanica thread “A point and a particle” [^] seemed to suggest otherwise. The issue again got raised, in a somewhat indirect manner, in relation to this comment [^] on yet another iMechanica thread today. In between, there also were a few message exchanges that I had at HBL last year, not all of which made it to the published HBL exchange. There, too, my own position was at odds with that of Dr. Harry Binswanger, an Objectivist professor of philosophy (and the way he sometimes describes himself, an amateur scientist).

The essential difference is this: People seem to think, for example, that:

(i) you can take a small but finite line-segment, subject it to an infinitely long limiting process, and what you get in the end is a point; or,

(ii) as the chord of a circle is systematically made ever smaller by bringing its two end-points closer, even as always keeping them on the circle, eventually, the circle, in comparison with the straight-chord, seems to get flattened out so much that eventually, in an infinite process, it becomes indistinguishable from a straight-line, and so, the circular arc becomes the chord (which is the same as saying that the chord becomes the arc); or,

(iii) a particle’s geometry is fully described by a point; etc.

All of these examples, in some way, touch on the title question. For instance, since a point does not have any parts, and if in an infinite process a line goes to a point, then, obviously, an infinitesimal cannot have parts. And so on…

Now, I seem to disagree with the views expressed by people, as above. I also think that some of the basic confusions arising in quantum mechanics (e.g. those concerning the quantum spin) in part arise out of this issue.

[Therefore, an immediate declaration: If someone gets a better idea of what QM really is like, after reading this post, thank me, and also, regardless of that and more importantly: give me appropriate and explicit intellectual credit. To my knowledge, the topic has not been treated so directly and in the following way anywhere else before.]

Background:

Consider an arbitrary but “nice” enough a function: y = f(x). Consider two points P(x_1,y_1) and Q(x_2,y_2) lying on the curve but a finite distance apart. The slope of the line-segment PQ is given by: m_f \equiv \dfrac{y_2 - y1}{x_2-x_1} \equiv \dfrac{\Delta y}{\Delta x}, where the subscript f put on m indicates that this is a finite-distance case.  As you know, there is an infinity of points in between the end-points of any finite line-segment.

To determine the slope of the curve at the point P, we take the limit of the ratio m_f as the distance between x_2 and x_1 approaches zero. In symbolic terms: m_P = \lim_{\Delta x \to 0} \dfrac{\Delta y}{\Delta x}, where m_P is supposed to be the slope of the curve at the point P.

Clarifications—The Idea of Slope:

The italicized parts in the above statement are important. Firstly, it is implicitly (and somewhat blithely) assumed that a curve can have a slope, which can be approximated by that of a line-segment such as PQ. Secondly, it is even more implicitly (and even more blithely) assumed that there exists something such as a slope at a point. Let’s examine both a bit more closely.

What does the notion of slope mean? The extreme case of 0^0 and the pathological case of 90^0 apart, what the notion basically means is that you are going to either gain or lose your current height as you travel (in some direction).

Notice that immediately implicit here is the idea of there being two different locations whose heights are being compared! You cannot define slope without there being two distinct reference points. Hence, you also should not use the term in those contexts where only one reference point is given. If so, how can we speak of a slope at a point?

Realize that the above objection applies as much to the points lying on a straight line-segment as those on a curved line-segment. Even single points on straight lines cannot, strictly speaking, can be said to have a slope—only the straight line-segment, as a whole (or any finite parts of it) may be said to have one. If so, what does it mean when we speak of a slope at a point?

My answer: Primarily, it means nothing! It’s just a loose way of putting things. What it really means is the entire limiting process, and the result of it (if there is any valid result coming off that limiting process).

The slope of a line at a point (whether that line is straight or curved, it does not matter) is just the definite “tendency” shown in the trends of the actual slopes of all the small but finitely long straight line-segments in the close neighbourhood of the given point. You cannot speak of a slope at a point in any other terms. Not even for straight-lines. Straight lines just happen to be a special case wherein all the slope values are the same, and so, determining the trend is a trivial matter. Yet, the principle of having to make a reference to an actual trend of certain property displayed by a definitely ordered sequence of finitely long segments, in an appropriate limiting process, does remain there. It is only in this sense that lines can have slopes at various points. Ditto, for the curved lines.

Clarifications—A Line “Going” “to” a Point:

Now, there is something even funnier. At least in applied science and engineering, we often speak of the above kind of a limiting process, in terms like the following:

Take Q close to P, as close as possible. In the limit, as the length of PQ “goes” “to” “zero,” the slope of the segment PQ “goes” “to” “the slope of the curve” “at” “P“.

All the words put in the scare-quotes (“”) are important.

What does it mean for a length of a straight line-segment PQ to go to zero? It means: P and Q are coincident—i.e. they are one and the same point. (There is no such a thing as two different points occupying the same point—it’s either two names for the same mathematical object, or a contradiction in terms.)

So, can a slope have a curve? The very idea is meaningless outside the context of a limiting process. Yes, you may gain or lose height as you traverse the curve, sure. But does it mean that the curve has a slope? Nope. Not unless your context has the right limiting process in it.

Clarifications—Points, Lines, and the Nature of Limiting Processes:

Now, a bit about the nature of limiting process.

Realize that there is a fundamental difference between a point and a line. (For our purposes, both may be taken as given axiomatically, as abstractions of the locations and the edges of the actually existing objects. That there also is suggested an infinite process in reaching such abstractions is a subtle point that we choose to ignore for the time being.)

The units of a point and a line are different. You cannot compare a point and a line in any commensurate manner whatsoever, full-stop. (Incommensurability is quite frequent in mathematics, more often than what most people realize.)

A line segment may be put in one:one correspondence with an (orderly) infinite set of points, and in this way, it may abstractly be seen to consist of points. However, realize that infinity does not exist. The one:one correspondence process, should you wish to conduct one in actuality, will never terminate, and hence, you will never get a line starting from points, or vice versa: a point, starting from a line. Incidentally, that’s just another way of realizing that a line is incommensurate with a point. Then how is it that we can talk meaningfully of such a process?

What we mean when we talk of a line as being made of an infinity of points is this:

Take a finite line-segment, say from the point P to Q. Take a point P lying on it. Find the finite lengths of M from each of its end-points.  (Aside: It is here that the defining processes of a point, a line, etc. that we have chose to ignore in this post, create some tricky issues. We will deal with them later, in another post.)

Now, take a sub-segment from any of the two end-points to the middle point (whose location, in the general case, is arbitrary; it need not exactly divide the segment into two equal halves.) Suppose we take the sub-segment PM. Now, conduct a limiting process by reducing the size of PM, while holding M fixed. (BTW, observe that every limiting process involves holding something the same even as varying something else.) Making the sub-segment monotonically smaller in size means that the end-point of the segment in the reduced size corresponding to P, say, P' monotonically increasingly gets closer to M. But, it never quite reaches M.

The only case in which P' could reach M is if it is coincident with—i.e. is the same point as—M. However, in this case, there cannot be two distinct end-points left to serve as the end-points of the diminishing sub-segment, and hence, no sub-segment left to speak of.

Hence, we have to say that the point P' never quite reaches M—not even in this infinite limiting process. The most crucial point of the logic is already thus given. The rest is a bit of house-keeping so that even if we revise the entire description here by expressing a point via a limiting process, the essential logic as spelt about remains unaffected.

Now, repeat the process for another, distinct, point N \neq M, lying on the same original line-segment. Since M and N are not one and the same point, and since the “getting closer” process for any arbitrary sub-part of the line-segment cannot terminate for either of them, and further, since both lie on the same original finite segment and thereby enjoy an ordering relation between them (e.g. that M < N etc.), we must conclude that there must be an infinity of N points corresponding to any arbitrarily given point M. Just make M coincident with (or the same as) Q, and the inevitable conclusion follows, namely, that there must be an infinity of such processes for them to span all the distinct points lying over the entire original line-segment.

The existence of this infinity of such “getting closer” processes is what we actually mean when we say a line is “made of” an infinity of points.

Emphatically, it does not mean that a point and a line are commensurate. It only means that the endpoints of a line can be made as close to a given point lying on that line as you wish. That’s all.

Clarifications—An Infinitesimal of a Finite:

Now, we are ready to tackle the idea of infinitesimal.

An infinitesimal of a line-segment is an imaginary projection of the result that would be had if a line-segment were to be made ever smaller in a limiting infinite (i.e. definite but unterminating) process.

Notice that we didn’t jump directly to what the term “infinitesimal” means in a general sense. We simply made a statement in respect of the infinitesimal of a line-segment. This distinction is important. The reason is that there is no such thing as a general infinitesimal!

You can have infinitesimals of (finite) lines, surfaces, volumes, etc. Or, of quantities that, essentially, are some kind of densities of some other quantities which have been defined in a “wholesale” manner over finite lines (or surfaces, volumes, etc.). But you cannot have infinitesimals “in general,” as such.

Infinitesimals not only acquire their meaning only in some definite kind of an infinite limiting process, but they also do so only in reference to the certain finite thing (and its associated properties) which is being subjected to that process. A process without an input or an output is a contradiction in terms. An infinitesimal can only result when you begin in the first place with a finite.

Since an infinitesimal must always refer to its input finite thing (be it a length, a surface, etc. or a density variable defined with respect to these), therefore, it must always carry some units—which are the same as that of the finite thing.

The “infinitesimal-izing” process (to coin a new word!) does not touch the units of the finite thing, and hence, neither does the end-result of that process—even if the result be only via an imaginary projection. Thus, the infinitesimal of a line always retains the units of, say, m, and that of a surface, m^2, etc.

The above precisely is the reason why we can “cancel out” dx dy with da where the first expression is a product of lengths, and the second one is an area—and wherein all the quantities are infinitesimals. Infinitesimals have units; equations formulated in infinitesimal terms must follow the law of dimensional homogeneity.

Clarifications—Can Infinitesimals Have Parts?

Now, having examined the nature of infinitesimals to (hopefully) sufficient extent, we are finally ready to answer the title question: “Can an infinitesimal have parts?”

I will not directly answer the question in yes or no terms. My answer should be obvious to you by now. (If not, kick yourself a couple of times, and proceed to read further or, equally well, abandon this blog forever.)

First, observe that it is only a finite line-segment which, when subject to an infinitesimal-izing process, becomes an infinitesimal.

Apart from its two end-points, you can always take a third point lying on that finite segment such that it divides the segment into two (not necessarily equal) parts. Say, L = L_1 + L_2. Now, observe that as you take L to an infinitesimally small quantity, you also thereby subject L_1 and L_2  to the same infinitesimal-izing process such that the equation dL = dL_1 + dL_2 holds as a result. (The reason we can directly put this relation in this way is that the rates with which each becomes small is identical. In contrast, the area gets smaller at a rate faster than that of the length—another way of seeing that an infinitesimal always has dimensions i.e. units.)

Now, returning back to today’s discussion. At iMechanica, I have raised a couple of points:

(i) Do we define stress in relation to a plane? Or do we do so in relation to a thin plate made infinitesimally small? The difference, now you can see, is this: a plane has no thickness. But a plate does. Its thickness has the units of length, which can’t be made zero. Hence the question.

(ii) Is the elemental cube (used for defining variations in stress, say to the first order) have a finite length? Or is it (or can it be) infinitesimal?

Once again, I will not provide a direct answer to these questions. However, I will leave you with a very very obvious clue (apart from what all I have mentioned above)—but one, which, nevertheless, raises further curious issues. These are essentially nothing but the same as the issues we have chosen to ignore today—what are points? lines? surfaces? do they exist? Anyway, the clue, presently, is the following.

Take a brick. You can always make its size ever smaller in a limiting process so as to get an infinitesimal Cartesian volume element. Agreed? OK.

Now, take a pack of playing cards. Subject it to a similar limiting process. And, ask yourself the above two questions.  The answer(s) should be obvious!! (As to the tricky part: Ask yourself: Can you assume zero thickness in between two adjacent playing cards in the same pack? Your answer to the question of whether stress is defined in relation to a plane or an infinitesimally thin plate, will in part differ depending on how you answer this question!)

[PS: I think I might edit this post a bit. If I do so, I will also note down any major change (e.g. that of the logic or of hierarchical precedence, etc.) that I make. For instance, I am not at all happy with the way I have explained the idea of “an infinity of points in a line, even though a line never goes to a point.” That part hasn’t at all come out well. I expect to make changes there—or, may be, perhaps, write another post to once again give a try to that part. … Hey, after all, this is not a paper on mathematics—just a blog post, OK? 🙂 ]

[A side note: I know that the limit notation as rendered here on the Web page does not look nice, but that’s an issue primarily with the WordPress support of LaTeX. I am not going to hack around with \dfrac etc. just to get the \lim look nice here!]

* * * * *   * * * * *   * * * * *

A Song I Like:
(Hindi) “dil beqaraar saa hai…”
Singer: Lata Mangeshkar (I like her version better than Rafi’s)
Music: Kalyanji-Anandji
Lyrics: Majrooh Sultanpuri

[E&OE]