Exactly what does this script show?

Update on 02 March 2018, 15:34 IST: I have now added another, hopefully better, version of the script (but also kept the old one intact); see in the post below. The new script too comes without comments.


Here is a small little Python script which helps you visualize something about a state of stress in 2D.

If interested in understanding the concept of stress, then do run it, read it, try to understand what it does, and then, if still interested in the concept of stress, try to answer this “simple” little question:

Exactly what does this script show? Exactly what it is that you are visualizing, here?


I had written a few more notes and inline comments in the script, but have deliberately deleted most of them—or at least the ones which might have given you a clue towards answering the above question. I didn’t want to spoil your fun, that’s why.

Once you all finish giving it a try, I will then post another blog-entry here, giving my answer to that question (and in the process, bringing back all the deleted notes and comments).

Anyway, here is the script:


'''
A simple script to help visualize *something* about
a 2D stress tensor.

--Ajit R. Jadhav. Version: 01 March 2018, 21:39 HRS IST.
'''

import math
import numpy as np
import matplotlib.pyplot as plt

# Specifying the input stress
# Note:
# While plotting, we set the x- and y-limits to -150 to +150,
# and enforce the aspect ratio of 1. That is to say, we do not
# allow MatPlotLib to automatically scale the axes, because we
# want to appreciate the changes in the shapes as well sizes in
# the plot.
#
# Therefore, all the input stress-components should be kept
# to within the -100 to +100 (both inclusive) range.
#
# Specify the stress state in this order: xx, xy; yx, yy
# The commas and the semicolon are necessary.

sStress = "-100, 45; 90, 25"

axes = plt.axes()
axes.set_xlim((-150, 150))
axes.set_ylim((-150, 150))
plt.axes().set_aspect('equal', 'datalim')
plt.title(
    "A visualization of *something* about\n" \
    "the 2D stress-state [xx, xy; yx, yy] = [%s]" \
    % sStress)

mStress = np.matrix(sStress)
mStressT = np.transpose(mStress)

mUnitNormal = np.zeros((2, 1))
mTraction = np.zeros((2, 1))

nOrientations = 18
dIncrement = 360.0 / float(nOrientations)
for i in range(0, nOrientations):
    dThetaDegrees = float(i) * dIncrement
    dThetaRads = dThetaDegrees * math.pi / 180.0
    mUnitNormal = [round(math.cos(dThetaRads), 6), round(math.sin(dThetaRads), 6)]
    mTraction = mStressT.dot(mUnitNormal)
    if i == 0:
        plt.plot((0, mTraction[0, 0]), (0, mTraction[0, 1]), 'black', linewidth=1.0)
    else:
        plt.plot((0, mTraction[0, 0]), (0, mTraction[0, 1]), 'gray', linewidth=0.5)
    plt.plot(mTraction[0, 0], mTraction[0, 1], marker='.',
             markeredgecolor='gray', markerfacecolor='gray', markersize=5)
    plt.text(mTraction[0, 0], mTraction[0, 1], '%d' % dThetaDegrees)
    plt.pause(0.05)

plt.show()


Update on 02 March 2018, 15:34 IST:

Here is a second version of a script that does something similar (but continues to lack explanatory comments). One advantage with this version is that you can copy-paste the script to some file, say, MyScript.py, and invoke it from command line, giving the stress components and the number of orientations as command-line inputs, e.g.,

python MyScript.py "100, 0; 0, 50" 12

which makes it easier to try out different states of stress.

The revised code is here:


'''
A simple script to help visualize *something* about
a 2D stress tensor.

--Ajit R. Jadhav. 
History: 
06 March 2018, 10:43 IST: 
In computeTraction(), changed the mUnitNormal code to make it np.matrix() rather than python array
02 March 2018, 15:39 IST; Published the code
'''

import sys
import math
import numpy as np
import matplotlib.pyplot as plt

# Specifying the input stress
# Note:
# While plotting, we set the x- and y-limits to -150 to +150,
# and enforce the aspect ratio of 1. That is to say, we do not
# allow MatPlotLib to automatically scale the axes, because we
# want to appreciate the changes in the shapes as well sizes in
# the plot.
#
# Therefore, all the input stress-components should be kept
# to within the -100 to +100 (both inclusive) range.
#
# Specify the stress state in this order: xx, xy; yx, yy
# The commas and the semicolon are necessary.
# If you run the program from a command-line, you can also
# specify the input stress string in quotes as the first
# command-line argument, and no. of orientations, as the
# second. e.g.:
# python MyScript.py "100, 50; 50, 0" 12
##################################################

gsStress = "-100, 45; 90, 25"
gnOrientations = 18


##################################################

def plotArrow(vTraction, dThetaDegs, clr, axes):
    dx = round(vTraction[0], 6)
    dy = round(vTraction[1], 6)
    if not (math.fabs(dx) < 10e-6 and math.fabs(dy) < 10e-6):
        axes.arrow(0, 0, dx, dy, head_width=3, head_length=9.0, length_includes_head=True, fc=clr, ec=clr)
    axes.annotate(xy=(dx, dy), s='%d' % dThetaDegs, color=clr)


##################################################

def computeTraction(mStressT, dThetaRads):
    vUnitNormal = [round(math.cos(dThetaRads), 6), round(math.sin(dThetaRads), 6)]
    mUnitNormal = np.reshape(vUnitNormal, (2,1))
    mTraction = mStressT.dot(mUnitNormal)
    vTraction = np.squeeze(np.asarray(mTraction))
    return vTraction


##################################################

def main():
    axes = plt.axes()
    axes.set_label("label")
    axes.set_xlim((-150, 150))
    axes.set_ylim((-150, 150))
    axes.set_aspect('equal', 'datalim')
    plt.title(
        "A visualization of *something* about\n" \
        "the 2D stress-state [xx, xy; yx, yy] = [%s]" \
        % gsStress)

    mStress = np.matrix(gsStress)
    mStressT = np.transpose(mStress)
    vTraction = computeTraction(mStressT, 0)
    plotArrow(vTraction, 0, 'red', axes)
    dIncrement = 360.0 / float(gnOrientations)
    for i in range(1, gnOrientations):
        dThetaDegrees = float(i) * dIncrement
        dThetaRads = dThetaDegrees * math.pi / 180.0
        vTraction = computeTraction(mStressT, dThetaRads)
        plotArrow(vTraction, dThetaDegrees, 'gray', axes)
        plt.pause(0.05)
    plt.show()


##################################################

if __name__ == "__main__":
    nArgs = len(sys.argv)
    if nArgs > 1:
        gsStress = sys.argv[1]
    if nArgs > 2:
        gnOrientations = int(sys.argv[2])
    main()



OK, have fun, and if you care to, let me know your answers, guess-works, etc…..


Oh, BTW, I have already taken a version of my last post also to iMechanica, which led to a bit of an interaction there too… However, I had to abruptly cut short all the discussions on the topic because I unexpectedly got way too busy in the affiliation- and accreditation-related work. It was only today that I’ve got a bit of a breather, and so could write this script and this post. Anyway, if you are interested in the concept of stress—issues like what it actually means and all that—then do check out my post at iMechanica, too, here [^].


… Happy Holi, take care to use only safe colors—and also take care not to bother those people who do not want to be bothered by you—by your “play”, esp. the complete strangers…

OK, take care and bye for now. ….


A Song I Like:

(Marathi [Am I right?]) “rang he nave nave…”
Music: Aditya Bedekar
Singer: Shasha Tirupati
Lyrics: Yogesh Damle

 

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Machine “Learning”—An Entertainment [Industry] Edition

Yes, “Machine ‘Learning’,” too, has been one of my “research” interests for some time by now. … Machine learning, esp. ANN (Artificial Neural Networks), esp. Deep Learning. …

Yesterday, I wrote a comment about it at iMechanica. Though it was made in a certain technical context, today I thought that the comment could, perhaps, make sense to many of my general readers, too, if I supply a bit of context to it. So, let me report it here (after a bit of editing). But before coming to my comment, let me first give you the context in which it was made:


Context for my iMechanica comment:

It all began with a fellow iMechanician, one Mingchuan Wang, writing a post of the title “Is machine learning a research priority now in mechanics?” at iMechanica [^]. Biswajit Banerjee responded by pointing out that

“Machine learning includes a large set of techniques that can be summarized as curve fitting in high dimensional spaces. [snip] The usefulness of the new techniques [in machine learning] should not be underestimated.” [Emphasis mine.]

Then Biswajit had pointed out an arXiv paper [^] in which machine learning was reported as having produced some good DFT-like simulations for quantum mechanical simulations, too.

A word about DFT for those who (still) don’t know about it:

DFT, i.e. Density Functional Theory, is “formally exact description of a many-body quantum system through the density alone. In practice, approximations are necessary” [^]. DFT thus is a computational technique; it is used for simulating the electronic structure in quantum mechanical systems involving several hundreds of electrons (i.e. hundreds of atoms). Here is the obligatory link to the Wiki [^], though a better introduction perhaps appears here [(.PDF) ^]. Here is a StackExchange on its limitations [^].

Trivia: Kohn and Sham received a Physics Nobel for inventing DFT. It was a very, very rare instance of a Physics Nobel being awarded for an invention—not a discovery. But the Nobel committee, once again, turned out to have put old Nobel’s money in the right place. Even if the work itself was only an invention, it did directly led to a lot of discoveries in condensed matter physics! That was because DFT was fast—it was fast enough that it could bring the physics of the larger quantum systems within the scope of (any) study at all!

And now, it seems, Machine Learning has advanced enough to be able to produce results that are similar to DFT, but without using any QM theory at all! The computer does have to “learn” its “art” (i.e. “skill”), but it does so from the results of previous DFT-based simulations, not from the theory at the base of DFT. But once the computer does that—“learning”—and the paper shows that it is possible for computer to do that—it is able to compute very similar-looking simulations much, much faster than even the rather fast technique of DFT itself.

OK. Context over. Now here in the next section is my yesterday’s comment at iMechanica. (Also note that the previous exchange on this thread at iMechanica had occurred almost a year ago.) Since it has been edited quite a bit, I will not format it using a quotation block.


[An edited version of my comment begins]

A very late comment, but still, just because something struck me only this late… May as well share it….

I think that, as Biswajit points out, it’s a question of matching a technique to an application area where it is likely to be of “good enough” a fit.

I mean to say, consider fluid dynamics, and contrast it to QM.

In (C)FD, the nonlinearity present in the advective term is a major headache. As far as I can gather, this nonlinearity has all but been “proved” as the basic cause behind the phenomenon of turbulence. If so, using machine learning in CFD would be, by the simple-minded “analysis”, a basically hopeless endeavour. The very idea of using a potential presupposes differential linearity. Therefore, machine learning may be thought as viable in computational Quantum Mechanics (viz. DFT), but not in the more mundane, classical mechanical, CFD.

But then, consider the role of the BCs and the ICs in any simulation. It is true that if you don’t handle nonlinearities right, then as the simulation time progresses, errors are soon enough going to multiply (sort of), and lead to a blowup—or at least a dramatic departure from a realistic simulation.

But then, also notice that there still is some small but nonzero interval of time which has to pass before a really bad amplification of the errors actually begins to occur. Now what if a new “BC-IC” gets imposed right within that time-interval—the one which does show “good enough” an accuracy? In this case, you can expect the simulation to remain “sufficiently” realistic-looking for a long, very long time!

Something like that seems to have been the line of thought implicit in the results reported by this paper: [(.PDF) ^].

Machine learning seems to work even in CFD, because in an interactive session, a new “modified BC-IC” is every now and then is manually being introduced by none other than the end-user himself! And, the location of the modification is precisely the region from where the flow in the rest of the domain would get most dominantly affected during the subsequent, small, time evolution.

It’s somewhat like an electron rushing through a cloud chamber. By the uncertainty principle, the electron “path” sure begins to get hazy immediately after it is “measured” (i.e. absorbed and re-emitted) by a vapor molecule at a definite point in space. The uncertainty in the position grows quite rapidly. However, what actually happens in a cloud chamber is that, before this cone of haziness becomes too big, comes along another vapor molecule, and “zaps” i.e. “measures” the electron back on to a classical position. … After a rapid succession of such going-hazy-getting-zapped process, the end result turns out to be a very, very classical-looking (line-like) path—as if the electron always were only a particle, never a wave.

Conclusion? Be realistic about how smart the “dumb” “curve-fitting” involved in machine learning can at all get. Yet, at the same time, also remain open to all the application areas where it can be made it work—even including those areas where, “intuitively”, you wouldn’t expect it to have any chance to work!

[An edited version of my comment is over. Original here at iMechanica [^]]


 

“Boy, we seem to have covered a lot of STEM territory here… Mechanics, DFT, QM, CFD, nonlinearity. … But where is either the entertainment or the industry you had promised us in the title?”

You might be saying that….

Well, the CFD paper I cited above was about the entertainment industry. It was, in particular, about the computer games industry. Go check out SoHyeon Jeong’s Web site for more cool videos and graphics [^], all using machine learning.


And, here is another instance connected with entertainment, even though now I am going to make it (mostly) explanation-free.

Check out the following piece of art—a watercolor landscape of a monsoon-time but placid sea-side, in fact. Let me just say that a certain famous artist produced it; in any case, the style is plain unmistakable. … Can you name the artist simply by looking at it? See the picture below:

A sea beach in the monsoons. Watercolor.

If you are unable to name the artist, then check out this story here [^], and a previous story here [^].


A Song I Like:

And finally, to those who have always loved Beatles’ songs…

Here is one song which, I am sure, most of you had never heard before. In any case, it came to be distributed only recently. When and where was it recorded? For both the song and its recording details, check out this site: [^]. Here is another story about it: [^]. And, if you liked what you read (and heard), here is some more stuff of the same kind [^].


Endgame:

I am of the Opinion that 99% of the “modern” “artists” and “music composers” ought to be replaced by computers/robots/machines. Whaddya think?

[Credits: “Endgame” used to be the way Mukul Sharma would end his weekly Mindsport column in the yesteryears’ Sunday Times of India. (The column perhaps also used to appear in The Illustrated Weekly of India before ToI began running it; at least I have a vague recollection of something of that sort, though can’t be quite sure. … I would be a school-boy back then, when the Weekly perhaps ran it.)]

 

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

 

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]

Explicit vs. implicit FDM: reference needed

The following is my latest post at iMechanica [^]:


“The context is the finite difference modeling (FDM) of the transient diffusion equation (the linear one: \dfrac{\partial T}{\partial t} = \alpha \dfrac{\partial^2 T}{\partial x^2}).

Two approaches are available for modeling the evolution of $T$ in time: (i) explicit and (ii) implicit (e.g., the Crank-Nicolson method).

It was obvious to me that the explicit approach has a local (or compact) support whereas the implicit approach has a global support.

However, with some simple Google searches (and browsing through some 10+ books I could lay my hands on), I could not find any prior paper/text to cite by way of a reference.

I feel sure that it must have appeared in some or the paper (or perhaps even in a text-book); it’s just that I can’t locate it.

So, here is a request: please suggest me a reference where this observation (about the local vs. global support of the solution) is noted explicitly. Thanks in advance.

Best,

–Ajit

[E&OE]”


Self-explanatory, right?


[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]

Errors in my CFD notes for the lecture no. 3

I have taken off the slides for the Lecture # 3 of my introductory course on CFD, because it contained some serious errors. (….Yes, those were the serious errors—not just typos.)

I had caught the errors right the next day after posting it (about 10 days ago), but it so happened that in the meanwhile, I ended up (literally) wasting my time (and money) attending to certain highly promising promises of interviews for a professor’s job, as well as also actually attending a couple of highly promising interviews for promising positions of a professor, and subsequently, also trying to interact with the  “management”s of these colleges.

I therefore couldn’t find the frame of the mind (and in fact also the necessary time at one stretch) to write down those lengthy equations in LaTeX, by way of rectifying the mistakes. I in fact was also traveling for these job-related matters. During these last 10 days, I visited some 4 places out of town, with two of them being more than a hundred km away. … No, I haven’t landed a professor’s job yet.

Anyway, back to the mistakes in the uploaded notes.

The mistakes in particular were concerned with the tensor calculus part of the slides. I caught the mistakes when I went a bit further in my notes, to the point of preparing the slides for the next couple of lectures—which would be: on the Navier-Stokes equations.

I had never worked out the full derivation of the Navier-Stokes equations in this way, i.e., for an infinitesimal CV, but using the Eulerian approach right from the beginning. … In the past, I had always relied on books, and never worked out my own derivations without referring to the proofs given in them. Many of these books are excellent. However, for the infinitesmal CV, they all always derive it only in the Lagrangian frame, and only later on do they use the vector calculus manipulations (or identities) to map the end-result to the Eulerian frame. Every one proceeds only that way. None does what I had unwittingly ended up attempting. …

… In fact, most of them use only the Reynolds’ Transport Theorem or RTT for short. (BTW, Reynolds had never himself stated this theorem in his entire life-time; the entire RTT movement was started only later, by an MIT professor.) Now, RTT is an integral approach, not differential. Usually, the books do the derivation using the RTT, and then proceed to get the differential form from this initial integral form. In the rare cases that they at all try to use an infinitesimal CV in an ab initio manner, they invariably use only the Lagrangian i.e. the non-conservative form.

Indeed, see the unanswered query on the Physics StackExchange here: [^]. … The first part of the question has gone unanswered for 3.5 years by now, after 10,000+ views. So, you know what I was getting at, here. And how, my errors, caught by me before engaging a single class based on these notes, therefore, might perhaps be excusable.

Anyway, what is more important is to note down the references which I found useful in working out this entire issue. These are the following two. (No, they of course don’t give you the derivation; they just deal with the basics of tensors and their calculus):

  • “A brief introduction to tensors and their properties,” by Prof. Allan Bower of Brown (a fellow iMechanician!) [^]
  • “Tensor derivative (continuum mechanics,” Wiki, section on divergence of a tensor [^] (and Prof. Piaras Kelly’s notes that it refers to, here [ (.PDF) ^] )

I then worked out the tensors appearing in the Navier-Stokes equations, in fully expanded components form. In this way, my path-way to the final Navier-Stokes equations now seems OK.

In other words, yes, I am now getting ready to answer that Physics StackExchange question, in my upcoming notes. … Give me a few days’ time, and both the components-wise worked out results, as well as the relevant portion excerpted for the slides of the Lecture # 3 of my CFD course, should be online.

But, also, please note, I haven’t run my work by anyone so far. So, it’s still an easy possibility that there are some elementary mistakes in it, too. At least, it would be easy enough for some unwarranted assumptions to creep in. (For instance, it was only during this recent phase of working out these things that I gathered for the first time in my life that there are some subtle pre-suppositions going into the Helmholtz decomposition theorem for the vector fields, too—assumptions like the field having to approach zero as distance tends to infinity—assumptions that I wasn’t at all aware of….) Therefore, I do plan to privately run my notes through a few mechanician friends/blogging acquaintances—even as I simultaneously post them here, within a few days’ time.

BTW, no, coming to those earlier errors in the Lecture #3 slides, even if someone had caught my errors (IMO, a low probability), none had pointed it out to me. None. I found it on my own. But only after publishing something else, in the first place!!

* * * * *   * * * * *   * * * * *

A Song I Like:

You are (or at least should be) well-familiar with the well-worn out story by now.

[E&OE]

An Important Comment I Just Made at iMechanica—And, (Much) More!

0. The title says it all!

Go, check out this comment I just made at iMechanica: [^].

1. Now, on to the “more” part of the title. Noted below are a few more things about my research.

2. My Researches on QM:

2.1 Since the publication of my QM-related results, I have moved on considerably further. As mentioned earlier on this blog, I have since then realized that my approach—the way I thought about it, as in contrast to what I (happened to have) published—always could handle the vector field equations of electromagnetism, including those for light. That is, including the angular momentum part of the EM fields. (Paddy, Suku, are you listening?) … However, I decided against publishing something in more detail to cover this aspect. A good decision, now it seems in retrospect.

(Yes, Jayant, you may now try your best to prod me towards publishing, including emphasizing how unpublished research is non-existent research. Just try it! Any which way you wish. … Precisely just the way I don’t give a damn to wannabe physicists turning JPBTIs turning entrepreneurs, I also don’t give a damn to the Statism-entrenching advices coming off the Statism-entrenching scientists, esp so if they also are the State-revered ones. So, just try it!! Also others, like, say, Sunil!!!)

2.2 I had also resolved the entanglement issue, and have chosen not to publish about it. As I stated earlier here [^], Louisa Guilder reports that Bell’s inequality paper has garnered the highest number of citations in physics literature so far, an astounding 2,500. The paper # 2,501 (or greater, as of today) must have concluded that the entanglement issue cannot be resolved—possibly out of the position/conviction that there was nothing to be resolved.

So, basically, I have resolved what an enormous number of misguided (and, possibly outright stupid) people could cite but not resolve.

Aside: Of the hundreds of papers on this topic I have come across, I know of Dr. Joy Christian’s position to be most reasonable—and in my knowledge, only his. Now, there are some minor differences between what he says and what I have always known and never published. But these differences are, in a sense, minor. The important part—and aren’t we concerned only with the important things here?—is that I knew about it, and have deliberately chosen not to publish about it. (If holding this position makes it possible to tick me off via certain lists such those maintained by a John Baez or a Scott Aaronson, I couldn’t care less about it—and both (and all) of them, I suppose, should know/could get to know, how (I care so less about those lists).)

BTW, as a matter of progression in time, I had thought that the issue would have to be first resolved in the context of photons, not of electrons. I am not very sure about it, though. In any case, that was the sequence in which I did it. First, photons; then, electrons.

Go, try your best to prod me towards publishing something on it! Just try it!! … BTW, my resolution had happened years before I had publicly offered an Indian PhD physicist on a “LinkedIn” group that I could explain my results if she (or anyone else) could meet me in person at Pune. This public offer of mine has just ended, right now!…. So, go ahead! Just try it!!!

3. My Researches on Other Topics

3.1 I have had some definite ideas for research on other topics from computational science and engineering and allied fields (including a numerics). I have kept these aside for the time being, because many of these are well-suited for guiding PhDs. Which brings me to the last couple of points for today (or at least, as of now, in the first version of this post).

3.2 As to student projects, I have decided not to accept anyone unless he is remarkably bright, and hard-working. (For those who seek to do truly independent PhD research, I cannot make myself available as a guide, as of now. Also see the point 3.3 below.) Roughly speaking, this means that rough level as would be understood by one or more of the following: GRE (V+Q) scores of at least 1350; GATE score of 95+P; throughout distinction class (or in at least 5 semesters out of 8) in BE of University of Pune (or equivalent).

3.3 The University of Pune has a stupid requirement for becoming a PhD guide: you (i.e. a fresh PhD graduate) must wait for at least 3 years after his own (successful) defense before he can become a PhD guide himself. The three years, in my case, end on September 20, 2012. (They—the Indian government(s)—probably arranged the date to numerically coincide with the date on which I first entered USA: 2nd September, 1990. Yes, the same government that whispered the UK government to give Rahul Gandhi’s brother-in-law all security clearance at UK airports, on par with the President and Prime Minister of India.)

Recently, someone reminded me a further requirement that I had forgotten. You also need to have two publications in those three years, before you can become a guide. Since I have mentioned the Gandhi’s and the defence-date here, I am sure that they would now interpret the sufficiently vague rules to imply that those two must be journal articles—peer-reviewed conference proceedings won’t do.

I, therefore, have decided to try to publish two journal articles in the near future of a few months. (Hey Elsevier, take notice!)

At least one, and probably both of these two articles would be on CFD.

Those of you who know me, would know that once I get going, I get going. I don’t disappoint (these of) you, not this time around at least: I have already installed Ubuntu 11.10 (natty) inside Oracle’s VirtualBox running on top of Windows (32 bit XP and 64-bit 7), and have already installed OpenFOAM v. 2.0.1 in that Ubuntu (32-bit, as of now). I also have installed other software. I have shortlisted the niche problems I could work on. I have contacted a couple of IIT Bombay professors, not for collaboration, but merely for sounding out. I knew that being employed by the IIT Bombay, there would be no collaboration, though a collaboration could have been perfectly OK by me. I also knew that once I wrote an email to them, it would get trapped (as all my emails are), and then, even the sounding things out over a 30 minute session would soon become impossible. And, that the impossibility would never be communicated explicitly via any means, esp. via an email. This  supposition of mine has indeed come to pass. (Congratulate me for being a good judge of the IIT Bombay, of the Indian government(s)—all of them, today’s and those of the past under the BJP regime as well, of Indians, and of humanity in general.) I knew all that, right in advance, and had prepared myself mentally for it. And, thought of plans B and C as well. I am executing on these.

And, no, I couldn’t care a hoot for how many freaking citations those two journal papers generate. As far as I am concerned, these two papers would allow me to fulfill the stupid requirements whereby I can become a PhD guide. And whereby, a slim chance does exist that I might get some good guy (gals included) for PhD supervision. (Chances are, it could be someone I already knew as a friend—numerically speaking, most of my friends are without PhDs.)

So, there. For the next few months, that’s the sort of research I am going to do—in my spare time, of course. Hey Elsevier, take notice (once again!!). As to others: If you consider yourself my friend, help me publish it in an easy and timely manner, ASAP.

That’s all for today. For this first version, anyway. As always, I might come back and correct or add a few things. …. Might as well add a few political comments right here.

4. A Few Comments on Politics and All:

Just noting down a few comments on politics (i.e. that politics which is “larger” than the one in S&T fields) in passing (and I will take liberties to pass comments on people without alerting them):

To ObjectivistMantra and Others:

Tavleen Singh’s article on the slap to Mr. Pawar was the best. However, it fell short on the count of completeness. On this count of completeness, she does far, far better (actually excellent) with her next article in the Indian Express’ Fifth Column. Why I say she fell short. In an entrenched mixed economy such as ours (i.e. India’s as in the past and as of today, and of USA’s in near future), the whole system has already become so statist, so mangling of individual rights, that it is impossible to systematically assign blame on any one systemic part of it. In my twenties and early thirty’s (i.e. 15–25 years ago), having known this, I used to argue that it would be impossible for the Indian army (i.e. defence services in general) or the Indian courts to be singled out as being clean. Time proved me right. Indeed, it’s at least since my X standard (i.e. for ~35 years now) that I have argued that you can’t blame politicians—indeed that far too many politicians, from the village through the national level enjoyed much more of esteem in my opinion than what salaried class (say, my “Brahmin” friends) would allow them. Sometime while I was in SF Bay Area, I further realized that the trend to say: “It’s all polltishuns; common people and businessmen are clean” had originated not in India, but in the middle-east and Pakistan etc., and that our Punjabi’s, Gujarathi’s etc. settled in the USA and UK (e.g. Kanwal Rehi, Vinod Khosla and their friends there and here) had been simply rubbing the characterization (actually applicable in the middle-east and Pakistan etc.), expectedly witlessly, on to India’s scene. Since Shobha Dey makes many frequent visits to Dubai, she was expected to have picked it up, too. And, she has shown over the years  that she has. Her latest column springs from that faulty position as regards India. Tavleen Singh is better. (That’s one basic reason why a link to her columns features in the my blogroll here.) Singh did stop short of stretching on that line. However, she did get overwhelmed by the dominant presence of that erroneous idea in our present culture. That’s why, she couldn’t think of a single example on the following lines: Taking a symmetrical case, should I be allowed to put a slap on the face of a Kanwal Rekhi or a Vinod Khosla, for not giving me a job in SF Bay Area in late 2000/early 2001, so that my green-card processing could have been completed? Should I be permitted—morally, even if not legally—to land a (Marathi) “saNsaNit thappaD” (nearest English: a resounding slap) on the face of a Ratan Tata, not just for never giving a job in his company (in Ratan’s case, Tatas) but even allowing my harassment (e.g. as stated on a LinkedIn thread re. VSNL/Tata Indicom Broadband)? Would it be morally justifiable? Why, Ms Tavleen, speak of the emotions of common man but refuse to discuss the issue on more clearly and more on specifically moral terms? So, you see, even if Ms. Singh is far better—and here I thankfully recall all her wonderful articles in the recent past, esp. the courage she has shown in taking on the urban twittering “middle” classes in the “Gandhian” Anna Hazaare “movement”—it is obvious that she overlooked something. Mind you, it’s just plain omission (and as far as I am concerned, it seems to be a very honest one). But still, an error is an error. On omission is an omission. Since I enjoy and admire her columns as much as you do, I hope that she addresses the moral aspects of the emotional issues rather than emotions. In any case, what she wrote was otherwise far better, far superior to what I could have written. This is exactly like Swapan Dasgupta’s recent article. Except for that one error, the rest of the article is excellent! But, hey, you don’t design or manufacture 99 components of an engine well, and leave 1 component out of either good design process or actual testing. As to Ms. Dey, I think I am going to stop reading her now. Some time ago, she was wondering when certain people had kissed last, in the context of—and who else: Indian “poltishun”s. (In case someone finds it intriguing, realize that she is a daughter of an Indian central bureucrat, and as far as I can make it out, has had no explicit rational philosophy to guide her writings, though she is a lady of enormous culture and composure in her own right too. Oh well, even explicit rational ideas do make a difference—think what a whole rational philosophy can do!)

I think I will stop here, and add possibly add other points via other blog posts. For the time being, as far as politics goes, I am enjoying (“loving it”) watching the BJP more than anyone else in the opposition/government, as far as the issue of retail FDI goes.  However, I am not going to support Walmart for the simple reasons that (i) their country has unreasonably failed me in the PhD and unreasonably denied me green-card/citizenship, (ii) they are too big to need my support anyway, and (iii) supporting a big company against government—Microsoft, in the DoJ case—was one among many things that got me a heart condition, I know. (How do I know? Well, it’s the same guy who has known how to resolve the QM wave-particle duality in the context of light, and about angular momentum in EM, and then, a resolution of the riddles of quantum entanglement, as well as many other unpublished, even un-discussed topics.)

One final point, again going back towards research. For the past several years I could not fathom the reason why people might be so unenthusiastic about my approach—I mean, honest people (apart from all the dirty things and “political” issues I have mentioned/indicated above.) Well, it was while reading Sean Carroll’s blog at Discovery magazine that I happened to realize one important (technical) reason why this might be (or must be) so! Hmmm…. Nice to know. It’s always great to know. Though, I am not going to divulge here what that thing was—or how it not only doesn’t contradict my approach but rather helps me be even more confident about my approach (if I ever needed such help, in this context!) And, as you know, I am not going to discuss it or publish about it either. Try to get me to do otherwise. … Just try!
Ok. Enough is enough. As usual, to be edited/streamlined later—perhaps!

* * * * *   * * * * *   * * * * * 
A Song I Like:
[RIP, Dev Anand!]
(Hindi) “gaataa rahe, meraa dil…”
Music: S. D. Burman (perhaps with R.D. looking after the orchestra (??) if not also the tune. (I have read somewhere that he was involved in “Aaraadhanaa,” but have no such idea when it comes to “Guide”)
Singers: Kishore Kumar, Lata Mangeshkar
Lyrics: Shailendra

[E&OE]

Where the Mind Stops—Not!

The way people use language, changes.

In the mid- and late-1990s, when the Internet was new, when blogs had yet to become widespread, when people would often use their own Web sites (or the feedback forms and “guestbooks” at others’ Web sites) to express their own personal thoughts, opinions and feelings—in short, when it still was Web 1.0—one would often run into expressions of the title sort. For example: XYZ is a very great course—NOT! XYZ university has a very great student housing—NOT! XYZ is a very cute product—NOT! … You get the idea—you really do! (NO not!!)… That’s the sense in which the title of this post is to be taken.

For quite some time, I had been thinking of a problem, a deceptively simple problem, from engineering sciences and mechanics. Actually, it’s not a problem, but a way of modeling problems.

Consider a body or a physical object, say a piece of chalk. Break it into two pieces. Easy to do so physically? … Fine. Now, consider how you would represent this scenario mathematically. That is the problem under consideration. … Let me explain further.

The problem would be a mere idle curiosity but for the fact that it has huge economic consequences. I shall illustrate it with just two examples.

Example 1: Consider hot molten metal being poured in sand molds, during casting. Though “thick,” the liquid metal does not necessarily flow very smoothly as it runs everywhere inside the mold cavity. It brushes against mold-walls, splashes, and forms droplets. These flying droplets are more effective than the main body of the molten metal in abrading (“scrubbing”) the mold-walls, and thereby dislodging sand particles off the mold walls. Further, the droplets themselves both oxidize fast, and cool down fast. Both the oxidized and solidified droplets, and the sand particles abraded or dislodged by the droplets, fall into the cooling liquid metal. Due to oxidized layer the solidified droplets (or due to the high melting point of silicates, the sand particles) do not easily remelt once they fall into the main molten metal. The particles remain separate, and thus get embedded into the casting, leading to defective castings. (Second-phase particles like oxidized droplets and sand particles adversely affect the mechanical load-carrying capacity of the casting, and also lead to easier corrosion.) We need the flow here to be smooth, not so much because laminar flow by itself is a wonderful to have (and mathematically easier to handle). We need it to remain smooth mainly in order to prevent splashing and to reduce wall-abrasion. The splashing part involves separation of a contiguous volume of liquid into several bodies (the main body of liquid, and all the splashed droplets). If we can accurately, i.e. mathematically, model how droplets separate out from a liquid, we would be better equipped to handle the task of designing the flow inside a mold cavity.

Example 2: Way back in mid-1980s, when I was doing my MTech at IIT Madras, I had already run into some report which had said that the economic losses due to unintended catastrophic fractures occurring in the US alone were estimated to be some $5 billion annually. … I quote the figure purely from my not-so-reliable memory. However, even today, I do think that the quoted figure seems reasonable. Just consider just one category of fractures: the loss of buildings and human life due to fractures occurring during earthquakes. Fracture mechanics has been an important field of research for more than half a century by now. The process of fracture, if allowed to continue unchecked, results in a component or an object fragmenting into many pieces.

It might surprise many of you (in fact, almost anyone who has not studied fluid mechanics or fracture mechanics) that there simply does not exist any good way to mathematically represent this crucial aspect of droplets formation or fracture: namely, the fact of one body becoming several bodies. More accurately, no one so far (at least to my knowledge) has ever proposed a neat mathematical way to represent such a simple physical fact. Not in any way that could even potentially prove useful in building a better mechanics of fluids or fracture.

Not very surprising. After all, right since Newton’s time, the ruling paradigm of building mathematical models has been: differential equations. Differential equations necessarily assume the existence of a continuum. The region over which a given differential equation is to be integrated, may itself contain holes. Now, sometimes, the existence of holes in a region of space by itself leads to some troubles in some areas of mechanics; e.g., consider how the compatibility criteria of elasticity lose simplicity once you let a body carry holes. Yet, these difficulties are nothing once you theoretically allow the original single body to split apart into two or more fragments. The main difficulty is the following:

A differential equation is nothing but an equation defined in terms of differentials. (That is some insight!) In the sense of its usage in physics/engineering, a differential equation is an equation defined over a differential element. A differential element (or an infinitesimal) is a mathematical abstraction. It begins with a mathematically demarcated finite piece of a continuum, and systematically takes its size towards zero. A “demarcated finite piece” here essentially means that it has boundaries. For example, for a 1D continuum, there would be two separate points serving as the end-points of the finite piece. Such a piece is, then, subjected to the mathematical limiting process, so as to yield a differential element. To be useful, the differential equation has to be integrated over the entire region, taking into consideration the boundary and initial values. (The region must be primarily finite, and it usually is so. However, sometimes, through certain secondary mathematical considerations and tricks involving certain specific kinds of boundary conditions, we can let the region to be indefinitely large in extent as well.)

Since the basic definition of the differential element itself refers to a continuum, i.e. to a continuous region of space, this entire paradigm requires that cuts or holes not existing initially in the region cannot at all be later introduced. A hole is, as I said above, mostly acceptable in mathematical physics. However, the hole cannot grow so as to actually severe a single contiguous region of space into two (or more) separate regions. A cut cannot be allowed to run all the way through. The reason is: (i) either the differential element spanning the two sides of the cut must be taken out of the model—which cannot be done under the differential equations paradigm, (ii) or the entire model must be rejected as being invalid.

Thus, no cut—no boundary—can be introduced within a differential element. A differential element may be taken to end on a boundary, in a sense. However, it can never be cut apart. (This, incidentally, is the reason why people fall silent when you ask them the question of one of my previous posts: can an infinitesimal carry parts?)

You can look at it as a simple logical consistency requirement. If you model anything with differential elements (i.e. using the differential equations paradigm), then, by the logic of the way this kind of mathematics has been built and works, you are not allowed to introduce a cut into a continuum and make fragments out of it, later on.  In case you are wondering about a logically symmetrical scenario: no, you can also not join two continua into one—the differential equation paradigm does not allow you to do that either. And, no, topology does not lead to any actual progress with this problem either. Topology only helps define some aspects of the problem in mathematically precise terms. But it does not even address the problem I am mentioning here.

Such a nature of continuum modeling is indeed was what I had once hinted at, in one of my comments at iMechanica [^]. I had said (and none contradicted me at that forum for it) that:

As an aside, I think in classical mathematics there is no solution to this issue, and there cannot be—you simply cannot model a situation like “one thing becomes two things” or “two infinitesimally close points become separated by a finite distance” within any continuum theory at all…

In other words, this is a situation where, if one wishes to think about it in mathematical terms, one’s mind stops.

Or does it?

Today, I happened to idly go over these thoughts once again. And then, a dim possibility of appending a NOT appeared.

The reason I say it’s a dim possibility is because: (i) I haven’t yet carefully thought it through; (ii) and so, I am not sure if it really does not carry philosophic inconsistencies (philosophy, here, is to be rather taken in the sense of philosophy of science, of physics and mathematics); (iii) I already know enough to know that this possibility would not in any way help at least that basic fracture mechanical problem which I have mentioned above; and (iv) I think an application simpler than the basic problem of fracture mechanics, should be possible—with some careful provisos in place. May be, just may be. (The reason I am being so tentative is that the idea struck me only this afternoon.)

I still need to go over the matter, and so, I will not provide any more details about that dim possibility, right here, right today. However, I think I have already provided a sufficiently detailed description of the problem (and the supposed difficulty about it) that, probably, anyone else (trained in basic engineering/physics and mathematics) could easily get it.

So, in the meanwhile, if you can think of any solution—or even a solution approach—that could take care of this problem, drop me a line or add a comment.  … If you are looking for a succinct statement of the problem out of this (as usual) verbose blog-post, then take the above-mentioned quote from my iMechanica comment, as the problem statement. … For years (two+ decades) I thought no solution/approach to that problem was possible, and even at iMechanica, it didn’t elicit any response indicating otherwise. … But, now, I think there could perhaps be a way out—if I am consistent by basic philosophic considerations, that is. It’s a simple thing, really speaking, a very obvious one too, and not at all a big deal… However, the point is, now the (or my) mind no longer comes to a complete halt when it comes to that problem…

Enough for the time being. I will consider posting about this issue at iMechanica after a little while. … And, BTW, if you are in a mode to think very deeply about it, also see something somewhat related to this problem, viz., the 2011 FQXi Essay Contest (and what its winners had to say about that problem): [^]. Though related, the two questions are a bit different. For the purpose of this post, the main problem is the one I mentioned above. Think about it, and have fun! And if you have something to say about it, do drop me a line! Bye for now!!

* * * * *   * * * * *   * * * * *
A Song I Like
(Hindi) “nahin nahin koee tumasaa hanseen…”
Singers: Kishore Kumar, Asha Bhosale
Music: Rajesh Roshan
Lyrics: Anand Bakshi

[PS: Perhaps, a revision to fix simple errors, and possibly to add a bit of content here and there, is still due.]
[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!]

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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]